Turán problems with bounded matching number in $k$-uniform hypergraphs

Abstract

For a family $\mathcal{F}$ of $k$-graphs, $\ex_k(n,\mathcal{F})$ denotes the maximum number of edges in an $n$-vertex $\mathcal{F}$-free $k$-graph. Let $M_{s+1}^k$ denote a matching of size $s+1$ in $k$-uniform hypergraphs. Recently, Alon and Frankl (JCTB, 2024) determined $\ex_2(n,\{M_{s+1}^2,K_{\ell+1}\})$ for all $n\geq 2s+1$ and $\ell\geq 2$. For every non-bipartite graph $F$, Gerbner (JGT, 2024) determined $\ex_2(n,\{M_{s+1}^2,F\})$ for sufficiently large $n$. In this paper, we investigate this problem for different ranges of the matching parameter. First we prove that for every graph $F$ with $χ(F)>3$, there exist constants $β>0$ and $s_0$ such that $\ex_k(n,\{M^2_{s+1}, F\})=\ex_2(2s+1,F)$ for $\max\{s_0,n/2-βn\}<s<n/2$. For integers $\ell\ge k\ge3$, let $\mathcal{K}_{\ell+1}^k$ be the family of all $k$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges for which there is an $(\ell+1)$-set $L$ such that every pair of vertices of $L$ is covered by an edge of $F$, and let $H_{\ell+1}^k$ be the $k$-uniform hypergraph obtained from the complete graph $K_{\ell+1}$ by enlarging each edge with a set of $k-2$ new vertices, which is a member of $\mathcal{K}_{\ell+1}^k$. We determine $\ex_k\bigl(n,\mathcal{K}_{\ell+1}^k\cup\{M_{s+1}^k\}\bigr)$ for $s\leq \frac{n}{8(k-1)^{k-2}(\ell-1)^3}$. For sufficiently large $s$, we also determine $\ex_k\bigl(n,\{M_{s+1}^k,H_{\ell+1}^k\}\bigr)$ for $\frac{n}{k}-βn<s<\frac{n}{k}$ and $s\leq \frac{n}{8(k-1)^{k-2}(\ell-1)^3}$, respectively.