Axiom of choice and chromatic number of \(R^n\)
type: Item
instance of: scholarly article
title: Axiom of choice and chromatic number of \(R^n\)
zbMATH Open document ID: 1063.03035
author: Alexander Soifer
published in: Journal of Combinatorial Theory. Series A
publication date: 2005-05-04T00:00:00Z
review text: The chromatic number of the following graph is considered. The vertices are the points of the Euclidean space \({\mathbb R}^n\), the points \({\mathbf x}\) and \({\mathbf y}\) are joined if \({\mathbf x}-{\mathbf y}-\sqrt{2}{\mathbf e}_i\in {\mathbb Q}^n\) holds for some \(1\leq i\leq n\) (here \({\mathbf e}_1,\dots,{\mathbf e}_n\) is the standard basis). Assuming AC, the axiom of choice, the chromatic number is 2, it is, however, uncountable if \(\text{ AC}_{\aleph_0}\) holds and every set of reals is measurable.
reviewed by: Péter Komjáth
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 05C15
zbMATH DE Number: 2164803
zbMATH Keywords: chromatic number
zbMATH Keywords: axiom of choice
zbMATH Keywords: combinatorial geometry
zbMATH Keywords: discrete geometry
zbMATH Keywords: Euclidean space
MaRDI profile type: Publication
full work available at URL: j.jcta.2004.10.004
OpenAlex ID: W1984210806
Wikidata QID: Q114013973
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DOI: 10.1016/J.JCTA.2004.10.004
description: scientific article; zbMATH DE number 2164803