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

2025/11/05 by Musin, Oleg R.
Computer Science · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Metric Geometry (math.MG) #Topological and Geometric Data Analysis

paper · doi:10.48550/arxiv.2511.03668

openalex publication_date 2025/11/05 · openalex created_date 2025/11/07 · openalex updated_date 2026/07/28

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.

Citations

• A 64-dimensional two-distance counterexample to Borsuk's conjecture
• On Borsuk's conjecture for two-distance sets

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• A 64-dimensional two-distance counterexample to Borsuk's conjecture
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