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.
Categories: math.CO