Erdos-Ko-Rado in Random Hypergraphs

Let 3 ≤ k < n/2. We prove the analogue of the Erdős-Ko-Rado theorem for the random k-uniform hypergraph Gk(n, p) when k < (n/2)1/3; that is, we show that with probability tending to 1 as n → ∞, the maximum size of an intersecting subfamily of Gk(n, p) is the size of a maximum trivial family. The analogue of the Erdős-Ko-Rado theorem does not hold for all p when k À n1/3. We give quite precise results for k < n1/2−ε. For larger k we show that the random Erdős-Ko-Rado theorem holds as long as p is not too small and fails to hold for a wide range of smaller values of p. Along the way, we prove that every nontrivial intersecting k-uniform hypergraph can be covered by k2−k+1 pairs, which is sharp as evidenced by projective planes. This improves upon a result of Sanders [7]. Several open questions remain.