On the 2-Colorability of Random Hypergraphs

A 2-coloring of a hypergraph is a mapping from its vertices to a set of two colors such that no edge is monochromatic. Let Hk(n,m) be a random k-uniform hypergraph on n vertices formed by picking m edges uniformly, independently and with replacement. It is easy to show that if r ≥ rc = 2k−1 ln 2− (ln 2)/2, then with high probability Hk(n,m = rn) is not 2-colorable. We complement this observation by proving that if r ≤ rc − 1 then with high probability Hk(n,m = rn) is 2-colorable.