# Vertex degree sums for matchings in 3-uniform hypergraphs

## TL;DR

This paper establishes a sharp degree sum condition in 3-uniform hypergraphs that guarantees the existence of a large matching, confirming a previously conjectured bound and extending prior results.

## Contribution

The paper proves a new optimal degree sum condition for matchings in 3-uniform hypergraphs, generalizing earlier specific cases and confirming a conjecture by the authors.

## Key findings

- Degree sum condition guarantees matchings of size s
- Condition is proven to be best possible
- Extends previous case when s = n/3

## Abstract

Let $n, s$ be positive integers such that $n$ is sufficiently large and $s\le n/3$. Suppose $H$ is a 3-uniform hypergraph of order $n$. If $H$ contains no isolated vertex and $deg(u)+ deg(v) > 2(s-1)(n-1)$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a matching of size $s$. This degree sum condition is best possible and confirms a conjecture of the authors [Electron. J. Combin. 25 (3), 2018], who proved the case when $s= n/3$.