Given two graphs $T$ and $F$, the maximum number of copies of $T$ in an $F$-free graph on $n$ vertices is called the generalized Tur\'{ a}n number, denoted by $ex(n,T,F)$. When $T=K_2$, it reduces to the classical Tur\'{ a}n number $ex(n,F)$. Let $M_{ k}$ be a matching with $k$ edges and $K^{ *}_{ s,t}$ a graph obtained from $K_{ s,t}$ by replacing the part of size $s$ by a clique of the same size. In this paper, we show that for any $s\geq 2$ and $n\geq 2k+1$, \[ ex(n,K_s,M_{ k+1})=\max\left\{ \binom{ 2k+1}{ s}, \binom{ k}{ s}+(n-k)\binom{ k}{ s-1}\right\}. \] For any $s\geq 1$, $t\geq 2$ and $n\geq 2k+1$, \[ ex(n,K_{ s,t}^*,M_{ k+1})=\max\left\{ \binom{ 2k+1}{ s+t}\binom{ s+t}{ t}, \binom{ k}{ s}\binom{ n-s}{ t}+(n-k)\binom{ k}{ s+t-1}\binom{ s+t-1}{ t}\right\}. \] Moreover, we also study the bipartite case of the problem. Let $ex_{ bip}(n,T,F)$ be the maximum possible number of copies of $T$ in an $F$-free bipartite graph with each part of size $n$. We prove that for any $s,t\geq 1$ and $n\geq k$, \[ ex_{ bip}(n,K_{ s,t},M_{ k+1})=\left\{ \begin{ aligned} &\binom{ k}{ s}\binom{ n}{ t}+\binom{ k}{ t}\binom{ n}{ s}, & \quad s\neq t, &\binom{ k}{ s}\binom{ n}{ s},&\quad s=t. \end{ aligned} \right. \] Our proof is mainly based on the shifting method.
Categories: math.CO