Advertisement

Mathematical Notes

, Volume 101, Issue 1–2, pp 265–276 | Cite as

On simplices in diameter graphs in ℝ4

  • A. B. KupavskiiEmail author
  • A. A. Polyanskii
Volume 101, Number 2, February, 2017
  • 31 Downloads

Abstract

Agraph G is a diameter graph in ℝ d if its vertex set is a finite subset in ℝ d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ℝ4 contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ℝ4, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

Keywords

diameter graphs Schur’s conjecture 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. Borsuk, “Drei Sätze über die n-dimensionale euklidische Sphäre,” Fund. Math. 20, 177–190 (1933).zbMATHGoogle Scholar
  2. 2.
    P. Brass, W. Moser, and J. Pach, Research Problems in Discrete Geometry (Springer, New York, 2005).zbMATHGoogle Scholar
  3. 3.
    A. M. Raigorodskii, “Borsuk’s problem and the chromatic numbers of some metric spaces,” Uspekhi Mat. Nauk 56 (1 (337)), 107–146 (2001) [RussianMath. Surveys 56 (1), 103–139 (2001)].MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. M. Raigorodskii, “Around Borsuk’s hypothesis,” in Contemporary Mathematics. Fundamental Directions, Vol. 23: Geometry and Mechanics (Ros. Univ. Druzhby Narodov, Moscow, 2007), pp. 147–164 [J. Math. Sci. (New York) 154 (4), 604–623 (2008)].Google Scholar
  5. 5.
    A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters,” in Discrete Geometry and Algebraic Combinatorics, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2014), Vol. 625, pp. 93–109.Google Scholar
  6. 6.
    A. M. Raigorodskii, “Coloring distance graphs and graphs of diameters,” in Thirty Essays on Geometric Graph Theory (Springer, New York, 2013), pp. 429–460.CrossRefGoogle Scholar
  7. 7.
    J. Kahn and G. Kalai, “A counterexample to Borsuk’s conjecture,” Bull. Amer. Math. Soc. (N. S.) 29 (1), 60–62 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    T. Jenrich and A. E. Brouwer, “A 64-dimensional counterexample to Borsuk’s conjecture,” Electron. J. Combin. 21 (Paper 4. 29) (2014).MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. V. Bondarenko, “On Borsuk’s conjecture for two-distance sets,” Discrete Comput. Geom. 51 (3), 509–515 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    H. Hopf and E. Pannwitz, “Aufgabe Nr. 167,” Jahr. Deutsch. Math.-Verein 43, 114 (1934).zbMATHGoogle Scholar
  11. 11.
    B. Grünbaum, “A proof of Vászonyi’s conjecture,” Bull. Res. Council Israel. Sec. A 6, 77–78 (1956).Google Scholar
  12. 12.
    A. Heppes, “Beweis einer Vermutung von A. Vázsonyi,” ActaMath. Acad. Sci. Hungar. 7, 463–466 (1957).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S. Straszewicz, “Sur un problème géométrique de P. Erdős,” Bull. Acad. Polon. Sci. Cl. III 5, 39–40 (1957).MathSciNetzbMATHGoogle Scholar
  14. 14.
    P. Erdős, “On sets of distances of n points in Euclidean space,” Magyar Tud. Akad. Mat. Kutató Int. Közl. 5, 165–169 (1960).MathSciNetzbMATHGoogle Scholar
  15. 15.
    K. J. Swanepoel, “Unit distances and diameters in Euclidean spaces,” Discrete Comput. Geom. 41 (1), 1–27 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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, Algorithms Combin. (Springer-Verlag, Berlin, 2003), Vol. 25, pp. 767–787.CrossRefGoogle Scholar
  17. 17.
    F. Morić and J. Pach, “Remarks on Schur’s conjecture,” in Computational Geometry and Graphs, Lecture Notes in Comput. Sci. (Springer-Verlag, Berlin, 2013), Vol. 8296, pp. 120–131.Google Scholar
  18. 18.
    A. Kupavskii, “Diameter graphs in R4,” Discrete Comput. Geom. 51 (4), 842–858 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    V. V. Bulankina, A. B. Kupavskii, and A. A. Polyanskii, “A remark on Schur’s conjecture in R4,” Dokl. Ross. Akad. Nauk 454 (5), 507–511 (2014) [Dokl. Math. 89 (1), 88–91 (2014)].zbMATHGoogle Scholar
  20. 20.
    A. Kupavskii and A. Polyanskii, “Proof of Schur’s conjecture in Rd,” Combinatorica (in press); https://arxiv.org/abs/1402.3694.Google Scholar
  21. 21.
    I. Bárány, “The densest (n + 2)-set in Rn,” in Intuitive Geometry, Coll. Math. Soc. János Bolyai (North-Holland, Amsterdam, 1994), Vol. 63, pp. 7–10.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Moscow Institute of Physics and Technology (State University)DolgoprudnyiRussia
  2. 2.École Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.Kharkevich Institute for Information Tranmission ProblemsRussian Academy of SciencesMoscowRussia
  4. 4.Technion – Israel Institute of TechnologyHaifaIsrael

Personalised recommendations