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Erdős’s Unit Distance Problem

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Open Problems in Mathematics

Abstract

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.

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Acknowledgements

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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  1. Renyi Alfred Mathematical Institute of the Hungarian Academy of Sciences, Reáltanoda utca 13-15, H-1053, Budapest, Hungary

    Endre Szemerédi

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  1. Endre Szemerédi
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Correspondence to Endre Szemerédi .

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Editors and Affiliations

  1. Department of Mathematics, Senior Research Mathematician, Princeton University, Princeton, New Jersey, USA

    John Forbes Nash, Jr.

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, USA

    Michael Th. Rassias

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Szemerédi, E. (2016). Erdős’s Unit Distance Problem. In: Nash, Jr., J., Rassias, M. (eds) Open Problems in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-32162-2_15

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