An extension of Mantel's theorem to random 4-uniform hypergraphs

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A sparse version of Mantel's Theorem is that, for sufficiently large p, with high probability (w.h.p.), every maximum triangle-free subgraph of G(n,p) is bipartite. DeMarco and Kahn proved this for p>K √ n/n for some constant K, and apart from the value of the constant, this bound is the best possible. Denote by T₃ the 3-uniform hypergraph with vertex set {a,b,c,d,e} and edge set {abc,ade,bde}. Frankl and Füredi showed that the maximum 3-uniform hypergraph on n vertices containing no copy of T₃ is tripartite for n> 3000. For some integer k, let Gᵏ(n,p) be the random k-uniform hypergraph. Balogh et al. proved that for p>K n/n for some constant K, every maximum T₃-free subhypergraph of G³(n,p) w.h.p. is tripartite and it does not hold when p=0.1 √ n/n. Denote by T₄ the 4-uniform hypergraph with vertex set {1,2,3,4,5,6,7} and edge set {1234,1235,4567}. Pikhurko proved that there is an n₀ such that for all n≥ n₀, the maximum 4-uniform hypergraph on n vertices containing no copy of T₄ is 4-partite. In this paper, we extend this type of extremal problem in random 4-uniform hypergraphs. We show that for some constant K and p>K n/n, w.h.p. every maximum T₄-free subhypergraph of G⁴(n,p) is 4-partite.