Some exact results for generalized Tur\'an problems

Fix a $k$-chromatic graph $F$. In this paper we consider the question to determine for which graphs $H$ does the Tur'an graph $T_{k-1}(n)$ have the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough). We say that such a graph $H$ is $F$-Tur'an-good. In addition to some general results, we give (among others) the following concrete results: (i) For every complete multipartite graph $H$, there is $k$ large enough such that $H$ is $K_k$-Tur'an-good. (ii) The path $P_3$ is $F$-Tur'an-good for $F$ with $\chi (F)\ge 4$. (iii) The path $P_4$ and cycle $C_4$ are $C_5$-Tur'an-good. (iv) The cycle $C_4$ is $F_2$-Tur'an-good where $F_2$ is the graph of two triangles sharing exactly one vertex.