Coloring H-free hypergraphs

Fix r ≥ 2 and a collection of r-uniform hypergraphs . What is the minimum number of edges in an -free r-uniform hypergraph with chromatic number greater than k? We investigate this question for various . 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.