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

Denote by ${\mathcal{H}}_k(n,p)$ the random $k$-graph in which each $k$-subset of $\{\mathrm{1...}n\}$ 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 $\epsilon >0$ such that if $n=2k+1$ and $p>1-\epsilon$, then w.h.p. (that is, with probability tending to 1 as $k\to \mathrm{\infty }$), ${\mathcal{H}}_k(n,p)$ has the “Erd\H{o}s-Ko-Rado property.” We also mention a similar random version of Sperner’s Theorem.