Proof of Schur’s Conjecture in ℝD

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 v1Δ1, v2Δ2 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, access via your institution.

References

  1. [1]

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

    MathSciNet  Article  MATH  Google Scholar 

  2. [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. [3]

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

    Google Scholar 

  4. [4]

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

    MathSciNet  Article  MATH  Google Scholar 

  5. [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. [6]

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

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

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

    MathSciNet  Article  MATH  Google Scholar 

  8. [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. [9]

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

    MATH  Google Scholar 

  10. [10]

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

    MathSciNet  MATH  Google Scholar 

  11. [11]

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

    Article  MATH  Google Scholar 

  12. [12]

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

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

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

  14. [14]

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

    Google Scholar 

  15. [15]

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

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

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

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

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

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

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

    MathSciNet  Article  MATH  Google Scholar 

  19. [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. [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. [21]

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

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

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

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

Mathematics Subject Classification (2000)

  • 52C10