Perfect Matchings in 4-uniform hypergraphs

TL;DR

This paper proves a tight bound for the existence of perfect matchings in large 4-uniform hypergraphs, confirming a conjecture and advancing understanding of hypergraph matchings.

Contribution

It establishes a precise minimum degree condition that guarantees perfect matchings in large 4-uniform hypergraphs, settling a longstanding conjecture.

Findings

Proves a tight degree condition for perfect matchings.

Confirms a conjecture by H{á}n, Person, and Schacht.

Provides a characterization of when perfect matchings exist in 4-uniform hypergraphs.

Abstract

A perfect matching in a 4-uniform hypergraph is a subset of $\lfloor\frac{n}{4}\rfloor$ disjoint edges. We prove that if $H$ is a sufficiently large 4-uniform hypergraph on $n=4k$ vertices such that every vertex belongs to more than ${n-1\choose 3} - {3n/4 \choose 3}$ edges then $H$ contains a perfect matching. This bound is tight and settles a conjecture of H{\'a}n, Person and Schacht.