Coloring Hfree hypergraphs
Fix r ≥ 2 and a collection of runiform hypergraphs . What is the minimum number of edges in an free runiform hypergraph with chromatic number greater than k? We investigate this question for various . Our results include the following:

An (r,l)system is an runiform 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 runiform 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 trianglefree graphs.

Let T be an runiform hypertree of t edges. Then every Tfree runiform hypergraph has chromatic number at most 2(r − 1)(t − 1) + 1. This generalizes the wellknown fact that every Tfree graph has chromatic number at most t.
Several open problems and conjectures are also posed.