On almost-equidistant sets - II

Abstract

A set in ${\mathbb{R}}^d$ is called almost-equidistant if for any three distinct points in the set, some two are at unit distance apart. First, we give a short proof of the result of Bezdek and Lángi claiming that an almost-equidistant set lying on a $(d-1)$-dimensional sphere of radius $r$, where $r<1/\sqrt{2}$, has at most $2d+2$ points. Second, we prove that an almost-equidistant set $V$ in ${\mathbb{R}}^d$ has $O(d)$ points in two cases: if the diameter of $V$ is at most $1$ or if $V$ is a subset of a $d$-dimensional ball of radius at most $1/\sqrt{2}+c{d}^{-2/3}$, where $c<1/2$. Also, we present a new proof of the result of Kupavskii, Mustafa and Swanepoel arXiv:1708.01590 that an almost-equidistant set in ${\mathbb{R}}^d$ has $O(d^{4/3})$ elements.

arXiv comment8 pages. All proofs are simplified. Open problems are added. Key words: equidistant sets, almost-equidistant sets, unit distance graph, diameter graph, triangle-free graph, Perron-Frobenius Theorem. To appear in the Electronic Journal of Combinatorics