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 $H_k(n,m)$ 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 $r \geq r_c = 2^{k-1} \ln 2 - (\ln 2) /2$, then with high probability $H_k(n,m=rn)$ is not 2-colorable. We complement this observation by proving that if $r \leq r_c - 1$ then with high probability $H_k(n,m=rn)$ is 2-colorable.