More distinct distances under local conditions



We establish the following result related to Erdős’s problem on distinct distances. Let V be an n-element planar point set such that any p members of V determine at least \(\left( {\begin{array}{*{20}{c}} p \\ 2 \end{array}} \right) - p + 6\) distinct distances. Then V determines at least \(n^{\tfrac{8} {7} - o(1)}\) distinct distances, as n tends to infinity.

Mathematics Subject Classification (2000)

52C10 05D10 


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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Stanford UniversityStanfordUSA
  2. 2.EPFLLausanneSwitzerland
  3. 3.Rényi InstituteBudapestHungary
  4. 4.University of Illinois at ChicagoChicagoUSA

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