Coloring H‐free hypergraphs

Abstract

Abstract Fix r ≥ 2 and a collection of r ‐uniform hypergraphs $\cal{H}$ . What is the minimum number of edges in an $\cal{H}$ ‐free r ‐uniform hypergraph with chromatic number greater than k ? We investigate this question for various $\cal{H}$ . Our results include the following: An ( r , l )‐system is an r ‐uniform hypergraph with every two edges sharing at most l vertices. For k sufficiently large, there is an ( r , l )‐system with chromatic number greater than k and number of edges at most c ( k r −1 log k ) l /( l −1) , where This improves on the previous best bounds of Kostochka et al. (Random Structures Algorithms 19 (2001), 87–98). The upper bound is sharp apart from the constant c as shown in (Random Structures Algorithms 19 (2001) 87–98). The minimum number of edges in an r ‐uniform hypergraph with independent neighborhoods and chromatic number greater than k is of order k r +1/( r −1) log O (1) k as k → ∞. This generalizes (aside from logarithmic factors) a result of Gimbel and Thomassen (Discrete Mathematics 219 (2000), 275–277) for triangle‐free graphs. Let T be an r ‐uniform hypertree of t edges. Then every T ‐free r ‐uniform hypergraph has chromatic number at most 2( r − 1)( t − 1) + 1. This generalizes the well‐known fact that every T ‐free graph has chromatic number at most t . Several open problems and conjectures are also posed. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010