Mantel's theorem for random hypergraphs

A classical result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle-free subgraph of K_n is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large p, every maximum triangle-free subgraph of G(n, p) is w.h.p. bipartite. Recently, DeMarco and Kahn proved this for p>K√logn/n for some constant K, and apart from the value of the constant this bound is best possible. We study an extremal problem of this type in random hypergraphs. Denote by F₅, which is sometimes called the generalized triangle, the 3-uniform hypergraph with vertex set (a,b,c,d,e) and edge set (abc,ade,bde). One of the first results in extremal hypergraph theory is by Frankl and Füredi, who proved that the maximum 3-uniform hypergraph on n vertices containing no copy of F₅ is tripartite for n>3000. A natural question is for what p is every maximum F₅-free subhypergraph of G³(n,p) w.h.p. tripartite. We show this holds for p>Klogn/n for some constant K and does not hold for p=0.1√logn/n.