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

TL;DR

This paper investigates the relationship between list-chromatic number and coloring number in uncountable graphs, establishing conditions under which they coincide and exploring their reflection principles.

Contribution

It proves the equivalence of list-chromatic and coloring numbers under the diamond principle and establishes a singular compactness theorem under GCH.

Findings

Coloring number equals list-chromatic number under the diamond principle.

Singular compactness theorem for list-chromatic number under GCH.

Analysis of reflection principles for these graph invariants.

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.