2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169459Fix 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 $$χ(H) <c (\frac{\D}{\log \D})^{\frac{1}{k-1}}.$$ This implies a classical result of Ajtai-Komlós-Pintz-Spencer-Szemerédi and its strengthening due to Duke-Lefmann-Rödl. The result is sharp apart from the constant $c$.34 pagesCombinatorics05D40text