Almost-equidistant sets

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 $f(5)\ge 16$. We further show that $12\le f(4)\le 13$, $f(5)\le 20$, $18\le f(6)\le 26$, $20\le f(7)\le 34$, and $f(9)\ge f(8)\ge 24$. Up to dimension $7$, our work is based on various computer searches, and in dimensions $6$ to $9$, we give constructions based on the known construction for $d=5$. For every dimension $d\ge 3$, we give an example of an almost-equidistant set of $2d+4$ points in the $d$-space and we prove the asymptotic upper bound $f(d)\le O(d^{3/2})$.

24 pages, 9 figures. Accepted by Graphs and Combinatorics