A stability result on matchings in 3-uniform hypergraphs

TL;DR

This paper proves a conjecture relating the maximum size of 3-uniform hypergraphs with certain matching and vertex cover constraints, establishing a stability result for these hypergraph configurations.

Contribution

The paper confirms Frankl and Kupavskii's conjecture for 3-uniform hypergraphs when the number of vertices is large, advancing understanding of extremal hypergraph structures.

Findings

Proves the conjecture for 3-uniform hypergraphs with large n.

Identifies extremal hypergraph configurations under given matching and cover constraints.

Establishes bounds on the number of edges in such 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 $n$.