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

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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.

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Acknowledgments

The author would like to thank the referee for many valuable comments and useful suggestions. This research was supported by JSPS KAKENHI Grant Nos. 18K03403 and 18K03404.

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Authors and Affiliations

  1. Faculty of Science and Engineering, Waseda University, Okubo 3-4-1, Shinjyuku, Tokyo, 169-8555, Japan

    Toshimichi Usuba

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  1. Toshimichi Usuba
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Correspondence to Toshimichi Usuba.

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Usuba, T. The list-chromatic number and the coloring number of uncountable graphs. Isr. J. Math. 259, 129–167 (2024). https://doi.org/10.1007/s11856-023-2483-y

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  • DOI: https://doi.org/10.1007/s11856-023-2483-y

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