Some sharp results on the generalized Tur\'an numbers

For graphs $T, H$, let $ex(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex $H$-free graph. In this paper we prove some sharp results on this generalization of Tur\'an numbers, where our focus is for the graphs $T,H$ satisfying $\chi(T)<\chi(H)$. This can be dated back to Erd\H{o}s, where he generalized the celebrated Tur\'an's theorem by showing that for any $r\geq m$, the Tur\'an graph $T_r(n)$ uniquely attains $ex(n,K_m,K_{r+1})$. For general graphs $H$ with $\chi(H)=r+1>m$, Alon and Shikhelman showed that $ex(n,K_m,H)=\binom{r}{m}(\frac{n}{r})^m+o(n^m)$. Here we determine this error term $o(n^m)$ up to a constant factor. We prove that $ex(n,K_m,H)=\binom{r}{m}(\frac{n}{r})^m+biex(n,H)\cdot\Theta(n^{m-2})$, where $biex(n,H)$ is the Tur\'an number of the decomposition family of $H$. As a special case, we extend Erd\H{o}s' result, by showing that $T_r(n)$ uniquely attains $ex(n,K_m,H)$ for any edge-critical graph $H$. We also consider $T$ being non-clique, where even the simplest case seems to be intricate. Following from a more general result, we show that for all $s\leq t$, $T_2(n)$ maximizes the number of $K_{s,t}$ in $n$-vertex triangle-free graphs if and only if $t<s+\frac12+\sqrt{2s+\frac14}$.