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Bounding the Size of an Almost-Equidistant Set in Euclidean Space

Part of: Discrete geometry Extremal combinatorics

Published online by Cambridge University Press:  13 June 2018

ANDREY KUPAVSKII ,
NABIL H. MUSTAFA  and
KONRAD J. SWANEPOEL
ANDREY KUPAVSKII
Affiliation:
Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141700, Russia and University of Birmingham, B15 2TT, Birmingham, UK (e-mail: kupavskii@yandex.ru)
NABIL H. MUSTAFA
Affiliation:
Université Paris-Est, Laboratoire d'Informatique Gaspard-Monge, ESIEE Paris, France (e-mail: mustafan@esiee.fr)
KONRAD J. SWANEPOEL
Affiliation:
Department of Mathematics, London School of Economics and Political Science, London WC2A 2AE, UK (e-mail: k.swanepoel@lse.ac.uk)
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Abstract

A set of points in d-dimensional Euclidean space is almost equidistant if, among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in ℝd has cardinality O(d4/3).

MSC classification

Primary: 52C10: Erdős problems and related topics of discrete geometry
Secondary: 05D10: Ramsey theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 2 , March 2019 , pp. 280 - 286
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

*

The work of Andrey Kupavskii was supported by Russian Foundation for Basic Research grant 18-01-00355.

†

The work of Nabil H. Mustafa in this paper has been supported by the grant ANR SAGA (JCJC-14-CE25-0016-01).

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