On [formula omitted]-uniform random hypergraphs without generalized fans

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Let the k-uniform hypergraph Fank consist of k edges that pairwise intersect exactly in one vertex x, plus one more edge intersecting each of these edges in a vertex different from x. Mubayi and Pikhurko (2007), determined the exact Turán number ex(n,Fank) of Fank for sufficiently large n, which provides a generalization of Mantel’s theorem. In this paper, we give a sparse version of Mubayi and Pikhurko’s result. For a fixed integer k(k≥3), let Gk(n,p) be a probability space consisting of k-uniform hypergraphs with n vertices, in which each element of [n]k occurring independently as an edge with probability p. We show that there exists a positive constant K such that with high probability the following is true. If p>K/n, then every maximum Fank-free subhypergraph of Gk(n,p) is k-partite for k≥4; and if p>K(logn)γ/n, where γ>0 is an absolute constant, then every maximum Fan3-free subhypergraph of G3(n,p) is tripartite. Our result is an exact version of a random analogue of the stability result of Fank-free k-graphs, which can be obtained by using the transference theorem given by Samotij (2014).