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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Citation information:

C. Lambie-Hanson and A. Rinot, Reflection on the coloring and chromatic numbers, Combinatorica, 39(1): 165-214, 2019.