Classification of three-distance sets in two dimensional Euclidean space

A subset X in k-dimensional Euclidean space Rk is called an s-distance set if there are exactly s distances between two distinct points in X. S.J. Einhorn–I.J. Schoenberg conjectured that there are only five maximal (i.e. cannot be contained in others) three-distance sets in R2 having five or more points. In this paper, we show that there are in fact twenty four maximal three-distance sets in R2 having five or more points and determine the largest possible cardinality of three-distance sets in R2.