Axiom of choice and chromatic number: Examples on the plane

Alexander Soifer*, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

Abstract

In our previous paper (J. Combin. Theory Ser. A 103 (2) (2003) 387) we formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed an example of a distance graph on the real line R whose chromatic number depends upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct a distance graph G2 on the plane R2, thus coming much closer to the setting of the chromatic number of the plane problem. The chromatic number of G2 is 4 in the Zermelo-Fraenkel-Choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. 92 Ser. 2 (1970) 1).

Original languageEnglish
Pages (from-to)359-364
Number of pages6
JournalJournal of Combinatorial Theory. Series A
Volume105
Issue number2
DOIs
StatePublished - Feb 2004

Keywords

  • Axiom of choice
  • Axiomatic set theory
  • Chromatic number
  • Erdös problems and related topics to discrete geometry
  • Euclidean Ramsey theory
  • Graph theory

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