On Turán-type problems for Berge matchings

Abstract: For a graph $F$, an $r$-uniform hypergraph ($r$-graph for short) $\mathcal{H}$ is a Berge-$F$ if there is a bijection $\phi:E(F)\rightarrow E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-graphs, an $r$-graph is $\mathcal{F}$-free if it does not contain any member in $\mathcal{F}$ 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. 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 $r\leq s-1$ or $r\geq 2s+2$, where $M_{s+1}$ denotes a matching of size $s+1$. In this paper, we settle the remaining case $s\le r\le 2s+1$. Moreover, we establish several exact and general results on the Tur\'{a}n numbers of Berge matchings together with a single $r$-graph, as well as of Berge matchings together with Berge bipartite graphs. Finally, we generalize the results on Tur\'{a}n problems for Berge hypergraphs proposed by Gerbner, Methuku, and Palmer [\textit{Eur. J. Comb. 86 (2020) 103082}].