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Borsuk’s problem and the chromatic numbers of some metric spaces. (English. Russian original) Zbl 1008.54018
Russ. Math. Surv. 56, No. 1, 103-139 (2001); translation from Usp. Mat. Nauk 56, No. 1, 107-146 (2001).
The paper contains various results pertaining to two problems of combinatorial geometry: Borsuk’s problem on partitions of an arbitrary bounded \(d\)-dimensional set of non-zero diameter into parts of smaller diameter and the problem of finding chromatic numbers of some metric spaces. The author presents an exposition of the general method for constructing counterexamples to Borsuk’s conjecture and obtaining lower bounds for the minimum number of parts of smaller diameter as well as for chromatic numbers of real and rational spaces.

54E35 Metric spaces, metrizability
05C15 Coloring of graphs and hypergraphs
51D20 Combinatorial geometries and geometric closure systems
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