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Semialgebraic graphs having countable list-chromatic numbers

Author: James H. Schmerl
Journal: Proc. Amer. Math. Soc. 144 (2016), 1429-1438
MSC (2010): Primary 05C15, 05C63
Published electronically: July 30, 2015
MathSciNet review: 3451221
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Abstract: For $ n \geq 1$ and a countable, nonempty set $ D$ of positive reals, the $ D$-distance graph $ {\bf X}_n(D)$ is the graph on Euclidean $ n$-space $ \mathbb{R}^n$ in which two points form an edge exactly when the distance between them is in $ D$. Each of these graphs is $ \sigma $-algebraic. Komjáth characterized those $ {\bf X}_n(D)$ having a countable list-chromatic number, easily implying a different, but essentially equivalent, noncontainment characterization. It is proved here that this noncontainment characterization extends to all $ \sigma $-algebraic graphs. We obtain, in addition, similar noncontainment characterizations for those $ \sigma $-semialgebraic graphs and those semialgebraic graphs having countable list-chromatic numbers.

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James H. Schmerl
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Received by editor(s): May 27, 2014
Received by editor(s) in revised form: April 13, 2015
Published electronically: July 30, 2015
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2015 American Mathematical Society

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