On the Maximum $F_5$-free Subhypergraphs of a Random Hypergraph

Abstract

Denote by $F_5$ the $3$-uniform hypergraph on vertex set $\{1,2,3,4,5\}$ with hyperedges $\{123,124,345\}$. Balogh, Butterfield, Hu, and Lenz proved that if $p>K\log n/n$ for some large constant $K$, then every maximum $F_5$-free subhypergraph of $G^3(n,p)$ is tripartite with high probability, and showed that if $p_0=0.1\sqrt{\log n}/n$, then with high probability there exists a maximum $F_5$-free subhypergraph of $G^3(n,p_0)$ that is not tripartite. In this paper, we sharpen the upper bound to be best possible up to a constant factor. We prove that if $p>C\sqrt{\log n}/n$ for some large constant $C$, then every maximum $F_5$-free subhypergraph of $G^3(n,p)$ is tripartite with high probability.

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