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Mat. Zametki, 2015, Volume 97, Issue 1, Pages 23–34 (Mi mz10375)  

This article is cited in 2 scientific papers (total in 2 papers)

On Schur's Conjecture in $\mathbb R^4$

V. V. Bulankinaa, A. B. Kupavskiib, A. A. Polyanskiib

a M. V. Lomonosov Moscow State University
b Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moskovskaya obl.

Abstract: A diameter graph in $\mathbb R^d$ is a graph in which vertices are points of a finite subset of $\mathbb R^d$ 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 $\mathbb R^d$ contains at most $n$ complete subgraphs of size $d$. It is known that Schur's conjecture is true in dimensions $d\le 3$. We prove this conjecture for $d=4$ and give a simple proof for $d=3$.

Keywords: diameter graph, Schur's conjecture, Borsuk's conjecture.

DOI: https://doi.org/10.4213/mzm10375

Full text: PDF file (510 kB)
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English version:
Mathematical Notes, 2015, 97:1, 21–29

Bibliographic databases:

Document Type: Article
UDC: 514.12+519.157
Received: 10.07.2013
Revised: 05.05.2014

Citation: V. V. Bulankina, A. B. Kupavskii, A. A. Polyanskii, “On Schur's Conjecture in $\mathbb R^4$”, Mat. Zametki, 97:1 (2015), 23–34; Math. Notes, 97:1 (2015), 21–29

Citation in format AMSBIB
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\paper On Schur's Conjecture in $\mathbb R^4$
\jour Mat. Zametki
\yr 2015
\vol 97
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\pages 23--34
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\jour Math. Notes
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. B. Kupavskii, A. Polyanskii, “Proof of Schur's conjecture in $\Bbb R^D$”, Combinatorica, 37:6 (2017), 1181–1205  crossref  mathscinet  zmath  isi  scopus
    2. L. E. Shabanov, A. M. Raigorodskii, “Turan-type bounds for distance graphs”, Dokl. Math., 96:1 (2017), 351–353  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
  • Математические заметки Mathematical Notes
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