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 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. We sharpen the upper bound to be best possible up to a constant factor. We prove that when p > C\sqrt{log n}/n for some large constant C, every maximum F_5-free subhypergraph of G^3(n,p) is tripartite with high probability. In this talk, I will introduce the main technique we use to improve this bound.
This is a joint work with Igor Araujo and Jozsef Balogh.
