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Erdős’s Unit Distance Problem and Rigidity

Authors János Pach , Orit E. Raz , József Solymosi

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
Keywords
  • Unit distance problem
  • Erdős
  • graph rigidity
  • incidences
  • polynomial partitioning technique

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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(n^{4/3}). This is far from Erdős’s lower bound, n^{1+O(1/log log n)}, which is conjectured to be optimal. We prove a structural result for point sets with nearly n^{4/3} 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.

Cite As Get BibTex

János Pach, Orit E. Raz, and József Solymosi. Erdős’s Unit Distance Problem and Rigidity. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 83:1-83:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.83

Author Details

János Pach
  • Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Orit E. Raz
  • Ben-Gurion University of the Negev, Beer Sheva, Israel
József Solymosi
  • University of British Columbia, Vancouver, Canada
  • Obuda University, Budapest, Hungary

Funding

  • Pach, János: This work was supported by ERC Advanced Grant no. 882971 "GeoScape" and by the National Research, Development and Innovation Office NKFIH Grant no. K-131529.
  • Raz, Orit E.: Part of this research was supported by the Charles Simonyi Endowment.
  • Solymosi, József: This research was supported by an NSERC Discovery grant and by the National Research Development and Innovation Office of Hungary, NKFIH, Grants no. KKP133819 and Excellence 151341.

Acknowledgements

The authors thank Zvi Shem-Tov, Joshua Zahl, and Frank de Zeeuw for their valuable discussions and assistance in shaping the paper. Part of this research was conducted while the second author was a Member at the Institute for Advanced Study in Princeton.

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