Reflection on the coloring and chromatic numbers

TL;DR

This paper explores the relationship between the coloring number and chromatic number in graphs, demonstrating their different behaviors regarding reflection and incompactness under various set-theoretic principles.

Contribution

It establishes the consistency of non-reflection of the chromatic number with reflection of the coloring number and shows incompactness for the chromatic number is compatible with several compactness principles.

Findings

Reflection of the coloring number can occur without reflection of the chromatic number.

Incompactness for the chromatic number can coexist with principles like Rado's conjecture and Martin's Maximum.

Coloring number does not admit arbitrarily large incompactness gaps.

Abstract

We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) is compatible with each of the following compactness principles: Rado's conjecture, Fodor-type reflection, $\Delta$-reflection, Stationary-sets reflection, Martin's Maximum, and a generalized Chang's conjecture. This is accomplished by showing that, under $\mathrm{GCH}$-type assumptions, instances of incompactness for the chromatic number can be derived from square-like principles that are compatible with large amounts of compactness. In addition, we prove that, in contrast to the chromatic number, the coloring number does not admit arbitrarily large incompactness gaps.