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(????) denote the expected number of edges in H. Let us say that a sequence of events En holds with high probability (w.h.p.) if limn → ∞ Pr[En] = 1. It is easy to show that if m = c2kn 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 = (c2k/k)n, then w.h.p H is 2-colorable.

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