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Combinatorica

, Volume 37, Issue 6, pp 1181–1205 | Cite as

Proof of Schur’s Conjecture in ℝ D

  • Andrey B. Kupavskii
  • Alexandr Polyanskii
Original Paper
  • 62 Downloads

Abstract

In this paper we prove Schur’s conjecture in ℝ d , which states that any diameter graph G in the Euclidean space ℝ d 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 S r d 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 ℝ d , 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 \(1/\sqrt{2}\).

Mathematics Subject Classification (2000)

52C10 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Ecole Polytechnique Fédérale de LausanneMoscow Institute of Physics and TechnologyMoscowRussia
  2. 2.Moscow Institute of Physics and TechnologyTechnion - Israel Institute of TechnologyMoscowRussia

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