Exact Gromov-Hausdorff distances are derived between arbitrary simplexes and 2-distance spaces, yielding a complete solution to the generalized Borsuk problem and expressions for graph clique cover and chromatic numbers.
On Borsuk's conjecture for two-distance sets
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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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math.MG 1years
2019 1verdicts
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The Gromov--Hausdorff Distance between Simplexes and Two-Distance Spaces
Exact Gromov-Hausdorff distances are derived between arbitrary simplexes and 2-distance spaces, yielding a complete solution to the generalized Borsuk problem and expressions for graph clique cover and chromatic numbers.