Proof of Schur’s Conjecture in ℝD


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 \(1/\sqrt{2}\).

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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.

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Kupavskii, A.B., Polyanskii, A. Proof of Schur’s Conjecture in ℝD. Combinatorica 37, 1181–1205 (2017).

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Mathematics Subject Classification (2000)

  • 52C10