Some extremal results on K_{s,t}-free graphs

For graphs $H$ and $F$, let $\text{ex}(n,H,F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. The study of this function, which generalizes the well-known Tur'{a}n number of graphs, was systematically studied by Alon and Shikhelman recently. In this paper, we show that for any $m$ and $t\ge 2m-3\ge 3$, $$\text{ex}(n,K_m,K_{2,t})=\mathrm{\Theta }(n^{\frac{3}{2}})\mathrm{.}$$ This result improves some results of Alon and Shikhelman (J. Combin. Theory Ser. B, 121:146-172, 2016). We also study the $k$-partite $K_{s,t}$-free graph, we show that for any $k\ge 3$ and $t\ge (k-1)(s-1)!+1$, $${\text{ex}}_{\chi \le k}(n,K_{s,t})\ge \frac{k-1}{2k}{n}^{2-1\mathrm{/}s}+o(n^{2-1\mathrm{/}s})\mathrm{.}$$ Moreover, we give a new construction of $3$-partite $K_{2,2t+1}$-free graphs with many edges.