Claims

../

1989_08_01_erdos_hickerson_pach: For every n and every distance strictly between 0 and 2, n points on the unit sphere can have a constant times n log* n pairs at that distance, so the answer is yes; refereed in the Monthly and credited by the site's curator.

2004_01_01_swanepoel_valtr: On every sphere in three-space of diameter above one, n points can span more than a constant times n root log n unit distances: a second proof of the answer yes with a stronger bound, credited by the site's curator.

---