2017/06/20 by Balko, Martin, Pór, Attila, Scheucher, Manfred +2
#05C55 #05C85 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Primary 52C10 #Secondary 05C30
paper · doi:10.48550/arxiv.1706.06375
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 to 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).