The unit distance problem on spheres
Konrad J. Swanepoel, Pável Valtr · Contemporary mathematics - American Mathematical Society · 2004
For any D > 1 and for any n 2 we construct a set of n points on a sphere in R of diameter D determining at least log n unit distances. This improves a previous lower bound of Erd}os, Hickerson and Pach (1989). We also construct a set of n points in the plane not containing collinear triples or the vertices of a parallelogram and determining at least cn log n unit distances.