Almost-Equidistant Sets

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 2d+4 points in the d -space and we prove the asymptotic upper bound f(d) ≤ O(d^3/2) .