List Coloring Hypergraphs

Let H be a hypergraph and let Lv : v ∈ V (H) be sets; we refer to these sets as lists and their elements as colors. A list coloring of H is an assignment of a color from Lv to each v ∈ V (H) in such a way that every edge of H contains a pair of vertices of different colors. The hypergraph H is k-list-colorable if it has a list coloring from any collection of lists of size k. The list chromatic number of H is the minimum k such that H is k-list-colorable. In this paper we prove that every d-regular three-uniform linear hypergraph has list chromatic number at least ( log d 5 log log d) 1/2 provided d is large enough. On the other hand there exist d-regular three-uniform linear hypergraphs with list chromatic number at most log3 d + 3. This leaves the question open as to the existence of such hypergraphs with list chromatic number o(log d) as d → ∞.