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)>=16. We further show that 12=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(d3/2).