The maximum number of cliques in hypergraphs without large matchings

Erica L.L. Liu¹ ; Jian Wang²

¹Center for Applied Mathematics. Tianjin University
²Department of Mathematics, Taiyuan University of Technology

The electronic journal of combinatorics, Tome 27 (2020) no. 4

Résumé

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 $n\geq 2rk$, 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 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 $\mathcal{F}^{(r)}_{n,k,a}$ 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 $s=r$.

Détail

DOI : 10.37236/9604

Classification : 05C15, 05C65, 05C69, 05C30
Mots-clés : matching number, Erdős conjecture, Katona's theorem on shadows of intersecting hypergraphs