Each graph–k box is one cell split into three diagonal bands — three independent facts about whether the balanced Turán graph maximizes this graph's count at this k. Hover a box for the full breakdown, including citations (and a row's name, below, for its exact definition). The chips below are also filters: click to toggle a state, double-click to isolate it; a row stays visible if at least one of its k-cells matches every active axis, and non-matching cells are dimmed. Toggling all of an axis's chips off matches nothing — reset filters brings everything back.
| g6 | graph | name | $\omega$ | $\chi$ | profile |
|---|---|---|---|---|---|
BW | $K_{1,2}$ | 2 | 2 | 3 k = 3 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown Complete bipartite K_{a,b} is 3-Turán-good when C(|a−b|,2) < min(a,b) [2019:MaQiu:SharpResultsGeneralizedTuran] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
Bw | $K_{3}$ (triangle) | 3 | 3 | 4 k = 4 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) Small book graphs (triangle, paw, bull, cricket, dart) are 4-Turán-good [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) 5k = 5 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) 6k = 6 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) 7k = 7 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) …every larger k · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) |
| g6 | graph | name | $\omega$ | $\chi$ | profile |
|---|---|---|---|---|---|
CU | $P_{4}$ | 2 | 2 | 3 k = 3 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown Double star S_{a,b} is 3-Turán-good when C(b-a,2) < a+1 [2021:Gerbner:GeneralizedTuranDoubleStars] · unique (algebraic) 4k = 4 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 5k = 5 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown 6k = 6 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown 7k = 7 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown …every larger k · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown | |
C] | $K_{2,2}$ | 2 | 2 | 3 k = 3 · (in part) covered by our results C₄ is k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · (in part) covered by our results C₄ is k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] · unique (algebraic) K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · (in part) covered by our results C₄ is k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · (in part) covered by our results C₄ is k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · (in part) covered by our results C₄ is k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results C₄ is k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
CF | $K_{1,3}$ (claw) | 2 | 2 | 3 k = 3 · (in part) covered by our results Complete bipartite K_{a,b} is not 3-Turán-good when C(|a−b|,2) ≥ min(a,b) (shape proven via R-monotonicity, every $k$) [2019:MaQiu:SharpResultsGeneralizedTuran] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · (in part) covered by our results The claw K_{1,3} is k-Turán-good for every k ≥ 6 (shape proven via R-monotonicity, every $k$) [2013:LiShi:TurantypeDegreeSequence] · unique (numeric) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · (in part) covered by our results The claw K_{1,3} is k-Turán-good for every k ≥ 6 (shape proven via R-monotonicity, every $k$) [2013:LiShi:TurantypeDegreeSequence] · unique (numeric) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results The claw K_{1,3} is k-Turán-good for every k ≥ 6 (shape proven via R-monotonicity, every $k$) [2013:LiShi:TurantypeDegreeSequence] · unique (numeric) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
CV | paw | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown Small book graphs (triangle, paw, bull, cricket, dart) are 4-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown K_3 with a pendant leaf on 1 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 5k = 5 · (in part) covered by our results K_3 with a pendant leaf on 1 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 6k = 6 · (in part) covered by our results K_3 with a pendant leaf on 1 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · (in part) covered by our results K_3 with a pendant leaf on 1 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · (in part) covered by our results K_3 with a pendant leaf on 1 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown | |
C^ | $K_{1,1,2}$ (diamond) | 3 | 3 | 4 k = 4 · (in part) covered by our results F is itself the balanced complete (k−1)-partite graph T_{k−1}(m); it is k-Turán-good by a Marcus-Lopes concavity argument (log-concave elementary symmetric polynomial) (shape proven via R-monotonicity, every $k$) · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
C~ | $K_{4}$ | 4 | 4 | 5 k = 5 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) 6k = 6 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) 7k = 7 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) …every larger k · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) |
| g6 | graph | name | $\omega$ | $\chi$ | profile |
|---|---|---|---|---|---|
DCw | chair, fork | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown Double star S_{a,b} is 3-Turán-good when C(b-a,2) < a+1 [2021:Gerbner:GeneralizedTuranDoubleStars] · unique (algebraic) 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
DQo | $P_{5}$ | 2 | 2 | 3 k = 3 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown 4k = 4 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown 5k = 5 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown 6k = 6 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown 7k = 7 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown …every larger k · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 (shape proven via R-monotonicity, every $k$) [2022:Gerbner:PathsTurangood] · uniqueness unknown | |
DEw | banner | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
DFw | $K_{2,3}$ | 2 | 2 | 3 k = 3 · (in part) covered by our results Complete bipartite K_{a,b} is 3-Turán-good when C(|a−b|,2) < min(a,b) (shape proven via R-monotonicity, every $k$) [2019:MaQiu:SharpResultsGeneralizedTuran] · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
D?{ | $K_{1,4}$ (star) | 2 | 2 | 3 k = 3 · (in part) covered by our results Complete bipartite K_{a,b} is not 3-Turán-good when C(|a−b|,2) ≥ min(a,b) (shape proven via R-monotonicity, every $k$) [2019:MaQiu:SharpResultsGeneralizedTuran] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
DUW | $C_{5}$ (pentagon, C5) | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] ex(n,C₅,K₃) is maximized by the balanced C₅ blow-up (≈(n/5)⁵), not T₂(n) (bipartite, zero C₅); exact for all n except n=8 — Grzesik 2012; Hatami–Hladký–Král'–Norin–Razborov 2013; Lidický–Pfender 2018 (exact all-n) [2017:LidickyPfender:PentagonsTrianglefreeGraphs] 4k = 4 C₅ is k-Turán-good for all k ≥ 4 [2021:LidickyMurphy:MaximizingFivecyclesfreeGraphs] · unique (algebraic) 5k = 5 C₅ is k-Turán-good for all k ≥ 4 [2021:LidickyMurphy:MaximizingFivecyclesfreeGraphs] · unique (algebraic) 6k = 6 C₅ is k-Turán-good for all k ≥ 4 [2021:LidickyMurphy:MaximizingFivecyclesfreeGraphs] · unique (algebraic) 7k = 7 C₅ is k-Turán-good for all k ≥ 4 [2021:LidickyMurphy:MaximizingFivecyclesfreeGraphs] · unique (algebraic) …every larger k C₅ is k-Turán-good for all k ≥ 4 [2021:LidickyMurphy:MaximizingFivecyclesfreeGraphs] · unique (algebraic) | |
DC{ | cricket | 3 | 3 | 4 k = 4 · established here Small book graphs (triangle, paw, bull, cricket, dart) are 4-Turán-good (shape proven via R-monotonicity, every $k$) [established here] · unique (numeric) A clique with a one-vertex pendant bundle is k-Turán-good when t(t−1) < r(r−1)² [established here] · unique (numeric) 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
DEk | bull | 3 | 3 | 4 k = 4 · (in part) covered by our results Small book graphs (triangle, paw, bull, cricket, dart) are 4-Turán-good (shape proven via R-monotonicity, every $k$) [2024:Gerbner:ExtremalValuesDegreebased] · unique (algebraic) K_3 with a pendant leaf on 2 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) [established here] · unique (algebraic) 5k = 5 · established here K_3 with a pendant leaf on 2 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) 6k = 6 · established here K_3 with a pendant leaf on 2 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) 7k = 7 · established here K_3 with a pendant leaf on 2 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) …every larger k · established here K_3 with a pendant leaf on 2 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) | |
DQw | tadpole, (3,2)-tadpole | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
DE{ | dart | 3 | 3 | 4 k = 4 · established here Small book graphs (triangle, paw, bull, cricket, dart) are 4-Turán-good (shape proven via R-monotonicity, every $k$) [established here] · unique (numeric) 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
DQ{ | butterfly, hourglass | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 6, exact): Turán shape for every k > ω; balance not decided 5k = 5 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown R-SOS-monotonicity certificate (base order 6, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 6, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 6, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 6, exact): Turán shape for every k > ω; balance not decided | |
DUw | house | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 6, exact): Turán shape for every k > ω; balance not decided 5k = 5 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown R-SOS-monotonicity certificate (base order 6, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 6, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 6, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 6, exact): Turán shape for every k > ω; balance not decided | |
DTw | co-chair, co-fork, kite | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
DF{ | $K_{1,1,3}$ (book, triangular book) | 3 | 3 | 4 k = 4 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
DU{ | gem | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
D]w | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
D]{ | $K_{1,2,2}$ (wheel, 4-wheel) | 3 | 3 | 4 k = 4 · (in part) covered by our results F is itself the balanced complete (k−1)-partite graph T_{k−1}(m); it is k-Turán-good by a Marcus-Lopes concavity argument (log-concave elementary symmetric polynomial) (shape proven via R-monotonicity, every $k$) · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
DT{ | (4,1)-lollipop | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown K_4 with a pendant leaf on 1 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 6k = 6 · (in part) covered by our results K_4 with a pendant leaf on 1 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · (in part) covered by our results K_4 with a pendant leaf on 1 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · (in part) covered by our results K_4 with a pendant leaf on 1 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown | |
DV{ | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 6k = 6 · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown | ||
D^{ | $K_{1,1,1,2}$ | 4 | 4 | 5 k = 5 · (in part) covered by our results F is itself the balanced complete (k−1)-partite graph T_{k−1}(m); it is k-Turán-good by a Marcus-Lopes concavity argument (log-concave elementary symmetric polynomial) (shape proven via R-monotonicity, every $k$) · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
D~{ | $K_{5}$ | 5 | 5 | 6 k = 6 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) 7k = 7 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) …every larger k · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) |
| g6 | graph | name | $\omega$ | $\chi$ | profile |
|---|---|---|---|---|---|
E?bo | cross | 2 | 2 | 3 k = 3 · (in part) covered by our results Double star S_{a,b} is 3-Turán-good when C(b-a,2) < a+1 (shape proven via R-monotonicity, every $k$) [2021:Gerbner:GeneralizedTuranDoubleStars] · unique (algebraic) 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
E?qo | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E?ow | 2 | 2 | 3 k = 3 · (in part) covered by our results Double star S_{a,b} is 3-Turán-good when C(b-a,2) < a+1 (shape proven via R-monotonicity, every $k$) [2021:Gerbner:GeneralizedTuranDoubleStars] · unique (algebraic) 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECR_ | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 4k = 4 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
ECZ? | $P_{6}$ | 2 | 2 | 3 k = 3 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 4k = 4 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · (in part) covered by our results Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | |
E?ro | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E?zO | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECr_ | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 4k = 4 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
ECZ_ | (4,2)-tadpole | 2 | 2 | 3 k = 3 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
EEh_ | $C_{6}$ (hexagon, C6) | 2 | 2 | 3 k = 3 · (in part) covered by our results Even cycles are 3-Turán-good [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] · unique (algebraic) R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 4k = 4 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | |
E?zo | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEr_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEj_ | domino | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 4k = 4 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | |
E?~o | $K_{2,4}$ | 2 | 2 | 3 k = 3 · (in part) covered by our results Complete bipartite K_{a,b} is 3-Turán-good when C(|a−b|,2) < min(a,b) (shape proven via R-monotonicity, every $k$) [2019:MaQiu:SharpResultsGeneralizedTuran] · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
EEz_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EFz_ | $K_{3,3}$ (utility graph) | 2 | 2 | 3 k = 3 · (in part) covered by our results Complete bipartite K_{a,b} is 3-Turán-good when C(|a−b|,2) < min(a,b) (shape proven via R-monotonicity, every $k$) [2019:MaQiu:SharpResultsGeneralizedTuran] · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
E?bw | 3 | 3 | 4 k = 4 · established here A clique with a one-vertex pendant bundle is k-Turán-good when t(t−1) < r(r−1)² (shape proven via R-monotonicity, every $k$) [established here] · unique (numeric) 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E?qw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECqg | net | 3 | 3 | 4 k = 4 · established here K_3 with a pendant leaf on 3 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) 5k = 5 · established here K_3 with a pendant leaf on 3 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) 6k = 6 · established here K_3 with a pendant leaf on 3 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) 7k = 7 · established here K_3 with a pendant leaf on 3 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) …every larger k · established here K_3 with a pendant leaf on 3 of its 3 vertices is k-Turán-good for EVERY k > 3 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) | |
ECZO | (3,3)-tadpole | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
ECYW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E?rw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E?zW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECrg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECZW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
ECfo | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEio | antenna | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
EEho | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEiW | co-twin-house | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
EEhW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EQj_ | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EQjO | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
ECrw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECZw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECzo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECzg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECzW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECvo | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEro | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEjo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEjW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEzO | co-fish | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
EQjo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
EQzO | co-domino | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | |
EUZ_ | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECzw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EC~o | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EErw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEjw | 4-fan | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
EEzo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEzg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEvo | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEno | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EElw | 3-sun | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
EQzo | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
EQzW | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
EUxo | prism, triangular prism | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
EEzw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EFzo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EFzW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EQ~o | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
EUzo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
EUzW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
EFzw | $K_{1,2,3}$ | 3 | 3 | 4 k = 4 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
E]zo | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E]~o | $K_{2,2,2}$ (octahedron, cocktail party graph) | 3 | 3 | 4 k = 4 · (in part) covered by our results F is itself the balanced complete (k−1)-partite graph T_{k−1}(m); it is k-Turán-good by a Marcus-Lopes concavity argument (log-concave elementary symmetric polynomial) (shape proven via R-monotonicity, every $k$) · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
E?Bw | $K_{1,5}$ (star) | 2 | 2 | 3 k = 3 · (in part) covered by our results Complete bipartite K_{a,b} is not 3-Turán-good when C(|a−b|,2) ≥ min(a,b) (shape proven via R-monotonicity, every $k$) [2019:MaQiu:SharpResultsGeneralizedTuran] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.6333, 0.1833, 0.1833] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
E?zw | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.5, 0.25, 0.25] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E?~w | $K_{1,1,4}$ (book, triangular book) | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.6333, 0.1833, 0.1833] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
ECfw | 4 | 4 | 5 k = 5 · established here A clique with a one-vertex pendant bundle is k-Turán-good when t(t−1) < r(r−1)² (shape proven via R-monotonicity, every $k$) [established here] · unique (numeric) 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ECuw | 4 | 4 | 5 k = 5 · established here K_4 with a pendant leaf on 2 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) 6k = 6 · established here K_4 with a pendant leaf on 2 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) 7k = 7 · established here K_4 with a pendant leaf on 2 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) …every larger k · established here K_4 with a pendant leaf on 2 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) | ||
EQjg | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 unknown — open …higher k not computed | ||
ECvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEuw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EQjw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 unknown — open …higher k not computed | ||
EQzg | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 unknown — open …higher k not computed | ||
EQyw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 unknown — open …higher k not computed | ||
EC~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EEnw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EQzw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 unknown — open …higher k not computed | ||
ETzo | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
ETzg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
ETno | co-cross | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
EE~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EQ~w | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
EUzw | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
ETzw | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E]zg | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E]yw | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
EF~w | $K_{1,1,1,3}$ (jewel) | 4 | 4 | 5 k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
EU~w | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E]zw | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
E]~w | $K_{1,1,2,2}$ | 4 | 4 | 5 k = 5 · (in part) covered by our results F is itself the balanced complete (k−1)-partite graph T_{k−1}(m); it is k-Turán-good by a Marcus-Lopes concavity argument (log-concave elementary symmetric polynomial) (shape proven via R-monotonicity, every $k$) · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
ETnw | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown K_5 with a pendant leaf on 1 of its 5 vertices is k-Turán-good for EVERY k > 5 (Schur-concave colouring polynomial, established here) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · (in part) covered by our results K_5 with a pendant leaf on 1 of its 5 vertices is k-Turán-good for EVERY k > 5 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · (in part) covered by our results K_5 with a pendant leaf on 1 of its 5 vertices is k-Turán-good for EVERY k > 5 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown | ||
ET~w | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown | ||
EV~w | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown | ||
E^~w | $K_{1,1,1,1,2}$ | 5 | 5 | 6 k = 6 · (in part) covered by our results F is itself the balanced complete (k−1)-partite graph T_{k−1}(m); it is k-Turán-good by a Marcus-Lopes concavity argument (log-concave elementary symmetric polynomial) (shape proven via R-monotonicity, every $k$) · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
E~~w | $K_{6}$ | 6 | 6 | 7 k = 7 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) …every larger k · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) | |
ECpo | 5-pan, (5,1)-tadpole | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
ECxo | twin-C5 | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
ECRo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
ECZG | 3 | 3 | 4 k = 4 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
ECRw | 3 | 3 | 4 k = 4 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
ECro | 3 | 3 | 4 k = 4 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
ECZo | fish | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
ECZg | 3 | 3 | 4 k = 4 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | ||
ECxw | twin-house | 3 | 3 | 4 k = 4 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | |
EEhw | co-antenna | 3 | 3 | 4 k = 4 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 5k = 5 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 6k = 6 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided 7k = 7 · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided …every larger k · established here R-SOS-monotonicity certificate (base order 7, exact): Turán shape for every k > ω; balance not decided | |
EUZO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
EUZo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
EUZw | wheel, 5-wheel | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed |
| g6 | graph | name | $\omega$ | $\chi$ | profile |
|---|---|---|---|---|---|
F?B@w | 2 | 2 | 3 k = 3 · (in part) covered by our results Double star S_{a,b} is 3-Turán-good when C(b-a,2) < a+1 (shape proven via R-monotonicity, every $k$) [2021:Gerbner:GeneralizedTuranDoubleStars] · unique (algebraic) 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?`F_ | 2 | 2 | 3 k = 3 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bB_ | 2 | 2 | 3 k = 3 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`e_ | skew-star | 2 | 2 | 3 k = 3 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
F?`cg | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCOf? | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQb? | $P_{7}$ | 2 | 2 | 3 k = 3 Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown 4k = 4 Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown 5k = 5 Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown 6k = 6 Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown 7k = 7 Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown …every larger k Paths are k-Turán-good for all k ≥ 3 [2022:Gerbner:PathsTurangood] · uniqueness unknown | |
F?bF_ | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`f_ | 2 | 2 | 3 k = 3 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`v? | 2 | 2 | 3 k = 3 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qb_ | 6-pan, (6,1)-tadpole | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
F?q_w | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?ov? | (4,3)-tadpole | 2 | 2 | 3 k = 3 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
F?opo | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCQf? | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bf_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?`v_ | 2 | 2 | 3 k = 3 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?rF_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qf_ | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?ov_ | 2 | 2 | 3 k = 3 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?re_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qr_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCpf? | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCXf? | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bv_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rf_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qv_ | 2 | 2 | 3 k = 3 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCZf? | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEhf? | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rv_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zf_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zV_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zv_ | 2 | 2 | 3 k = 3 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?~v_ | $K_{3,4}$ | 2 | 2 | 3 k = 3 · (in part) covered by our results Complete bipartite K_{a,b} is 3-Turán-good when C(|a−b|,2) < min(a,b) (shape proven via R-monotonicity, every $k$) [2019:MaQiu:SharpResultsGeneralizedTuran] · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
F?AFo | 2 | 2 | 3 k = 3 · (in part) covered by our results Double star S_{a,b} is not 3-Turán-good when C(b-a,2) >= a+1 (shape proven via R-monotonicity, every $k$) [2019:MaQiu:SharpResultsGeneralizedTuran] 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?BFo | 2 | 2 | 3 k = 3 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.6667, 0.3333] 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?Bfo | 2 | 2 | 3 k = 3 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.6667, 0.3333] 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?Bvo | 2 | 2 | 3 k = 3 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.6667, 0.3333] 4k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?B~o | $K_{2,5}$ | 2 | 2 | 3 k = 3 · (in part) covered by our results Complete bipartite K_{a,b} is not 3-Turán-good when C(|a−b|,2) ≥ min(a,b) (shape proven via R-monotonicity, every $k$) [2019:MaQiu:SharpResultsGeneralizedTuran] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
F?BDw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?Bcw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bDg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?aN_ | 3 | 3 | 4 k = 4 · established here — numeric only, not a publication claim Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (generalized seed: remainder established good via coloring-poly:numeric (numeric)) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`uO | 3 | 3 | 4 k = 4 · established here — numeric only, not a publication claim Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (generalized seed: remainder established good via coloring-poly:numeric (numeric)) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?otO | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?osW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCQeO | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQbO | (3,4)-tadpole | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
FCQeG | eiffeltower | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
F?Bew | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bFg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?aNo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?beg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bcw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?`uW | 3 | 3 | 4 k = 4 · established here — numeric only, not a publication claim Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (generalized seed: remainder established good via coloring-poly:numeric (numeric)) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bN_ | 3 | 3 | 4 k = 4 · established here — numeric only, not a publication claim Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (generalized seed: remainder established good via coloring-poly:numeric (numeric)) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?rDo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qeo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qdo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qcw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?ovO | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?ouW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCQeo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQeW | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQV_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQVO | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRV? | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQrO | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpe_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpeG | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCXe_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCde_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCdeG | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCdcg | 3 | 3 | 4 k = 4 · established here — numeric only, not a publication claim Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (generalized seed: remainder established good via coloring-poly:numeric (numeric)) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bFw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?`fw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bfg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bew | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bNo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bNg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rFo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qfo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qew | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?ovo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?ovW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?reo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rdo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?reg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qto | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qvG | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?quW | 3 | 3 | 4 k = 4 · established here — numeric only, not a publication claim Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (generalized seed: remainder established good via coloring-poly:numeric (numeric)) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?rN_ | 3 | 3 | 4 k = 4 · established here — numeric only, not a publication claim Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (generalized seed: remainder established good via coloring-poly:numeric (numeric)) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qlo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?o~O | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?o}W | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?o|W | co-rising sun | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
FCQVo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCReg | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRdg | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRV_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRVO | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQv_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQvO | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQuo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQuW | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQrW | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpeo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXeo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZbO | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZV? | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZUO | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCdf_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCdfG | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCdeg | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQhV? | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bfw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bvW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bno | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rFw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qfw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?ovw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rfo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rfg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rew | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rdw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rNo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qno | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qjw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?o~o | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?o~W | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zcw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zUo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCRVo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRVW | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQvo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQvW | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRuo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCreW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrRo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrVG | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrHw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZeg | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZV_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZVO | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZUo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZMo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZMg | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZLW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZKw | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZHw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXn_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXnO | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXmo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXkw | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCY^O | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCY]o | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCdfg | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQhV_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?b~o | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rfw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?o~w | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rno | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rng | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?q~o | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?q~g | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qzw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zfo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zew | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCR^o | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrfg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrfW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrVo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrVW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrNo | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCqno | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZVo | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZVW | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZLw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXno | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCY^o | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCY^W | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzfO | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCzeW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCzcw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvfO | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvfG | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCveg | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvdg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvbg | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvdW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvbW | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvaw | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvRW | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCuuW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEjfG | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEjeg | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEjdg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEitW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEhtg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjV_ | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjVG | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?r~o | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zfw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zno | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?z^o | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCrfw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrno | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCrng | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCr^o | co-parapluie, parachute | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
FCZ^o | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCfvW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzfo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzfW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCzew | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvfg | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvfW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvbw | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvVW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCuvW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEjfg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEivo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEivW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEzfO | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzeW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzV_ | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzVO | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEzUo | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEnf_ | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEnaw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQjew | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQzUo | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQzUW | 3 | 3 | 4 k = 4 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQyuW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?~vW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCr~o | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCZ~o | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCzfw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCzvo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCzno | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCz^o | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FErvo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FErvW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEr^o | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (chained: 2 gluing steps) (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEjfw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEjvo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEjvW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEj^o | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzfo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzfW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzVo | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEnfg | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FFzf_ | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FFzeo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FUZv_ | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FUxv_ | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FC~vW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEr~o | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEj~o | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzfw | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzvo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzno | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEvvW | 3 | 3 | 4 k = 4 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEnvW | co-longhorn | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
FFzfo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FE~vW | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FFzfw | $K_{1,3,3}$ | 3 | 3 | 4 k = 4 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
FFzvo | 3 | 3 | 4 k = 4 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FFz~o | $K_{2,2,3}$ | 3 | 3 | 4 k = 4 · (in part) covered by our results F is itself the balanced complete (k−1)-partite graph T_{k−1}(m); it is k-Turán-good by a Marcus-Lopes concavity argument (log-concave elementary symmetric polynomial) (shape proven via R-monotonicity, every $k$) · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
F??Fw | $K_{1,6}$ (star) | 2 | 2 | 3 k = 3 · (in part) covered by our results Complete bipartite K_{a,b} is not 3-Turán-good when C(|a−b|,2) ≥ min(a,b) (shape proven via R-monotonicity, every $k$) [2019:MaQiu:SharpResultsGeneralizedTuran] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 4k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.7, 0.15, 0.15] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
F?AFw | 3 | 3 | 4 k = 4 · established here A clique with a one-vertex pendant bundle is not k-Turán-good when t(t−1) ≥ r(r−1)² (shape proven via R-monotonicity, every $k$) [established here] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?BFw | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.4, 0.3, 0.3] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?Bfw | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.6, 0.2, 0.2] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?BvW | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.6, 0.2, 0.2] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?Bvw | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.6667, 0.1667, 0.1667] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bvg | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.3667, 0.3167, 0.3167] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bvw | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.3667, 0.3167, 0.3167] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qvw | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.3667, 0.3167, 0.3167] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rvg | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.3667, 0.3167, 0.3167] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rvw | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.3667, 0.3167, 0.3167] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zVw | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.3667, 0.3167, 0.3167] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zvg | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.3667, 0.3167, 0.3167] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zvw | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.3667, 0.3167, 0.3167] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?~vo | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.3667, 0.3167, 0.3167] 5k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?~vw | $K_{1,2,4}$ | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.3667, 0.3167, 0.3167] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
F?aNw | 4 | 4 | 5 k = 5 · established here A clique with a one-vertex pendant bundle is k-Turán-good when t(t−1) < r(r−1)² (shape proven via R-monotonicity, every $k$) [established here] · unique (numeric) 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bLw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qkw | 4 | 4 | 5 k = 5 · established here K_4 with a pendant leaf on 3 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) 6k = 6 · established here K_4 with a pendant leaf on 3 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) 7k = 7 · established here K_4 with a pendant leaf on 3 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) …every larger k · established here K_4 with a pendant leaf on 3 of its 4 vertices is k-Turán-good for EVERY k > 4 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) | ||
FCdeo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCdcw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bNw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?bmw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rLw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qmw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCrLW | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCZUg | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCY]g | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCY[w | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCdew | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQhVO | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bnw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rNw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?qnw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rmw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?q~W | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?q|w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCrNW | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCrLw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCqnW | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCZUw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCY]w | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCe^o | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCveo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvcw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCuto | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCusw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQhVo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjfG | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQjdg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQjVO | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjUg | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQinO | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?b~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?rnw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?q~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?zmw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?z\w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCrNw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCqnw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCrnW | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCrmw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCrlw | rising sun | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | |
FCY^w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCZ]w | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCfvg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCfuw | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCf^o | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvew | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvVo | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvTw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCuvo | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCuuw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEivg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEitw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzUg | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzTg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzSw | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEncw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQhVw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjfg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQjfW | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQjdw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQjVo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjVg | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQino | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzVO | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzTo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?r~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?znw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F?z^w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCrnw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCr^w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCfvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCf~o | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCznW | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCzmw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCz]w | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCz\w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvfw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvVw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCuvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvvo | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvvg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvtw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCv^o | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FErvg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEruw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FErtw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEivw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEhvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEjvg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEjuw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEjtw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEj]w | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEj\w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzVg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzUw | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzTw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEnfo | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEnew | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQjfw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQjVw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjvo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjvg | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjvW | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjuw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjno | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzVo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzVW | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzvO | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUxvO | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?z~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCr~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCZ~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCzvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCznw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCz^w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCvvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FC~vo | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FC~uw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FErvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEr^w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEjvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEj^w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzVw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzvg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEznW | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEvvo | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEvvg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEv^o | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEu~o | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEu~g | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEnfw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEnvo | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEnvg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEl}w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQjvw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQj~o | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzVw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzvo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzvg | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzno | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzmw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQz^o | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQy~o | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQy}w | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUxvo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCz~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FC~vw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEr~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEj~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEzvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEznw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEvvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEnvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEl~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FE~vo | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FE~uw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FE~tw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FFzvg | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQzvw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQz^w | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQ~vo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQ~vW | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUxvw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUzvo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUzvW | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUz^o | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUz]w | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FTzvo | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FTzvW | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEz~w | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FE~vw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FFzvw | 4 | 4 | 5 k = 5 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQ~vw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUzvw | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUz^w | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FU~vo | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FU~vW | 4 | 4 | 5 k = 5 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F]zno | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FFz~w | $K_{1,1,2,3}$ | 4 | 4 | 5 k = 5 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
FU~vw | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F]~vo | 4 | 4 | 5 k = 5 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 6k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F]~vw | $K_{1,2,2,2}$ | 4 | 4 | 5 k = 5 · (in part) covered by our results F is itself the balanced complete (k−1)-partite graph T_{k−1}(m); it is k-Turán-good by a Marcus-Lopes concavity argument (log-concave elementary symmetric polynomial) (shape proven via R-monotonicity, every $k$) · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
F?B~w | $K_{1,1,5}$ (book, triangular book) | 3 | 3 | 4 k = 4 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.7, 0.15, 0.15] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 5k = 5 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.5, 0.1667, 0.1667, 0.1667] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
F?~~w | $K_{1,1,1,4}$ | 4 | 4 | 5 k = 5 · established here Provably not k-Turán-good at this k: an explicit unbalanced host wins (shape proven via R-monotonicity, every $k$) · optimum ≈ [0.55, 0.15, 0.15, 0.15] Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 6k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
FCe^w | 5 | 5 | 6 k = 6 · established here A clique with a one-vertex pendant bundle is k-Turán-good when t(t−1) < r(r−1)² (shape proven via R-monotonicity, every $k$) [established here] · unique (numeric) 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCf\w | 5 | 5 | 6 k = 6 · established here K_5 with a pendant leaf on 2 of its 5 vertices is k-Turán-good for EVERY k > 5 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) 7k = 7 · established here K_5 with a pendant leaf on 2 of its 5 vertices is k-Turán-good for EVERY k > 5 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) …every larger k · established here K_5 with a pendant leaf on 2 of its 5 vertices is k-Turán-good for EVERY k > 5 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) | ||
FQinW | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FCf^w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCv\w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQinw | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FQjnW | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FQjlw | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FCf~w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCv^w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FCu~w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEv\w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEu|w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQjnw | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FQznW | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FQzlw | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FCv~w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEv^w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEu~w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQj~w | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FQznw | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FQy~w | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FTzvg | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FTnvo | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FTnvg | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FTm~o | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FC~~w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEv~w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FEn~w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQz~w | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FTzvw | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FTznw | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FTnvw | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F]znW | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F]zlw | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FE~~w | 5 | 5 | 6 k = 6 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown 7k = 7 · established here Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FQ~~w | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FUz~w | 5 | 5 | 6 k = 6 Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …higher k not computed | ||
FTz~w | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F]znw | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F]y~w | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
FF~~w | $K_{1,1,1,1,3}$ | 5 | 5 | 6 k = 6 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · established here — numeric only, not a publication claim Numerically k-Turán-good at this k: the balanced host is optimal among (k−1)-partite (shape proven via R-monotonicity, every $k$) · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
FU~~w | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F]z~w | 5 | 5 | 6 k = 6 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown 7k = 7 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · established here R-monotone graphs have a Turán-shaped (possibly unbalanced) optimum for every k > ω(G) [established here] | ||
F]~~w | $K_{1,1,1,2,2}$ | 5 | 5 | 6 k = 6 · (in part) covered by our results F is itself the balanced complete (k−1)-partite graph T_{k−1}(m); it is k-Turán-good by a Marcus-Lopes concavity argument (log-concave elementary symmetric polynomial) (shape proven via R-monotonicity, every $k$) · unique (algebraic) Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] 7k = 7 · (in part) covered by our results K_{k−2} and an edge joined arbitrarily (≠ K_k): k-Turán-good at k = v(G) (Qian–Xie–Ge 2021, Prop 2.6) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω (shape proven via R-monotonicity, every $k$) [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
FTm~w | 6 | 6 | 7 k = 7 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown K_6 with a pendant leaf on 1 of its 6 vertices is k-Turán-good for EVERY k > 6 (Schur-concave colouring polynomial, established here) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · (in part) covered by our results K_6 with a pendant leaf on 1 of its 6 vertices is k-Turán-good for EVERY k > 6 (Schur-concave colouring polynomial, established here) (shape proven via R-monotonicity, every $k$) [established here] · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown | ||
FTn~w | 6 | 6 | 7 k = 7 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown | ||
FT~~w | 6 | 6 | 7 k = 7 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown | ||
FV~~w | 6 | 6 | 7 k = 7 · (in part) covered by our results Gluing a K_{k−1} onto a k-Turán-good seed stays k-Turán-good (shape proven via R-monotonicity, every $k$) [2020:GerbnerPalmer:ExactResultsGeneralizedTuran] · uniqueness unknown K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown …every larger k · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown | ||
F^~~w | $K_{1,1,1,1,1,2}$ | 6 | 6 | 7 k = 7 · (in part) covered by our results F is itself the balanced complete (k−1)-partite graph T_{k−1}(m); it is k-Turán-good by a Marcus-Lopes concavity argument (log-concave elementary symmetric polynomial) (shape proven via R-monotonicity, every $k$) · unique (algebraic) K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] …every larger k · (in part) covered by our results K_{v−1} plus one arbitrarily-joined extra vertex (≠ K_v): k-Turán-good for every k ≥ v(G) (Qian–Xie–Ge 2021, Prop 2.1) (shape proven via R-monotonicity, every $k$) [2021:QianEtAl:ResultsturangoodGraphs] · uniqueness unknown Complete multipartite graphs have a complete (k−1)-partite optimum for every k > ω [1991:GyoriEtAl:MaximalNumberCertainSubgraphs] | |
F~~~w | $K_{7}$ | 7 | 7 | 8 k = 8 · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) …every larger k · (in part) covered by our results Complete graphs are k-Turán-good for every k above their order (shape proven via R-monotonicity, every $k$) [1949:Zykov:SomePropertiesLinearComplexes] · unique (algebraic) | |
F?BDo | 2 | 2 | 3 k = 3 unknown — open 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?Bco | 2 | 2 | 3 k = 3 unknown — open 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?Beo | 2 | 2 | 3 k = 3 unknown — open 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bBo | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bao | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?q`o | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQb_ | (5,2)-tadpole | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
FCp`_ | $C_{7}$ (heptagon, C7) | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
F?BvO | 2 | 2 | 3 k = 3 unknown — open 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bbo | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qpo | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCR`o | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpb_ | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpV? | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bro | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?o~_ | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpv? | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZb_ | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCxv? | 2 | 3 | 3 k = 3 χ(G)=3 > ω(G)=2: no complete 2-partite host contains a copy of G (folklore) [folklore] 4k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`Fo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bDo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`eo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`eg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`cw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCOf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQe_ | longhorn | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
F?`Fw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bFo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`fo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`fg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`ew | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?beo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?baw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`vO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`vG | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bLo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qbo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qaw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?oto | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?otW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qrO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCOfo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQfO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQbo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQfG | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpd_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpeO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpdO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpbO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpdG | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bfo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bbw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`vo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`vg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`vW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bvO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bmo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qbw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qvO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qro | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qtg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qrg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qn_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qmo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qjo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?o|o | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCOfw | friendship graph | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
FCQfo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQfg | co-parachute, parapluie | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
FCQfW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCReo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRdo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRfG | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRbg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRcw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCR`w | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRTo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpfO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpdo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpbo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpfG | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpeg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpdg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpeW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpV_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpVO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCptO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXfO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZeO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZco | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZTO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZSo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhe_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhd_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?`vw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?bvo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?brw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qvo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qvg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qvW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qtw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?qrw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?rn_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?q~_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?zVO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?zTo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?zUW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?zTW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?zPw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQfw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRfo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRfg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRew | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRdw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRv_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRvO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRto | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpfo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpfg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpfW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpVo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrfO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCreo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrdo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrbo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpv_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpvO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpuo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpug | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCprg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrJo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCqn_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXfo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXfW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZfO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZeo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZbo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZfG | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZbg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZTo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZN_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZLo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZJo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZNG | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZLg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZJg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCY^_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCY^G | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEheo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhbo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhv? | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhuO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjR_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?rvo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?zVo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?zVW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?zTw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?zvO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
F?zuo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRfw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRvo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRvW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpfw | 5-fan | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
FCpVw | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrfo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpvo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpvg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpvW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpuw | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrro | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXfw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZfo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZfg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZfW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZew | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZbw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZNo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZNg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZv_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZn_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZ^_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzbo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzbW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzaw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCxv_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCxvO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCxuW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCxrW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCxsw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvbo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCv`w | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCurW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhfo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEjf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEjeo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEjbo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEiro | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhvO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhuo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhto | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhro | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjRo | Moser spindle | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
F?zvo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCR~o | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCpvw | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrvo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrrw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZfw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZvo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZno | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZng | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzbw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCxvo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCxvW | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCxuw | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzro | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhfw | wheel, 6-wheel | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
FEjfo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhvo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhtw | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEh}o | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhzo | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEzf_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEzPw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEnbo | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEndg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEnbg | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzV_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQyv_ | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQyuo | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQyqw | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCxvw | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzrw | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEh~o | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEzvO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEnbw | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQyuw | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQzuo | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUxuo | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FFzvO | 3 | 3 | 4 k = 4 unknown — open 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUZvo | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUzro | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUZ~o | 3 | 4 | 4 k = 4 χ(G)=4 > ω(G)=3: no complete 3-partite host contains a copy of G (folklore) [folklore] 5k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQVg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQug | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQVw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRVg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRTw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQvg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQuw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZTg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZMW | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZIw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXmW | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCdfo | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRVw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCQvw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRvg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRuw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRtw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrVg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrJw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZVg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZTw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZNW | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZMw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZJw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXnW | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXmw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCY^g | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCdfw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvdo | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjdo | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCRvw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCR^w | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrVw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrvg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCruw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrjw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZVw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZNw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCXnw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZvg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZnW | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZmw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZjw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZ^g | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZ\w | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCfvo | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvfo | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCvdw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhvg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhuw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQjfo | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQyvO | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQytW | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCR~w | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCrvw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZvw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZnw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCZ^w | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzvg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCzjw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCz^g | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCx}w | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEjrw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEh}w | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEhzw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQyvo | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FQyvW | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FCx~w | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEh~w | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FEl~o | co-eiffeltower | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | |
FQyvw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUZvg | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUZvW | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUZuw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUZvw | 4 | 4 | 5 k = 5 unknown — open 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed | ||
FUZ~w | 4 | 5 | 5 k = 5 χ(G)=5 > ω(G)=4: no complete 4-partite host contains a copy of G (folklore) [folklore] 6k = 6 unknown — open 7k = 7 unknown — open …higher k not computed |
1941:Turan:ExtremalGraph — Turán 19411949:Zykov:LinearComplexes — Zykov 19491949:Zykov:SomePropertiesLinearComplexes — 1949:Zykov:SomePropertiesLinearComplexes1991:GyoriEtAl:MaximalNumberCertainSubgraphs — 1991:GyoriEtAl:MaximalNumberCertainSubgraphs1991:GyoriEtAl:MaximalSubgraphs — Győri–Pach–Simonovits 19911998:SchelpThomason:EmptySubgraphs — Schelp–Thomason 19982011:Grzesik:CyclesTriangle — Grzesik 20112011:HatamiEtAl:PentagonsGraphs — Hatami–Hladký–Král–Norine–Razborov 20112013:LiShi:TuranSequence — Li–Shi 20132013:LiShi:TurantypeDegreeSequence — 2013:LiShi:TurantypeDegreeSequence2017:LidickyPfender:PentagonsGraphs — Lidický–Pfender 20172017:LidickyPfender:PentagonsTrianglefreeGraphs — 2017:LidickyPfender:PentagonsTrianglefreeGraphs2019:MaQiu:GeneralizedTuran — Ma–Qiu 20192019:MaQiu:SharpResultsGeneralizedTuran — 2019:MaQiu:SharpResultsGeneralizedTuran2020:Gerbner:GeneralizedTuran — Gerbner 20202020:GerbnerPalmer:ExactResultsGeneralizedTuran — 2020:GerbnerPalmer:ExactResultsGeneralizedTuran2020:GerbnerPalmer:GeneralizedTuran — Gerbner–Palmer 20202021:Gerbner:GeneralizedTuran — Gerbner 20212021:Gerbner:GeneralizedTuranDoubleStars — 2021:Gerbner:GeneralizedTuranDoubleStars2021:GyoriEtAl:ExtremalStars — Győri–Wang–Woolfson 20212021:LidickyMurphy:MaximizingFivecyclesfreeGraphs — 2021:LidickyMurphy:MaximizingFivecyclesfreeGraphs2021:LidickyMurphy:MaximizingGraphs — Lidický–Murphy 20212021:MurphyNir:PathsTuran — Murphy–Nir 20212021:QianEtAl:ResultsturangoodGraphs — 2021:QianEtAl:ResultsturangoodGraphs2021:QianEtAl:TuranGraphs — Qian–Xie–Ge 20212022:Gerbner:PathsTuran — Gerbner 20222022:Gerbner:PathsTurangood — 2022:Gerbner:PathsTurangood2022:Gerbner:WeaklyTuran — Gerbner 20222022:MorrisonEtAl:GraphTuran — Morrison–Nir–Norin–Rzążewski–Wesolek 20222024:Gerbner:ExtremalSubgraph — Gerbner 20242024:Gerbner:ExtremalValuesDegreebased — 2024:Gerbner:ExtremalValuesDegreebased2025:GerbnerPalmer:TuranSubgraphs — Gerbner–Palmer 2025Shape for R-monotone graphs is exact (Thm 3.5). Balance is decided exactly where a certificate exists (AM–GM, GPS tiling, decorated inequality, cited theorem, or an exact rational witness for the unbalanced cases) and numerically otherwise (balanced iff the optimum is the centroid); $C_5$ is literature (Lidický–Murphy). The open set stays grey until settled. A solid outline marks a cell we settle here with a certificate; a dashed outline marks one where the literature is not improved on except by numeric (simplex) evidence — suggestive of Turán-goodness but not something we prove or claim. Morrison et al. guarantee every graph is eventually $k$-Turán-good.