Almost-Equidistant Sets

Abstract

AbstractFor 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(2)=7, $$f(3)=10$$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 $$f(5) \ge 16$$f(5)≥16. We further show that $$12\le f(4)\le 13$$12≤f(4)≤13, $$f(5)\le 20$$f(5)≤20, $$18\le f(6)\le 26$$18≤f(6)≤26, $$20\le f(7)\le 34$$20≤f(7)≤34, and $$f(9)\ge f(8)\ge 24$$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 $$d=5$$d=5. For every dimension $$d \ge 3$$d≥3, we give an example of an almost-equidistant set of $$2d+4$$2d+4 points in the d-space and we prove the asymptotic upper bound $$f(d) \le O(d^{3/2})$$f(d)≤O(d3/2).