ErdsKoRado in random hypergraphs

Let 3 k < n/2. We prove the analogue of the ErdsKoRado theorem for the random k-uniform hypergraph G^k(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 G^k(n, p) is the size of a maximum trivial family. The analogue of the ErdsKoRado theorem does not hold for all p when k ≫ n^1/3. We give quite precise results for k < n^1/2. For larger k we show that the random ErdsKoRado 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 non-trivial 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.