A stability result on matchings in 3-uniform hypergraphs

Abstract: Let $n,s,k$ be three positive integers such that $1\leq s\leq(n-k+1)/k$ and let $[n]={1,\ldots,n}$. Let $H$ be a $k$-graph with vertex set ${1,\ldots,n}$, and let $e(H)$ denote the number of edges of $H$. Let $\nu(H)$ and $\tau(H)$ denote the size of a largest matching and the size of a minimum vertex cover in $H$, respectively. Define $A^k_i(n,s):={e\in\binom{[n]}{k}:|e\cap[(s+1)i-1]|\geq i}$ for $2\leq i\leq k$ and $HM^k_{n,s}:=\big{e\in\binom{[n]}{k}:e\cap[s-1]\neq\emptyset\big} \cup\big{S\big}\cup \big{e\in\binom{[n]}{k}: s\in e, e\cap S\neq \emptyset}$, where $S={s+1,\ldots,s+k}$. Frankl and Kupavskii conjectured that if $\nu(H)\leq s$ and $\tau(H)>s$, then $e(H)\leq \max{|A^k_2(n,s)|,\ldots ,|A^k_k(n,s)|,|HM^k_{n,s}|}$. In this paper, we prove this conjecture for $k=3$ and sufficiently large $n$.