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Partitioning the plane into denumerably many sets without repeated distances

Published online by Cambridge University Press:  24 October 2008

Roy. O. Davies
Roy. O. Davies
Affiliation:
University of Leicester
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Ceder(1) proved (assuming the axiom of choice, as we do throughout this paper) that the Euclidean plane can be partitioned into ℵ0 sets none of which contains an equilateral triangle; indeed he proved that given any denumerable set of triangles, the plane can be partitioned into ℵ0 sets, none containing a triangle similar to one of the given triangles. Erdős and Kakutani(2) proved that the continuum hypothesis implies that the real line can be partitioned into ℵ0, rationally independent sete, and that the existence of such a partition implies the continuum hypothesis. Erdős asked (private communication) whether there is a partition of the plane into ℵ0 sets not containing an isosceles triangle, or more generally in which any four points determine six different distances. Assuming the continuum hypothesis, it will be shown here (Theorem 1) that a, partition of the latter kind does exist. (I communicated this result to Erdős and others some years ago, but subsequently noticed that the argument was incomplete.) Conversely (Theorem 2) the existence of such a partitition, even for the line, implies the continuum hypothesis. This strengthening of the converse half of the Erdős-Kakutani theorem is proved by what is essentially their method (actually in a rather simplified form).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 2 , September 1972 , pp. 179 - 183
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Ceder, J.Finite subsets and countable decompositions of Euclidean spaces. Rev. Roumaine Math. Pures Appl. 14 (1969), 1247–1251.Google Scholar
(2)Erdős, P. and Kakutani, S.On non-denumerable graphs. Bull. Amer. Math. Soc. 49 (1943), 457–461.CrossRefGoogle Scholar
(3)Davies, Roy O.Equivalence to the continuum hypothesis of a certain proposition of elementary plane geometry. Z. Math. Logik Grundlagen Math. 8 (1962), 109–111.CrossRefGoogle Scholar
(4)Davies, Roy O.The power of the continuum and some propositions of plane geometry. Fund. Math. 52 (1963), 277–281.CrossRefGoogle Scholar
(5)Davies, Roy O.On a problem of Erdős concerning decompositions of the plane. Proc. Cambridge Philos. Soc. 59 (1963), 33–36.CrossRefGoogle Scholar
(6)Davies, Roy O.On a denumerable partition problem of Erdós. Proc. Cambridge Philos. Soc. 59 (1963), 501–502.CrossRefGoogle Scholar
(7)Davies, Roy O.Covering the plane with denumerably many curves. J. London Math. Soc. 38 (1963), 433–348.CrossRefGoogle Scholar
(8)Erdős, P. and Hajnal, A.On a property of families of sets. Acta Math. Acad. Sci. Hungar. 12 (1961), 87–123.CrossRefGoogle Scholar
(9)Miller, E. W.On a property of families of sets. C. R. Soc. Sci. Varsovie 30 (1937), 31–38.Google Scholar
(10)Erdős, P. and Hajnal, A.On chromatic numbers of graphs and set systems. Acta Math. Acad. Sci. Hungar. 17 (1966), 61–99.CrossRefGoogle Scholar