Coloring H‐free hypergraphs

Abstract

AbstractFix 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(kr−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 kr+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