Discrete & Computational Geometry

, Volume 5, Issue 4, pp 325–331 | Cite as

Countable decompositions ofR2 andR3

  • P. Erdős
  • P. Komjáth


If the continuum hypothesis holds,R2 is the union of countably many sets, none spanning a right triangle. Some partial results are obtained concerning the following conjecture of the first author:R2 is the union of countably many sets, none spanning an isosceles triangle. Finally, it is shown thatR3 can be colored with countably many colors with no monochromatic rational distance.


Discrete Comput Geom Isosceles Triangle Continuum Hypothesis Density Zero Countable Collection 
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  1. 1.
    J. Ceder, Finite subsets and countable decompositions of Euclidean spaces,Rev. Roumaine Math. Pures Appl. 14 (1969), 1247–1251.MathSciNetzbMATHGoogle Scholar
  2. 2.
    R. O. Davies, Covering the plane with denumerably many curves,J. London Math. Soc. 38 (1963), 433–438.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    R. O. Davies, Partitioning the plane into denumerably many sets without repeated distances,Math. Proc. Cambridge Philos. Soc. 72 (1972), 179–183.CrossRefzbMATHGoogle Scholar
  4. 4.
    P. Erdős and A. Hajnal, On chromatic number of graphs and set-systems,Acta Math. Hungar. 17 (1966), 61–99.CrossRefGoogle Scholar
  5. 5.
    P. Erdős, Geometrical and set-theoretical properties of subsets of the Hilbert space,Mat. Lapok 19 (1968), 255–258 (in Hungarian).MathSciNetGoogle Scholar
  6. 6.
    P. Erdős, Problems and results in chromatic graph theory, inProof Techniques in Graph Theory (F. Harary, ed.), 27–35, Academic Press, New York, 1969.Google Scholar
  7. 7.
    P. Erdős, Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space,Real Anal. Exchange 4 (1978–1979), 113–138.MathSciNetGoogle Scholar
  8. 8.
    P. Erdős, Problems and results in combinatorial analysis,Proceedings of the Eighth South-East Conference on Combinatorics, Graph Theory, and Computer Science, 3–12, 1985.Google Scholar
  9. 9.
    K. Kunen,Set Theory, North-Holland, Amsterdam, 1980.zbMATHGoogle Scholar
  10. 10.
    K. Kunen, On decomposingR n, to appear.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • P. Erdős
    • 1
  • P. Komjáth
    • 2
    • 3
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary
  2. 2.Department of MathematicsLehigh UniversityBethlehemUSA
  3. 3.Department of Computer ScienceR. Eötvös UniversityBudapestHungary

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