A hypergraph analogue of Alon-Frankl Theorem

Abstract

Recently, Alon and Frankl (JCTB, 2024) determined the maximum number of edges in $K_{\ell +1}$-free $n$-vertex graphs with bounded matching number. For integers $\ell \ge r\ge 2$, the family ${\mathcal{K}}_{\ell +1}^r$ consists of all $r$-graphs $F$ with at most $\left(\genfrac{}{}{0ex}{}{\ell +1}{2}\right)$ edges such that, for some $(\ell +1)$-set $K$, every pair $\{x,y\}\subseteq K$ is covered by an edge in $F$. In this paper, we study the maximum number of edges in ${\mathcal{K}}_{\ell +1}^r$-free $r$-uniform hypergraphs that have the matching number at most $s$, that is, ${\mathrm{ex}}_r(n,\{{\mathcal{K}}_{\ell +1}^r,M_{s+1}^r\})$, and obtain the exact value for sufficiently large $n$, along with the corresponding extremal hypergraph. This result can be viewed as a hypergraph extension of the work of Alon and Frankl. In addition, for the $3$-uniform Fano plane $\mathbb{F}$, we determine the exact value of ${\mathrm{ex}}_3(n,\{\mathbb{F},M_{s+1}^3\})$, and characterize the corresponding extremal hypergraph.

12 pages