On Borsuk's conjecture for two-distance sets

2013/05/12 by Andriy Bondarenko, Andriy V. Bondarenko, Bondarenko, Andriy V. · 2 citations
Computer Science · Mathematics · #Advanced Topology and Set Theory #Digital Image Processing Techniques #Limits and Structures in Graph Theory #math.MG #msc:05C50 #msc:41A55 #msc:41A63 #msc:52C35

paper · pdf · doi:10.48550/arxiv.1305.2584

9 pages

arxiv created 2013/08/29 · arxiv updated 2013/08/30

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.

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• Borsuk's conjecture for two-distance sets and its equivalent formulation for graphs
• Some old and new problems in combinatorial geometry I: Around Borsuk's problem

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