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Countable partitions of Euclidean space

  • James H. Schmerl (a1)

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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 ‖xy‖ is the distance between x and y.

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[1]Ceder, J.Finite subsets and countable decompositions of Euclidean spaces. Rev. Roumaine Math. Pures Appl. 14 (1969), 12471251.
[2]Davies, R. O.Partitioning the plane into denumerably many sets without repeated distances. Proc. Cambridge Philos. Soc. 72 (1972), 179183.
[3]Erdös, P. Problems and results in chromatic number theory; in Proof Techniques in graph theory (ed. Harary, F.) (Academic Press, 1969), 4755.
[4]Erdös, P.Problems and results in discrete mathematics. Discrete Math. 136 (1994), 5373.
[5]Erdös, P. and Komjáth, P.Countable decompositions of ℝ2 and ℝ3. Discrete and Comp. Geom. 5 (1990), 325331.
[6]Gunning, R. C. and Rossi, H.Analytic functions of several complex variables (Prentice-Hall, 1965).
[7]Komjáth, P.Tetrahedron free decomposition of ℝ3. Bull. London Math. Soc. 23 (1991), 116120.
[8]Komjáth, P.The master coloring. Comptes Rendus Mathématique de l'academie des Sciences, la Société Royale du Canada 14 (1992), 181182.
[9]Komjáth, P.A decomposition theorem for ℝn. Proc. Amer. Math. Soc. 120 (1994), 921927.
[10]Kunen, K.Partitioning Euclidean space. Math. Proc. Camb. Phil. Soc. 102 (1987), 379383.
[11]Schmerl, J. H.Partitioning Euclidean space. Discrete and Comp. Geom. 10 (1993), 101106.
[12]Schmerl, J. H.Triangle-free partitions of Euclidean space. Bull. London Math. Soc., 26 (1994), 483486.

Countable partitions of Euclidean space

  • James H. Schmerl (a1)

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