Two-coloring random hypergraphs

A 2-coloring of a hypergraph is a mapping from its vertex set to a set of two colors such that no edge is monochromatic. Let H = H(k, n, p) be a random k-uniform hypergraph on a vertex set V of cardinality n, where each k-subset of V is an edge of H with probability p, independently of all other k-subsets. Let m = p((n)(k)) denote the expected number of edges in H. Let us say that a sequence of events xi(n) holds with high probability (w.h.p.) if lim(n-->infinity) Pr[xi(n)] = 1. It is easy to show that if m = c2(k)n then w.h.p H is not 2-colorable for c > ln 2/2. We prove that there exists a constant c > 0 such that if M = (c2(k)/k)n, then w.h.p H is 2-colorable. (C) 2002 Wiley Periodicals, Inc.