Advertisement

Mathematical Notes

, Volume 97, Issue 1–2, pp 21–29

# On Schur’s conjecture in ℝ4

• V. V. Bulankina
• A. B. Kupavskii
• A. A. Polyanskii
Article

## Abstract

A diameter graph in ℝ d is a graph in which vertices are points of a finite subset of ℝ d and two vertices are joined by an edge if the distance between them is equal to the diameter of the vertex set. This paper is devoted to Schur’s conjecture, which asserts that any diameter graph on n vertices in ℝ d contains at most n complete subgraphs of size d. It is known that Schur’s conjecture is true in dimensions d ≤ 3. We prove this conjecture for d = 4 and give a simple proof for d = 3.

## Keywords

diameter graph Schur’s conjecture Borsuk’s conjecture

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
P. Brass, W. Moser, and J. Pach, Research Problems in Discrete Geometry (Springer-Verlag, Berlin, 2005).
2. 2.
A. M. Raigorodskii, “Borsuk’s problem and the chromatic numbers of some metric spaces,” Uspekhi Mat. Nauk 56(1), 107–146 (2001) [Russian Math. Surveys 56 (1), 103–139 (2001)].
3. 3.
A. M. Raigorodskii, “Around Borsuk’s hypothesis,” in Contemporary Mathematics: Fundamental Directions, Geometry and Mechanics, Vol. 23: Geometry and Mechanics (Ross. Univ. Druzhby Narodov, Moscow, 2007), pp. 147–164 [J. Math. Sci. 154 (4), 604–623 (2008)].Google Scholar
4. 4.
A. M. Raigorodskii, “Three lectures on the Borsuk partition problem,” in London Math. Soc. Lecture Note Ser., Vol. 347: Surveys in Contemporary Mathematics(Cambridge Univ. Press, Cambridge, 2008), pp. 202–247.Google Scholar
5. 5.
K. Borsuk, “Drei Sätzeüber die n-dimensionale euklidische Sphäre,” Fund. Math. 20, 177–190 (1933).Google Scholar
6. 6.
J. Kahn and G. Kalai, “A counterexample to Borsuk’s conjecture,” Bull. Amer. Math. Soc. (N. S.) 29(1), 60–62 (1993).
7. 7.
V. L. Dol’nikov, “Some properties of graphs of diameters,” Discrete Comput. Geom. 24(2–3), 293–299 (2000).
8. 8.
A. Heppes and P. Révész, “Zum Borsukschen Zerteilungsproblem,” Acta Math. Acad. Sci. Hungar 7(2), 159–162 (1956).
9. 9.
H. Hopf and E. Pannwitz, “Aufgabe 167,” Jahresbericht Deutsch. Math.-Verein. 43, 114 (1934).Google Scholar
10. 10.
B. Grünbaum, “A proof of Vászonyi’s conjecture,” Bull. Res. Council Israel. Sec. A 6, 77–78 (1956).Google Scholar
11. 11.
A. Heppes, “Beweis einer Vermutung von A. Vázsonyi,” Acta Math. Acad. Sci. Hungar. 7(3–4), 463–466 (1956).
12. 12.
S. Straszewicz, “Sur un problème géométrique de P. Erdős,” Bull. Acad. Polon. Sci. Cl. III 5, 39–40 (1957).
13. 13.
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 Algorithms Combin., Vol. 25: Discrete and Computational Geometry (Springer-Verlag, Berlin, 2003), pp. 767–787.
14. 14.
F. Morić and J. Pach, “Remarks on Schur’s conjecture,” in Lecture Notes in Comput. Sci., Vol. 8296: Computational Geometry and Graphs (Springer-Verlag, Berlin, 2013), pp. 120–131.Google Scholar
15. 15.
K. J. Swanepoel, “Unit distances and diameters in Euclidean spaces,” Discrete Comput. Geom. 41(1), 1–27 (2009).

## Copyright information

© Pleiades Publishing, Ltd. 2015

## Authors and Affiliations

• V. V. Bulankina
• 1
• A. B. Kupavskii
• 2
• A. A. Polyanskii
• 3
1. 1.Moscow State UniversityMoscowRussia
2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyi, Moscow RegionRussia
3. 3.Moscow Institute of Physics and TechnologyState UniversityDolgoprudnyiRussia