Hypergraph Turan with bounded matching number

For a fixed graph $G$, an $r$-uniform hypergraph is said to contain a Berge-$G$ if there exists a bijection $f\colon E(G)\to E(\mathcal{H})$ for some subhypergraph $\mathcal{H}$ such that $e\subseteq f(e)$ for every $e\in E(G)$.

Motivated by Alon and Frankl's study of Turán problems under bounded matching constraints, we investigate the maximum number of edges in $r$-uniform Berge-$K_3$-free hypergraphs with matching number at most~$s$.

We determine the exact Turán numbers for the cases $r=3$ and $r=4$.

For $r=3$ and $n \geq 3 s$, we prove that every $n$-vertex Berge- $K_3$-free 3-graph with matching number $s$ has at most $s(n-2 s)$ edges, and we characterize the unique extremal hypergraph attaining equality.

For $r=4$ and $n \geq 4 s$, the maximum number of edges is $s\lfloor(n-2 s) / 2\rfloor$, except for the exceptional case $s=1$ and $n \equiv 1(\bmod 4)$, in which the bound is $(n-1) / 2$.

As a corollary, our results recover the classical theorem of Győri on Berge-$K_3$-free hypergraphs.