Erdős’s Unit Distance Problem

  • Endre SzemerédiEmail author


We survey some problems and results around one of Paul Erdős’s favorite questions, first published 70 years ago: What is the maximum number of times that the unit distance can occur among n points in the plane? This simple and beautiful question has generated a lot of important research in discrete geometry, in extremal combinatorics, in additive number theory, in Fourier analysis, in algebra, and in other fields, but we still do not seem to be close to a satisfactory answer.


Unit Circle Unit Distance Geometric Graph Combinatorial Geometry Diameter Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I would like to thank János Pach and József Solymosi for many interesting discussions on the topic of this survey, and for their help in preparing the manuscript. Work was supported by ERC-AdG. 321104, and OTKA NK 104183 grants.


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Renyi Alfred Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

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