A hypergraph analogue of Alon-Frankl Theorem

TL;DR

This paper extends the Alon-Frankl theorem to hypergraphs, determining maximum edge counts in hypergraphs avoiding certain substructures while having bounded matching number, including specific results for 3-uniform hypergraphs like the Fano plane.

Contribution

It provides the exact maximum edge counts for hypergraphs avoiding hypergraph analogues of cliques and Fano planes with bounded matching number, generalizing previous graph results.

Findings

Exact maximum edge counts for hypergraphs avoiding $oldsymbol{ ext{K}_{oldsymbol{ ext{l+1}}}^{oldsymbol{r}}}$-free structures.

Characterization of extremal hypergraphs achieving these bounds.

Extension of Alon-Frankl theorem from graphs to hypergraphs.

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 $\binom{\ell+1}{2}$ 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, $\mathrm{ex}_r(n, \{\mathcal{K}_{\ell+1}^r, M^r_{s+1}\})$, 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 $\mathrm{ex}_3(n, \{\mathbb{F}, M^3_{s+1}\})$, and characterize the corresponding extremal hypergraph.