In this paper we answer Larman's question on Borsuk's conjecture for
two-distance sets. We find a two-distance set consisting of 416 points on the
unit sphere in the dimension 65 which cannot be partitioned into 83 parts of
smaller diameter. This also reduces the smallest dimension in which Borsuk's
conjecture is known to be false. Other examples of two-distance sets with large
Borsuk's numbers will be given.Comment: 9 page