The maximum distance in 3-D can occur no more than times. Also, there exists a fixed number such that no distance determined by a set of points in 3-D space occurs more than times. The maximum distance can occur no more than times in 4-D, where is the Floor Function.
Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 122-123, 1976.