ON WEAKLY TURAN-GOOD GRAPHS

Given graphs H and F with chi(H)<chi(F), we say that H is weakly F-Tur & aacute;n-good if among n-vertex F-free graphs, a (chi(F)-1)-partite graph contains the most copies of H. Let H be a bipartite graph that contains a complete bipartite subgraph K such that each vertex of H is adjacent to a vertex of K. We show that H is weakly K-3-Tur & aacute;n-good, improving a very recent asymptotic bound due to Grzesik, Gyori, Salia and Tompkins. They also showed that for any r there exist graphs that are not weakly K-r-Tur & aacute;n-good. We show that for any non-bipartite F there exists graphs that are not weakly F-Tur & aacute;n-good. We also show examples of graphs that are C2k+1-Tur & aacute;n-good but not C2 & ell;+1-Tur & aacute;n-good for every k>& ell;.