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

TL;DR

This paper determines the threshold probability for which the largest $F_5$-free subhypergraphs of a random 3-uniform hypergraph are almost surely tripartite, refining previous bounds to be nearly optimal.

Contribution

It sharpens the known upper bound on the probability threshold, establishing that above a certain constant times $rac{ ootlog n}{n}$, maximum $F_5$-free subhypergraphs are tripartite with high probability.

Findings

Established the threshold $p > C rac{ ootlog n}{n}$ for tripartiteness.

Proved the sharpness of the bound up to a constant factor.

Extended previous results to nearly optimal bounds.

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.