A stability result on matchings in 3-uniform hypergraphs

Abstract

Let $n,s,k$ be three positive integers such that $1\le s\le (n-k+1)/k$ and let $[n]=\{1,\dots ,n\}$. Let $H$ be a $k$-graph with vertex set $\{1,\dots ,n\}$, 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 $A_i^k(n,s):=\{e\in (\genfrac{}{}{0ex}{}{[n]}{k}):|e\cap [(s+1)i-1]|\ge i\}$ for $2\le i\le k$ and $H{M}_{n,s}^k:=\{e\in (\genfrac{}{}{0ex}{}{[n]}{k}):e\cap [s-1]\ne \varnothing \}\cup \left\{S\right\}\cup \{e\in (\genfrac{}{}{0ex}{}{[n]}{k}):s\in e,e\cap S\ne \varnothing \}$, where $S=\{s+1,\dots ,s+k\}$. Frankl and Kupavskii conjectured that if $\nu (H)\le s$ and $\tau (H)>s$, then $e(H)\le \max \{|A_2^k(n,s)|,\dots ,|A_k^k(n,s)|,|H{M}_{n,s}^k|\}$. In this paper, we prove this conjecture for $k=3$ and sufficiently large $n$.