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- W3094179932 abstract "Let $[n]$ denote the set ${1, 2, ldots, n}$ and $mathcal{F}^{(r)}_{n,k,a}$ 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 $ngeq 2rk$, Frankl proved that $mathcal{F}^{(r)}_{n,k,1}$ 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 ErdH{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 $mathcal{F}^{(r)}_{n,k,a}$ 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 $a=lfloor frac{s-r}{k} rfloor+1$. We also obtain a condition on the number of $s$-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 $s=r$." @default.
- W3094179932 created "2020-10-29" @default.
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- W3094179932 date "2020-10-16" @default.
- W3094179932 modified "2023-10-12" @default.
- W3094179932 title "The Maximum Number of Cliques in Hypergraphs without Large Matchings" @default.
- W3094179932 doi "https://doi.org/10.37236/9604" @default.
- W3094179932 hasPublicationYear "2020" @default.
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