[Paper Review] On Borsuk's conjecture for two-distance sets

Andriy Bondarenko|arXiv (Cornell University)|May 12, 2013

Digital Image Processing Techniques4 references4 citations

TL;DR

This paper disproves Borsuk's conjecture for two-distance sets by constructing a 416-point set on the 64-dimensional unit sphere in ℝ⁶⁵ with pairwise angles corresponding to inner products of 1/5 or -1/15, which cannot be partitioned into 83 parts of smaller diameter. The result establishes that the Borsuk number for two-distance sets in dimension 65 is at least 84, and extends this to infinitely many dimensions via recursive constructions based on strongly regular graphs of G₂(4) and Fi₂₃.

ABSTRACT

In this paper we answer Larman's question on Borsuk's conjecture for two-distance sets. We find a two-distance set consisting of 416 points on the unit sphere in the dimension 65 which cannot be partitioned into 83 parts of smaller diameter. This also reduces the smallest dimension in which Borsuk's conjecture is known to be false. Other examples of two-distance sets with large Borsuk's numbers will be given.

Motivation & Objective

• To resolve Larman's question on whether Borsuk's conjecture holds for two-distance sets.
• To construct explicit two-distance sets in high-dimensional spheres with large Borsuk numbers.
• To reduce the smallest dimension where Borsuk's conjecture fails for two-distance sets.
• To extend the result to infinitely many dimensions using recursive constructions from strongly regular graphs.

Proposed method

• Construct two-distance sets via Euclidean representations of the adjacency matrices of strongly regular graphs.
• Use the spectral properties of the adjacency matrix A of a strongly regular graph with parameters (v,k,λ,μ) to derive vectors in ℝ^f with two distinct inner products.
• Apply the transformation z_i = y_i - (1/v)∑y_j and normalize to obtain unit vectors x_i on the sphere S^{f-1}.
• Leverage the fact that the diameter of the configuration corresponds to the minimum distance between non-adjacent vertices in the graph.
• Estimate the minimal number of parts of smaller diameter by bounding the size of the largest clique in the graph.
• Use recursive constructions in higher dimensions by combining n copies of the base configuration and adding a new vector at unit distance to all others.

Experimental results

Research questions

• RQ1 Can Borsuk's conjecture be disproven for two-distance sets in low-dimensional Euclidean spaces?
• RQ2 What is the maximal possible Borsuk number b₂(n) for two-distance sets in dimension n?
• RQ3 Do strongly regular graphs provide a systematic way to construct two-distance sets with high Borsuk numbers?
• RQ4 Can the construction be extended to infinitely many dimensions while preserving the non-partitionability property?
• RQ5 What is the smallest dimension n for which b₂(n) > n+1?

Key findings

• A two-distance set of 416 points on S⁶⁴ ⊂ ℝ⁶⁵ with inner products 1/5 and -1/15 cannot be partitioned into 83 parts of smaller diameter, so b₂(65) ≥ 84.
• The configuration arises from the strongly regular graph G₂(4) with parameters (416,100,36,20), whose spectral properties yield the desired two-distance structure.
• A larger two-distance set of 31,671 points on S⁷⁸¹ ⊂ ℝ⁷⁸² with inner products 1/10 and -1/80 cannot be partitioned into 1,376 parts, so b₂(781) ≥ 1,377.
• Using the Fi₂₃ strongly regular graph (31671,3510,693,351), the construction yields b₂(781) ≥ 1377, and tighter bounds for lower dimensions: b₂(780) ≥ 1102 and b₂(779) ≥ 1002.
• The result is extended to infinitely many dimensions: b₂(66n + k) ≥ 84n + k + 1 and b₂(783n + k) ≥ 1377n + k + 1 for all n ≥ 1 and k ≥ 0.
• The constructions rely on clique size bounds in subconstituents of the graphs, particularly showing that the largest clique in the Fi₂₃ graph's second subconstituent is at most 21, limiting the size of any clique in the full graph to 23.