Discrete & Computational Geometry

, Volume 43, Issue 2, pp 393–401 | Cite as

On a Question of Erdős and Ulam

  • Jozsef SolymosiEmail author
  • Frank de Zeeuw


Ulam asked in 1945 if there is an everywhere dense rational set, i.e., 1 a point set in the plane with all its pairwise distances rational. Erdős conjectured that if a set S has a dense rational subset, then S should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erdős’ conjecture for algebraic curves by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.


Rational distances Erdős problems in discrete geometry Rational points 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsUBCVancouverCanada

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