On Erdős–Ko–Rado for Random Hypergraphs II

Abstract

Denote by${\mathcal H}_k$(n,p) the randomk-graph in which eachk-subset of {1,. . .,n} is present with probabilityp, 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 ifn= 2k+ 1 andp> 1 - ε, then w.h.p. (that is, with probability tending to 1 ask→ ∞),${\mathcal H}_k$(n,p) has the ‘Erdős–Ko–Rado property’. We also mention a similar random version of Sperner's theorem.