Hypergraph Tur\'an problem of the generalized triangle with bounded matching number

TL;DR

This paper determines the maximum number of edges in 3-graphs avoiding a generalized triangle with a bounded matching number, extending Turán-type results with new combinatorial bounds.

Contribution

It introduces a bound on edges for F_5-free 3-graphs with limited matching number and proves a 2-colored Mantel's theorem for the first time.

Findings

Maximum edges in F_5-free 3-graphs with matching number s

Bound of s * floor((n-s)^2/4) for n ≥ 30(s+1) and s ≥ 3

A new 2-colored Mantel's theorem

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 showed that an $F_5$-free 3-graph $\mathcal{H}$ with matching number at most $s$ has at most $s\lfloor (n-s)^2/4\rfloor$ edges for $n\geq 30(s+1)$ and $s\geq 3$. For the proof, we establish a 2-colored version of Mantel's theorem, which may be of independent interests.