# Reflection on the coloring and chromatic numbers

Joint work with Chris Lambie-Hanson.

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 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.

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### 2 Responses to Reflection on the coloring and chromatic numbers

1. saf says:

Update Feb/2017: Added a new section, entitled “Realizing all closed intervals”.

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2. saf says:

Submitted to Combinatorica, December 2016.
Accepted, June 2017.

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