Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph G = G n,p , we show, roughly speaking, that (with high probability) the triangles of G span its cycle space whenever each of its edges lies in a triangle (which happens (w.h.p.) when p is at least about (3/2) ln n/n, and not below this unless p is very small). We give two related proofs of this statement, together with a fundamental "stability" theorem for triangle-free subgraphs of G n,p , originally due to Kohayakawa, Luczak and Rödl, that underlies the first of our proofs.

Denote by H k (n, p) the random k-graph in which each k-subset of {1, . . . , n} is present with probability p, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed ε > 0 such that if n = 2k + 1 and p > 1 − ε, then w.h.p. (that is, with probability tending to 1 as k → ∞), H k (n, p) has the "Erdős-Ko-Rado property." We also mention a similar random version of Sperner's Theorem.

A family of sets is intersecting if no two of its members are disjoint, and has the Erdős-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has nonempty intersection.Denote by H k (n, p) the random family in which each k-subset of {1, . . . , n} is present with probability p, independent of other choices.

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