Erdős–Ko–Rado in Random Hypergraphs

Abstract

Let 3 ≤k<n/2. We prove the analogue of the Erdős–Ko–Rado theorem for the randomk-uniform hypergraphGk(n,p) whenk< (n/2)1/3; that is, we show that with probability tending to 1 asn→ ∞, the maximum size of an intersecting subfamily ofGk(n,p) is the size of a maximum trivial family. The analogue of the Erdős–Ko–Rado theorem does not hold for allpwhenk≫n1/3.We give quite precise results fork<n1/2−ϵ. For largerkwe show that the random Erdős–Ko–Rado theorem holds as long aspis not too small, and fails to hold for a wide range of smaller values ofp. Along the way, we prove that every non-trivial intersectingk-uniform hypergraph can be covered byk2−k+ 1 pairs, which is sharp as evidenced by projective planes. This improves upon a result of Sanders [7]. Several open questions remain.