Borsuk's conjecture for two-distance sets and its equivalent formulation for graphs

Abstract

Every graph G can be embedded in a Euclidean space as a two-distance set. This allows us to reformulate the analogue of Borsuk's conjecture for two-distance sets in terms of graphs. This conjecture remains open for dimensions from 4 to 63. This short note also discusses an approach for finding counterexamples using graphs, as well as its generalization for s-distance sets.