On Erd\H{o}s-Ko-Rado for random hypergraphs I

A family of sets is intersecting if no two of its members are disjoint, and has the Erd\H{o}s-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has nonempty intersection. Denote by ${\mathcal{H}}_k(n,p)$ the random family in which each $k$-subset of $\{1\dots n\}$ 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 $\mathcal{H_k(n,p)$ likely to be EKR?} Here, for fixed $c<1\mathrm{/}4$, and $k<\sqrt{cn\log n}$ we give a precise answer to this question, characterizing those sequences $p=p(n,k)$ for which $$\mathrm{Pr}({\mathcal{H}}_k(n,p)\text{ is EKR})\to 1\text{ as }n\to \mathrm{\infty }\mathrm{.}$$