On k-uniform random hypergraphs without generalized fans

Let the k-uniform hypergraph Fan(k) 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 Turan number ex(n, Fan(k)) of Fan(k) 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 G(k) (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 Fan(k)-free subhypergraph of G(k)(n, p) is k-partite for k >= 4; and if p > K(logn)(gamma)/n, where gamma > 0 is an absolute constant, then every maximum Fan(3)-free subhypergraph of G(3)(n, p) is tripartite. Our result is an exact version of a random analogue of the stability result of Fan(k)-free(k)-graphs, which can be obtained by using the transference theorem given by Samotij (2014). (C) 2021 Elsevier B.V. All rights reserved.