The list-chromatic number and the coloring number of uncountable graphs

Abstract

We study the list-chromatic number and the coloring number of graphs, especially uncountable graphs. We show that the coloring number of a graph coincides with its list-chromatic number provided that the diamond principle holds. Under the GCH assumption, we prove the singular compactness theorem for the list-chromatic number. We also investigate reflection principles for the list-chromatic number and the coloring number of graphs.