Reflection on the coloring and chromatic numbers

DOI10.1007/S00493-017-3741-6zbMATH Open1438.03017arXiv1708.06929OpenAlexW2962720150MaRDI QIDQ2416518

Chris Lambie-Hanson, Assaf Rinot

Publication date: 23 May 2019

Published in: Combinatorica (Search for Journal in Brave)

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 $mathrmGCH$-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.

Full work available at URL: https://arxiv.org/abs/1708.06929

zbMATH Keywords

reflectionchromatic numbercoloring numberincompactness

Mathematics Subject Classification ID

Consistency and independence results (03E35) Coloring of graphs and hypergraphs (05C15) Other combinatorial set theory (03E05) Infinite graphs (05C63)

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