Borsuk's conjecture for two-distance sets and its equivalent formulation for graphs
Oleg R. Musin
TL;DR
The paper reframes Borsuk's conjecture, which bounds the Borsuk number $B(S)$ by $d+1$, in terms of graph theory: every graph $G$ on $n$ vertices can be realized as a two-distance set in $\mathbb{R}^d$ with $d=n-\mu(G)-1$ and $B(S)=\theta(G)$, where $\mu(G)$ is the multiplicity of the smallest root $t>1$ of the Cayley-Menger determinant $C_G(t)$ and $\theta(G)$ is the clique-cover number. Therefore, a counterexample arises exactly when $\theta(G)+\mu(G)>n$; this bridges geometric combinatorics with graph parameters and provides a concrete route to search for counterexamples via graphs. The paper also extends the framework to spherical two-distance sets and discusses a strategy to find counterexamples by optimizing the graph structure (via $\mathrm{Clq}(G)$ and $\mu(G)$) and by generalizing to $s$-distance sets using colored edge decompositions of complete graphs. Together, these results offer a graph-theoretic path to identify, analyze, and potentially generate counterexamples to Borsuk's conjecture, and they sketch how discrete optimization and multi-distance representations might aid in discovering such examples. The work thus connects geometric embedding theory with combinatorial graph parameters, offering practical avenues for future counterexample searches and generalizations to higher-distance frameworks.
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.