On Some Applications of Graph Theory to Geometry

Let [Pn (k)] be the class of all subsets Pn (k) of the k-dimensional Euclidean space consisting of n distinct points and having diameter 1. Denote by dk(n, r) the maximum number of times a given distance r can occur among points of a set Pn (k) .Put In other words Dk(n) denotes the maximum number of times the same distance can occur between n suitably chosen points in k-dimensional space.