Coloring Simple Hypergraphs

Abstract

Fix an integer $k \ge 3$. 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 $\D$ has chromatic number satisfying $$\chi(H) <c (\frac{\D}{\log \D})^{\frac{1}{k-1}}.$$ This implies a classical result of Ajtai-Koml\'os-Pintz-Spencer-Szemer\'edi and its strengthening due to Duke-Lefmann-R\"odl. The result is sharp apart from the constant $c$.