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