The Tur\'{a}n number of Berge matchings

Abstract

Given a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a {\em Berge-$F$} if there is a bijection $\phi:E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs, an $r$-uniform hypergraph is $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Tur\'{a}n number of $\mathcal{F}$ is the maximum number of hyperedges in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Let $M_{s+1}$ denote a matching of size $s+1$, i.e., the graph consisting of $s+1$ independent edges. Khormali and Palmer [\textit{European J. Combin.} 102 (2022) 103506] completely determined the Tur\'{a}n number of Berge matchings for sufficiently large $n$. Subsequently, Kang, Ni, and Shan [\textit{Discrete Math.} 345 (2022) 112901] determined the exact value of the Tur\'{a}n number of Berge-$M_{s+1}$ for all $n$ when $r \le s-1$ or $r \ge 2s+2$. In this paper, we settle the final open case $s \le r \le 2s+1$, thereby completing the determination of the Tur\'{a}n number of Berge matchings.