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 $t\ge 2m-3\ge3$, $$\text{ex}(n,K_{m},K_{2,t})=\Theta(n^{\frac{3}{2}}).$$ 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 $k\ge3$ and $t\ge(k-1)(s-1)!+1$, $$\text{ex}_{\chi\le k}(n,K_{s,t})\ge\frac{k-1}{2k}n^{2-1/s}+o(n^{2-1/s}).$$ Moreover, we give a new construction of $3$-partite $K_{2,2t+1}$-free graphs with many edges.