Hypergraph Turán Problem of the Generalized Triangle with Bounded Matching Number

Abstract

Let $\mathcal{H}$ be a 3-graph on $n$ vertices. The matching number $\nu(\mathcal{H})$ is defined as the maximum number of disjoint edges in $\mathcal{H}$. The generalized triangle $F_5$ is a 3-graph on the vertex set $\{a,b,c,d,e\}$ with the edge set $\{abc, abd,cde\}$. In this paper, we determine the maximum number of edges in an $F_5$-free $3$-graph $\mathcal{H}$ satisfying $\nu(\mathcal{H})\leq s$ when $n\geq 30(s+1)$. More precisely, the maximum is $\binom{n-1}{2}$ for $s=1,2$ and is $s\bigl\lfloor (n-s)^2/4\bigr\rfloor$ for $s\geq 3$. We also determine the unique extremal graph for each case. For the proof, we establish a 2-colored version of Mantel's theorem, which may be of independent interest.