New Maximal Two-Distance Sets

A two-distance set in Edis a point setXin thed-dimensional Euclidean space such that the distances between distinct points inXassume only two different non-zero values. Based on results from classical distance geometry, we develop an algorithm to classify, for a givend, all maximal (largest possible) two-distance sets in Ed. Using this algorithm we have completed the full classification for alld⩽7, and we have found one set in E8whose maximality follows from Blokhuisʹ upper bound on sizes ofs-distance sets. While in the dimensionsd⩽6 our classifications confirm the maximality of previously known sets, the results in E7and E8are new. Their counterpart in dimensiond⩾10 is a set of unit vectors with only two values of inner products in the Lorentz space Rd, 1. The maximality of this set again follows from a bound due to Blokhuis.