Two-coloring random hypergraphs

Authors:
Dimitris Achlioptas;Jeong Han Kim;Michael Krivelevich;Prasad Tetali
Affiliations:
Microsoft Research, One Microsoft Way, Redmond, WA;Microsoft Research, One Microsoft Way, Redmond, WA;Department of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel;School of Mathematics, Georgia Institute of Technology, Atlanta, GA
Venue:
Random Structures & Algorithms
Year:
2002

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Abstract

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.