The shifting method and generalized Tur\'{a}n number of matchings

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.