On the Maximum F₅-free Subhypergraphs of a Random Hypergraph

Denote by F₅ 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 F₅-free subhypergraph of G³(n, p) is tripartite with high probability, and showed that if (Formula present) then with high probability there exists a maximum F₅-free subhy-pergraph of G³(n, p₀) 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 (Formula present) for some large constant C, then every maximum F₅-free subhypergraph of G³(n, p) is tripartite with high probability.