Turán-goodness of small connected graphs

Turán goodAmong K_k-free hosts, the number of copies of G is uniquely maximised by the balanced Turán graph T_k−1(n). Equivalently the colouring polynomial P_G(α)=Σ_{proper (k−1)-colourings c} ∏_v α_c(v) is maximised at the centroid α=(1/(k−1),…).numerically goodAs Turán good, but established only numerically: the simplex optimum of P_G lies at the centroid (within tolerance). No proof yet.weakly goodThe optimum over K_k-free hosts is a complete (k−1)-partite host, but an unbalanced one — some part-vector α ≠ centroid strictly beats T_k−1. So G is weakly k-Turán-good but not k-Turán-good.numerically weakly goodAs weakly good, but only numerically: the simplex optimum lies off the centroid, with no exact witness.Turán shapeThe optimum is a complete (k−1)-partite (Turán-shaped) host, but whether the balanced T_k−1 attains it is undetermined. R-monotonicity gives this for every k > ω(G) (Thm 3.5); it is also the literature notion of weakly k-Turán-good.non-TuránNo complete (k−1)-partite host is optimal — the maximiser is a genuinely non-Turán structure (e.g. a balanced blow-up of G). G is not even weakly k-Turán-good.unknownNot settled by the current exact or numeric methods (flag-algebra territory).oursThe only certificate at this k is R-monotonicity (this work); no prior result settles it. These cells are our contribution.