On weakly Tur\'an-good graphs

Given graphs $H$ and $F$ with $\chi(H)<\chi(F)$, we say that $H$ is weakly $F$-Tur\'an-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\'an-good, improving a very recent asymptotic bound due to Grzesik, Gy\H ori, Salia and Tompkins. They also showed that for any $r$ there exist graphs that are not weakly $K_r$-Tur\'an-good. We show that for any non-bipartite $F$ there exists graphs that are not weakly $F$-Tur\'an-good. We also show examples of graphs that are $C_{2k+1}$-Tur\'an-good but not $C_{2\ell+1}$-Tur\'an-good for every $k>\ell$.