Hypergraph Turán Problem of the Generalized Triangle with Bounded Matching Number

Let H be a 3-graph on n vertices. The matching number ν(H) is defined as the maximum number of disjoint edges in H. The generalized triangle F₅ is a 3-graph on the vertex set ,b,c,d,e\ with the edge set , abd,cde\. In this paper, we determine the maximum number of edges in an F₅-free $3$-graph H satisfying ν(H)≤ s when n≥ 30(s+1). More precisely, the maximum is n-12 for $s=1,2$ and is s (n-s)²/4 for s≥ 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.