The maximum number of unit distances among \(n\) points in dimension four
type: Item
instance of: scholarly article
title: The maximum number of unit distances among \(n\) points in dimension four
zbMATH Open document ID: 0962.52002
author: Paul van Wamelen
published in: Beiträge zur Algebra und Geometrie
publication date: 2000-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/229042
full work available at URL: 15.html
review text: This paper completes the determination of the exact maximum number of unit distances among \(n\) points in euclidean fourdimensional space. This problem, which is a very difficult open problem in dimensions two and three, becomes much simpler in higher dimensions and was asymptotically solved long ago by \textit{P. Erdős} [Can. J. Math. 19, 968-971 (1967; Zbl 0161.20603)] and \textit{P. Erdős} and \textit{J. Pach} [Combinatorica 10, No. 3, 261-269 (1990; Zbl 0722.52009)]. In [\textit{P. Braß}, Bolyai Soc. Math. Stud. 6, 277-290 (1997; Zbl 0885.52015)] the reviewer showed that it is even possible to solve the problem exactly, for dimension four, where he obtained the structure of the extremal sets and their edge-number, depending on the existence of a solution of a trigonometric diophantine equation (see the last cited paper for a correction of one case). This diophantine equation is solved in this paper by van Wamelen, so that the complete answer is: the maximum number of unit distances among \(n\) points in dimension four is \(\lfloor{1\over 4}n^2\rfloor +n\) if \(n\) is divisible by 8 or 10, and one less for all other \(n\).
Mathematics Subject Classification ID: 52C10
zbMATH DE Number: 1460291
zbMATH Keywords: number of unit distances
zbMATH Keywords: Lenz construction
zbMATH Keywords: Erdős problem
zbMATH Keywords: repeated distances
reviewed by: Peter Brass
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1460291