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Mat. Zametki, 2017, Volume 101, Issue 2, Pages 232–246 (Mi mz10611)  

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

On Simplices in Diameter Graphs in $\mathbb R^4$

A. B. Kupavskiiab, A. A. Poljanskijacd

a Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
b École Polytechnique Fédérale de Lausanne
c Technion – Israel Institute of Technology
d Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow

Abstract: A graph $G$ is a diameter graph in $\mathbb R^d$ if its vertex set is a finite subset in $\mathbb R^d$ of diameter $1$ and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph $G$ in $\mathbb R^4$ contains the complete subgraph $K$ on five vertices, then any triangle in $G$ shares a vertex with $K$. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in $\mathbb R^4$, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than $1$.

Keywords: diameter graphs, Schur's conjecture.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-03530
15-01-99563 A
15-31-20403 (мол_a_вед)
Swiss National Science Foundation 200021-137574
200020-14453
The work of the first-named author was supported by the Swiss National Foundation for Scientific Research under grants nos. 200021-137574 and 200020-14453 and by the Russian Foundation for Basic Research under grant no. 15-01-03530. The work of the second-named author was supported in part by the Russian Foundation for Basic Research under grants nos. 15-01-03530, 15-01-99563 A, and 15-31-20403 (mol_a_ved).


DOI: https://doi.org/10.4213/mzm10611

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English version:
Mathematical Notes, 2017, 101:2, 265–276

Bibliographic databases:

UDC: 519.112.7
Received: 04.03.2014
Revised: 13.03.2016

Citation: A. B. Kupavskii, A. A. Poljanskij, “On Simplices in Diameter Graphs in $\mathbb R^4$”, Mat. Zametki, 101:2 (2017), 232–246; Math. Notes, 101:2 (2017), 265–276

Citation in format AMSBIB
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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. Kupavskii A.B., Polyanskii A., “Proof of Schur'S Conjecture in R-D”, Combinatorica, 37:6 (2017), 1181–1205  crossref  mathscinet  zmath  isi  scopus
    2. Polyanskii A., “On Almost-Equidistant Sets II”, Electron. J. Comb., 26:2 (2019), P2.14  isi
  • Математические заметки Mathematical Notes
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