On Erdos-Ko-Rado for random hypergraphs I

A family of sets is intersecting if no two of its members are disjoint, and has the Erdos-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has non-empty 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. A question first studied by Balogh, Bohman and Mubayi asks: For what p = p(n, k) is H-k(n, p) likely to be EKR? Here, for fixed c < 1/4, and k < root cn log n we give a precise answer to this question, characterizing those sequences p = p(n, k) for which P(H-k(n, p) is EKR) -> 1 as n -> infinity.