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DOI: 10.4230/LIPIcs.SoCG.2026.83

Erdős’s Unit Distance Problem and Rigidity

Authors: János Pach, Orit E. Raz, and József Solymosi

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

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.

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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)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2026.83,
  author =	{Pach, J\'{a}nos and Raz, Orit E. and Solymosi, J\'{o}zsef},
  title =	{{Erd\H{o}s’s Unit Distance Problem and Rigidity}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{83:1--83:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.83},
  URN =		{urn:nbn:de:0030-drops-258906},
  doi =		{10.4230/LIPIcs.SoCG.2026.83},
  annote =	{Keywords: Unit distance problem, Erd\H{o}s, graph rigidity, incidences, polynomial partitioning technique}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2020.65

Dense Graphs Have Rigid Parts

Authors: Orit E. Raz and József Solymosi

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract

While the problem of determining whether an embedding of a graph G in ℝ² is infinitesimally rigid is well understood, specifying whether a given embedding of G is rigid or not is still a hard task that usually requires ad hoc arguments. In this paper, we show that every embedding (not necessarily generic) of a dense enough graph (concretely, a graph with at least C₀n^{3/2}(log n)^β edges, for some absolute constants C₀>0 and β), which satisfies some very mild general position requirements (no three vertices of G are embedded to a common line), must have a subframework of size at least three which is rigid. For the proof we use a connection, established in Raz [Discrete Comput. Geom., 2017], between the notion of graph rigidity and configurations of lines in ℝ³. This connection allows us to use properties of line configurations established in Guth and Katz [Annals Math., 2015]. In fact, our proof requires an extended version of Guth and Katz result; the extension we need is proved by János Kollár in an Appendix to our paper. We do not know whether our assumption on the number of edges being Ω(n^{3/2}log n) is tight, and we provide a construction that shows that requiring Ω(n log n) edges is necessary.

Cite as

Orit E. Raz and József Solymosi. Dense Graphs Have Rigid Parts. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 65:1-65:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{raz_et_al:LIPIcs.SoCG.2020.65,
  author =	{Raz, Orit E. and Solymosi, J\'{o}zsef},
  title =	{{Dense Graphs Have Rigid Parts}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{65:1--65:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.65},
  URN =		{urn:nbn:de:0030-drops-122236},
  doi =		{10.4230/LIPIcs.SoCG.2020.65},
  annote =	{Keywords: Graph rigidity, line configurations in 3D}
}

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Erdős’s Unit Distance Problem and Rigidity

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Dense Graphs Have Rigid Parts

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