Erdős’s Unit Distance Problem and Rigidity

János Pach; Orit E. Raz; József Solymosi

doi:10.4230/LIPIcs.SoCG.2026.83

Erdős’s Unit Distance Problem and Rigidity

János Pach^*
, Orit E. Raz^*
, József Solymosi^*

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

According to a classical result of Spencer, Szemerédi, and Trotter (1984), the maximum number of times the unit distance can occur among n points in the plane is O(n⁴/³). This is far from Erdős’s lower bound, n¹⁺O(1/ log log n⁾, which is conjectured to be optimal. We prove a structural result for point sets with nearly n⁴/³ unit distances and use it to reduce the problem to a conjecture on rigid frameworks. This conjecture, if true, would yield the first improvement on the bound of Spencer et al. A weaker version of this conjecture has been established by Raz and Solymosi.

Original language	English
Title of host publication	42nd International Symposium on Computational Geometry, SoCG 2026
Editors	Hee-Kap Ahn, Michael Hoffmann, Amir Nayyeri
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959774185
DOIs	https://doi.org/10.4230/LIPIcs.SoCG.2026.83
State	Published - 27 May 2026
Externally published	Yes
Event	42nd International Symposium on Computational Geometry, SoCG 2026 - New Brunswick, United States Duration: 2 Jun 2026 → 5 Jun 2026

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	367
ISSN (Print)	1868-8969

Conference

Conference	42nd International Symposium on Computational Geometry, SoCG 2026
Country/Territory	United States
City	New Brunswick
Period	2/06/26 → 5/06/26

Bibliographical note

Publisher Copyright:
© János Pach, Orit E. Raz, and József Solymosi;

Keywords

Erdős
graph rigidity
incidences
polynomial partitioning technique
Unit distance problem

Access to Document

10.4230/LIPIcs.SoCG.2026.83

Cite this

Pach, J., Raz, O. E., & Solymosi, J. (2026). Erdős’s Unit Distance Problem and Rigidity. In H.-K. Ahn, M. Hoffmann, & A. Nayyeri (Eds.), 42nd International Symposium on Computational Geometry, SoCG 2026 Article 83 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 367). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2026.83

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abstract = "According to a classical result of Spencer, Szemer{\'e}di, and Trotter (1984), the maximum number of times the unit distance can occur among n points in the plane is O(n4/3). This is far from Erd{\H o}s{\textquoteright}s lower bound, n1+O(1/ log log n), which is conjectured to be optimal. We prove a structural result for point sets with nearly n4/3 unit distances and use it to reduce the problem to a conjecture on rigid frameworks. This conjecture, if true, would yield the first improvement on the bound of Spencer et al. A weaker version of this conjecture has been established by Raz and Solymosi.",

keywords = "Erd{\H o}s, graph rigidity, incidences, polynomial partitioning technique, Unit distance problem",

author = "J{\'a}nos Pach and Raz, \{Orit E.\} and J{\'o}zsef Solymosi",

note = "Publisher Copyright: {\textcopyright} J{\'a}nos Pach, Orit E. Raz, and J{\'o}zsef Solymosi;; 42nd International Symposium on Computational Geometry, SoCG 2026 ; Conference date: 02-06-2026 Through 05-06-2026",

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doi = "10.4230/LIPIcs.SoCG.2026.83",

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publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

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Pach, J, Raz, OE & Solymosi, J 2026, Erdős’s Unit Distance Problem and Rigidity. in H-K Ahn, M Hoffmann & A Nayyeri (eds), 42nd International Symposium on Computational Geometry, SoCG 2026., 83, Leibniz International Proceedings in Informatics, LIPIcs, vol. 367, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 42nd International Symposium on Computational Geometry, SoCG 2026, New Brunswick, United States, 2/06/26. https://doi.org/10.4230/LIPIcs.SoCG.2026.83

Erdős’s Unit Distance Problem and Rigidity. / Pach, János; Raz, Orit E.; Solymosi, József.
42nd International Symposium on Computational Geometry, SoCG 2026. ed. / Hee-Kap Ahn; Michael Hoffmann; Amir Nayyeri. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2026. 83 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 367).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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ER -

Erdős’s Unit Distance Problem and Rigidity

Abstract

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Bibliographical note

Keywords

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