Matching of given sizes in hypergraphs

TL;DR

This paper establishes new minimum degree conditions in hypergraphs that guarantee the existence of large matchings, improving previous bounds and answering open questions in hypergraph matching theory.

Contribution

It provides improved degree threshold results for matchings in hypergraphs, extending and strengthening prior work and addressing open problems.

Findings

New degree conditions guarantee large matchings in hypergraphs.

Improved bounds extend to the entire range of possible matching sizes.

Counterexamples show the limits of the new conditions.

Abstract

For all integers $k,d$ such that $k \geq 3$ and $k/2\leq d \leq k-1$, let $n$ be a sufficiently large integer {\rm(}which may not be divisible by $k${\rm)} and let $s\le \lfloor n/k\rfloor-1$. We show that if $H$ is a $k$-uniform hypergraph on $n$ vertices with $\delta_{d}(H)>\binom{n-d}{k-d}-\binom{n-d-s+1}{k-d}$, then $H$ contains a matching of size $s$. This improves a recent result of Lu, Yu, and Yuan and also answers a question of K\"uhn, Osthus, and Townsend. In many cases, our result can be strengthened to $s\leq \lfloor n/k\rfloor$, which then covers the entire possible range of $s$. On the other hand, there are examples showing that the result does not hold for certain $n, k, d$ and $s= \lfloor n/k\rfloor$.