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Mathematical Notes

, Volume 97, Issue 1–2, pp 21–29 | Cite as

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 

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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

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