The Maximum Number of Cliques in Hypergraphs without Large Matchings

Let $[n]$ denote the set $\{1,2,\dots ,n\}$ and ${\mathcal{F}}_{n,k,a}^{(r)}$ be an $r$-uniform hypergraph on the vertex set $[n]$ with edge set consisting of all the $r$-element subsets of $[n]$ that contains at least $a$ vertices in $[ak+a-1]$. For $n\ge 2rk$, Frankl proved that ${\mathcal{F}}_{n,k,1}^{(r)}$ maximizes the number of edges in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$. Huang, Loh and Sudakov considered a multicolored version of the Erd\H{o}s matching conjecture, and provided a sufficient condition on the number of edges for a multicolored hypergraph to contain a rainbow matching of size $k$. In this paper, we show that ${\mathcal{F}}_{n,k,a}^{(r)}$ maximizes the number of $s$-cliques in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$ for sufficiently large $n$, where $a=\lfloor \frac{s-r}{k} \rfloor+1$\mathrm{.}Wealsoobtainaconditiononthenumberof$s$-clques for a multicolored $r$-uniform hypergraph to contain a rainbow matching of size $k$, which reduces to the condition of Huang, Loh and Sudakov when $s=r$.