2014/02/15 by Andrey Kupavskii, Kupavskii, Andrey B., Alexandr Polyanskii +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1402.3694
openalex publication_date 2014/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove Schur's conjecture in \mathbb Rd, which states that any diameter graph G in the Euclidean space \mathbb 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 Sdr of radius r>1/√ 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 \mathbb Rd, either they share d-2 vertices, or there are vertices v1∈ Δ1,v2∈ Δ2 such that ‖v1-v2‖>1. The same holds for unit simplices on a d-dimensional sphere of radius greater than 1/√ 2.