On counterexamples to Borsuk's conjecture on a sphere

The classical Borsuk’s conjecture is the statement that any set of diameter 1 in the Euclidean space Rd can be divided into d + 1 parts of smaller diameter. This conjecture is proved wrong for d > 64. In this paper, a generalization of Borsuk’s conjecture on the sphere Sd-1r is considered. In particular, we study the function fr(d) defined as the smallest number of parts of diameter smaller than 1 into which any set A ⊂ Sd-1r of diameter 1 canbe divided. Using the linear algebraic method, we obtain new lower bounds of this functionthat improves the results of other authors. The optimal choice of parameters in the presented theorems is considered.