Erdős-Ko-Rado for Random Hypergraphs: Asymptotics and Stability

We investigate the asymptotic version of the Erdős-Ko-Rado theorem for the random kuniform hypergraph H(n, p). For 2 ≤ k(n) ≤ n/2, let N = ( n k ) and D = ( n−k k ) . We show that with probability tending to 1 as n→∞, the largest intersecting subhypergraph of H(n, p) has size (1 + o(1))p k nN , for any p n k ln 2 ( n k ) D−1. This lower bound on p is asymptotically best possible for k = Θ(n). For this range of k and p, we are able to show stability as well. A different behavior occurs when k = o(n). In this case, the lower bound on p is almost optimal. Further, for the small interval D−1 p ≤ (n/k)1−εD−1, the largest intersecting subhypergraph of H(n, p) has size Θ(ln(pD)ND−1), provided that k √ n lnn. Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in H(n, p), for essentially all values of p and k.