Towards the Erd\H{o}s matching conjecture for 4-uniform hypergraphs: stability and applications

TL;DR

This paper advances the understanding of Erd ext{"o}s' matching conjecture for 4-uniform hypergraphs by proving it for large n and s, and introduces a stability result with applications to degree threshold conjectures.

Contribution

It proves the Erd ext{"o}s matching conjecture for 4-uniform hypergraphs when n is sufficiently large, and establishes a new stability theorem with applications to degree threshold problems.

Findings

Proved the conjecture for 4-uniform hypergraphs with n ≥ 5s and large n.

Developed a stability result of independent interest.

Resolved new cases of degree threshold conjectures for 5- and 6-uniform hypergraphs.

Abstract

A famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem has been central in extremal combinatorics, with substantial progress in the literature, including a complete solution for $k=3$ due to the first author. In this paper, we make progress towards the $4$-uniform case, proving the conjecture for $n\ge 5s$ and sufficiently large $n$, thereby taking a first step analogous to the $3$-uniform case. The main technical contribution is a stability result of independent interest. We further apply this stability to resolve two new instances of conjectures on the minimum $d$-degree threshold for matchings in $5$- and $6$-uniform hypergraphs, in a strengthened form.