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

TL;DR

This paper explores the reformulation of Borsuk's conjecture for two-distance sets via graph embeddings, discusses potential counterexample approaches, and extends the ideas to s-distance sets, highlighting open problems in dimensions 4 to 63.

Contribution

It provides a graph-theoretic reformulation of Borsuk's conjecture for two-distance sets and proposes methods to find counterexamples, extending the framework to s-distance sets.

Findings

Reformulation of Borsuk's conjecture in terms of graph embeddings.

Discussion of approaches to find counterexamples using graphs.

Extension of the conjecture to s-distance sets.

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.