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Bounds on Borsuk Numbers in Distance Graphs of a Special Type

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Abstract

In 1933, Borsuk stated a conjecture, which has become classical, that the minimum number of parts of smaller diameter into which an arbitrary set of diameter 1 in \(\mathbb{R}^n\) can be partitioned is \(n+1\). In 1993, this conjecture was disproved using sets of points with coordinates 0 and 1. Later, the second author obtained stronger counterexamples based on families of points with coordinates \(-1\), \(0\), and \(1\). We establish new lower bounds for Borsuk numbers in families of this type.

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Funding

The research was supported in part by the Russian Foundation for Basic Research, project no. 18-01-00355, and the President of the Russian Federation Council for State Support of Leading Scientific Schools, grant no. NSh-2540.2020.1.

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Translated from Problemy Peredachi Informatsii, 2021, Vol. 57, No. 2, pp. 44–50 https://doi.org/10.31857/S0555292321020030.

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Berdnikov, A., Raigorodskii, A. Bounds on Borsuk Numbers in Distance Graphs of a Special Type. Probl Inf Transm 57, 136–142 (2021). https://doi.org/10.1134/S0032946021020034

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  • DOI: https://doi.org/10.1134/S0032946021020034

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