Skip to main content
SpringerLink
Log in

Proof of Schur’s Conjecture in ℝD

  • Original Paper
  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

In this paper we prove Schur’s conjecture in ℝd, which states that any diameter graph G in the Euclidean space ℝd on n vertices may have at most n cliques of size d. We obtain an analogous statement for diameter graphs with unit edge length on a sphere Srd of radius \(r>1/\sqrt{2}\). The proof rests on the following statement, conjectured by F. Morić and J. Pach: given two unit regular simplices Δ1, Δ2 on d vertices in ℝd, either they share d-2 vertices, or there are vertices v1∈Δ1, v2∈Δ2 such that ‖v1-v2‖>1. The same holds for unit simplices on a d-dimensional sphere of radius greater than \(1/\sqrt{2}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. V. Akopyan and A. S. Tarasov: A Constructive Proof of Kirszbraun’s Theorem, Mathematical Notes 84 (2008), 725–728.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. V. Alekseevskij, E. B. Vinberg and A. S. Solodovnikov: Geometry of spaces of constant curvature Geometry II: Spaces of constant curvature, Encycl. Math. Sci. 29 (1993), 1–138. Translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 29 (1988), 5–146.

    MATH  Google Scholar 

  3. P. Brass, W. Moser and J. Pach: Research problems in discrete geometry, Springer, Berlin, 2005.

    MATH  Google Scholar 

  4. V. V. Bulankina, A. B. Kupavskii and A. A. Polyanskii: Note on Schur’s conjecture in R4, Doklady Mathematics 89 (2014), 88–91.

    Article  MathSciNet  MATH  Google Scholar 

  5. V. V. Bulankina, A. B. Kupavskii and A. A. Polyanskiy: On Schur’s conjecture in R4, Mathematical notes 97 (2015), 21–29.

    Article  MATH  Google Scholar 

  6. V. L. Dol’nikov: Some properties of graphs of diameters, Discrete Comput. Geom. 24 (2000), 293–299.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Erdős: On a set of distances of n points, Amer. Math. Monthly 53 (1946), 248–250.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Erdős: On sets of distances of n points in Euclidean space, Magyar Tud. Akad. Mat. Kut. Int. Közl. 5 (1960), 165–169.

    MathSciNet  MATH  Google Scholar 

  9. H. Hopf and E. Pannwitz: Aufgabe Nr. 167, Jahresbericht Deutsch. Math.-Verein. 43 (1934), 114.

    MATH  Google Scholar 

  10. B. Grünbaum: A proof of Vázsonyi’s conjecture, Bull. Res. Council Israel, Sect. A 6 (1956), 77–78.

    MathSciNet  MATH  Google Scholar 

  11. A. Heppes: Beweis einer Vermutung von A. Vázsonyi, Acta Math. Acad. Sci. Hungar. 7 (1957), 463–466.

    Article  MATH  Google Scholar 

  12. A. Kupavskii: Diameter graphs in R4, Discrete and Computational Geometry 51 (2014), 842–858.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Kupavskii and A. Polyanskii: On simplices in diameter graphs in R4, Mathematical Notes, accepted.

  14. L. Fejes Tóth: Lagerungen in der Ebene, auf der Kugel und im Raum, Springer-Verlag, 1972.

    Book  MATH  Google Scholar 

  15. H. Maehara: Dispersed points and geometric embedding of complete bipartite graphs, Discrete and Computational Geometry 6 (1991), 57–67.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Morić and J. Pach: Remarks on Schur’s conjecture, Comput. Geom. 48 (2015), 520–527.

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Morić and J. Pach: Large simplices determined by finite point sets, Beitr. Algebra Geom. 54 (2013), 45–57.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. M. Raigorodskii: Around Borsuk’s conjecture, J. of Math. Sci. 154 (2008), 604–623.

    Article  MathSciNet  MATH  Google Scholar 

  19. Z. Schur, M. A. Perles, H. Martini and Y. S. Kupitz: On the number of maximal regular simplices determined by n points in Rd, in: Discrete and Computational Geometry, The Goodman-Pollack Festschrift, B. Aronov, S. Basu, J. Pach, M. Sharir eds., Springer, 2003.

    Google Scholar 

  20. S. Straszewicz: Sur un problème géométrique de P. Erdős, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 39–40.

    MathSciNet  MATH  Google Scholar 

  21. K. J. Swanepoel: Unit distances and diameters in Euclidean spaces, Discrete and Computational Geometry 41 (2009), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ecole Polytechnique Fédérale de Lausanne, Moscow Institute of Physics and Technology, Moscow, Russia

    Andrey B. Kupavskii

  2. Moscow Institute of Physics and Technology, Technion - Israel Institute of Technology, Moscow, Russia

    Alexandr Polyanskii

Authors
  1. Andrey B. Kupavskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandr Polyanskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrey B. Kupavskii.

Additional information

Research supported in part by the Swiss National Science Foundation Grants 200021-137574 and 200020-14453 and by the grant N 15-01-03530 of the Russian Foundation for Basic Research.

Research suppoted in part by the Presedent Grant MK-3138.2014.1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kupavskii, A.B., Polyanskii, A. Proof of Schur’s Conjecture in ℝD. Combinatorica 37, 1181–1205 (2017). https://doi.org/10.1007/s00493-016-3340-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-016-3340-y

Mathematics Subject Classification (2000)

Search

Navigation