 
Summary: Dense uniform hypergraphs have high list chromatic number
Noga Alon
and Alexandr Kostochka
October 29, 2010
Mathematics Subject Classification: 05C15, 05C35.
Keywords: List coloring, hypergraph, codegree
Abstract
The first author showed that the list chromatic number of every graph with average degree
d is at least (0.5  o(1)) log2 d. We prove that for r 3, every runiform hypergraph in which
at least half of the (r  1)vertex subsets are contained in at least d edges has list chromatic
number at least ln d
100r3 . When r is fixed, this is sharp up to a constant factor.
1 Introduction
A list for a hypergraph G is an assignment L that provides a subset Lv of a set S (called the set
of colors) to every vertex v of G. A list L for a hypergraph G is an slist if L(v) = s for every
v V (G). Given a list L for G, an Lcoloring of G is a proper (that is, with no monochromatic
edges) coloring f of the vertices of G such that f(v) Lv for every v V (G). The list chromatic
number (or choice number) (G) of a hypergraph G is the minimum integer s such that for
every slist L for G, there exists an Lcoloring of G. These notions were introduced (for graphs)
independently by Vizing in [10] and by Erdos, Rubin and Taylor in [5]. It turned out that list
