Stability on Matchings in 3-Uniform Hypergraphs

Abstract

Given a positive integer r, let $[r]=\{1,\ldots ,r\}$. Let n, m be positive integers such that n is sufficiently large and $1\le m\le \lfloor n/3\rfloor -1$. Let H be a 3-graph with vertex set [n], and let $\delta _1(H)$ denote the minimum vertex degree of H. The size of a maximum matching of H is denoted by $\nu (H)$. Kühn, Osthus and Treglown (2013) proved that there exists an integer $n_0\in \mathbb {N}$ such that if H is a 3-graph with $n\ge n_0$ vertices and $\delta _1(H)>{n-1\atopwithdelims ()2}-{n-m\atopwithdelims ()2}$, then $\nu (H)\ge m$. In this paper, we show that there exists an integer $n_1\in \mathbb {N}$ such that if $|V(H)|\ge n_1$, $\delta _1(H)>{n-1\atopwithdelims ()2}-{n-m\atopwithdelims ()2}+3$ and $\nu (H)\le m$, then H is a subgraph of $H^*(n,m)$, where $H^*(n,m)$ is a 3-graph with vertex set [n] and edge set $E(H^*(n,m))=\{e\subseteq [n]: |e|=3 \text{ and } e\cap [m] \ne \emptyset \}$. The minimum degree condition is best possible.