Almost-equidistant sets

Balko, Martin, Por, Attila, Scheucher, Manfred, Swanepoel, Konrad ORCID: 0000-0002-1668-887X and Valtr, Pavel (2020) Almost-equidistant sets. Graphs and Combinatorics, 36 (3). 729 - 754. ISSN 0911-0119

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Abstract

For a positive integer d, a set of points in d-dimensional Euclidean space is called almost-equidistant if for any three points from the set, some two are at unit distance. Let f(d) denote the largest size of an almost-equidistant set in d-space. It is known that f(2) = 7 , f(3) = 10 , and that the extremal almost-equidistant sets are unique. We give independent, computer-assisted proofs of these statements. It is also known that f(5) ≥ 16. We further show that 12 ≤ f(4) ≤ 13 , f(5) ≤ 20 , 18 ≤ f(6) ≤ 26 , 20 ≤ f(7) ≤ 34 , and f(9) ≥ f(8) ≥ 24. Up to dimension 7, our work is based on various computer searches, and in dimensions 6–9, we give constructions based on the known construction for d= 5. For every dimension d≥ 3 , we give an example of an almost-equidistant set of 2 d+ 4 points in the d-space and we prove the asymptotic upper bound f(d) ≤ O(d 3 / 2).

Item Type:	Article
Official URL:	https://link.springer.com/journal/373
Additional Information:	© 2020 The Authors
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	21 Feb 2020 11:45
Last Modified:	17 Oct 2024 16:04
URI:	http://eprints.lse.ac.uk/id/eprint/103533

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