On Erdős-Ko-Rado for random hypergraphs I

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 non-empty intersection. Denote by ${{\cal H}_k}(n,p)$ 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: \begin{equation} {\rm{For what }}p = p(n,k){\rm{is}}{{\cal H}_k}(n,p){\rm{likely to be EKR}}? \end{equation} Here, for fixed c $k \lt \sqrt {cn\log n} $ we give a precise answer to this question, characterizing those sequences p = p ( n , k ) for which \begin{equation} {\mathbb{P}}({{\cal H}_k}(n,p){\rm{is EKR}}{\kern 1pt} ) \to 1{\rm{as }}n \to \infty . \end{equation}