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Countable partitions of Euclidean space

Published online by Cambridge University Press:  24 October 2008

James H. Schmerl
James H. Schmerl
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269, U.S.A
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Erdös has asked whether the plane ℝ2, or more generally n-dimensional Euclidean space ℝn, can be partitioned into countably many sets none of which contains the vertices of an isosceles triangle. Assuming the Continuum Hypothesis (CH), Davies[2] (for n = 2) and Kunen[10] (for arbitrary n) proved that such partitions exist. Assuming Martin's Axiom, Erdös and Komjáth proved in [5] that such partitions exist for n = 2. We will prove here, without additional set-theoretic hypotheses, that there are such partitions in all dimensions.

Let ‖x‖ denote the usual Euclidean norm of a point x ∈ ℝn, so that ‖x − y‖ is the distance between x and y.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 1 , July 1996 , pp. 07 - 12
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

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