Equitable coloring of k-uniform hypergraphs

Let H be a k-uniform hypergraph with n vertices. A strong r-coloring is a partition of the vertices into r parts such that each edge of H intersects each part. A strong r-coloring is called equitable if the size of each part is [n/r] or [n/r]. We prove that for all a greater than or equal to 1, if the maximum degree of H satisfies Delta(H) less than or equal to k(a), then H has an equitable coloring with k/ln k( 1 - o(k)(1)) parts. In particular, every k-uniform hypergraph with maximum degree O(k) has an equitable coloring with k/ln k (1 - o(k)(1)) parts. The result is asymptotically tight. The proof uses a double application of the nonsymmetric version of the Lovasz local lemma.