Coloring H-free hypergraphs

Fix r ≥ 2 and a collection of r-uniform hypergraphs H. What is the minimum number of edges in an H-free r-uniform hypergraph with chromatic number greater than k? We investigate this question for various 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 c = 2 ( 100(r)l l! )1/(l−1) ( 10(r − 1) l − 1 )l/(l−1) . This improves on the previous best bounds of Kostochka-Mubayi-Rödl-Tetali [10]. The upper bound is sharp apart from the constant c as shown in [10]. • 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 k as k → ∞. This generalizes (aside from logarithmic factors) a result of Gimbel and Thomassen [6] 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. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890; Supported in part by NSF grant DMS 0701183 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890; Supported in part by NSF grant CCF0502793 Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 606077045; Supported in part by NSF grant DMS 0653946