Coloring simple hypergraphs

Fix an integer k >= 3. A k-uniform hypergraph is simple if every two edges share at most one vertex. We prove that there is a constant c depending only on k such that every simple k-uniform hypergraph H with maximum degree Delta has chromatic number satisfying chi(H) < c (Delta/log Delta)(1/k-1). This implies a classical result of Ajtai, Komlos, Pintz, Spencer and Szemeredi and its strengthening due to Duke, Lefmann and Rodl. The result is sharp apart from the constant c. (C) 2013 Elsevier Inc. All rights reserved.