Extremal Problem for Matchings and Rainbow Matchings on Direct Products

Let $n_1,\dots,n_\ell,k_1,\dots,k_\ell$ be integers and let $V_1,\dots,V_\ell$ be disjoint sets with $|V_i|=n_i$ for $i=1,\dots,\ell$. Define $\sqcup_{i=1}^\ell \binom{V_i}{k_i}$ as the collection of all subsets $F$ of $\cup_{i=1}^\ell V_i$ with $|F\cap V_i| =k_i$ for each $i=1,\dots,\ell$. In this paper, we show that if the matching number of $\mathcal{F}\subseteq \sqcup_{i=1}^\ell \binom{V_i}{k_i}$ is at most $s$ and $n_i\geq 4\ell^2 k_i^2s$ for all $i$, then $|\mathcal{F}| \leq \max_{1\leq i\leq \ell}[\binom{n_i}{k_i}-\binom{n_i-s}{k_i}]\prod_{j\neq i}\binom{n_j}{k_j}$. Let $\mathcal{F}_1,\mathcal{F}_2,\dots,\mathcal{F}_s\subseteq\sqcup_{i=1}^\ell \binom{V_i}{k_i}$ with $n_i\geq 8\ell^2k_i^2s$ for all $i$. We also prove that if $\mathcal{F}_1,\mathcal{F}_2,\dots,\mathcal{F}_s$ are rainbow matching free, then there exists $t$ in $[s]$ such that $|\mathcal{F}_t|\leq \max_{1\leq i\leq \ell}\left[\binom{n_i}{k_i}-\binom{n_i-s+1}{k_i}\right]\prod_{j\neq i}\binom{n_j}{k_j}.$