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Mat. Sb., 2014, Volume 205, Number 8, Pages 95–138 (Mi msb8316)  

Covering sets in $\mathbb{R}^m$

V. P. Filimonov

Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region

Abstract: The paper investigates questions related to Borsuk's classical problem of partitioning a set in Euclidean space into subsets of smaller diameter, as well as to the well-known Nelson-Erdős-Hadwiger problem on the chromatic number of a Euclidean space.
The results of the work are obtained using combinatorial and geometric methods alike. A new approach to the investigation of such problems is suggested; it leads to a collection of results which significantly improve all results known so far.
Bibliography: 58 titles.

Keywords: chromatic number, Borsuk's problem, diameter of a set, covering of a plane set, universal covering sets and systems.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00759-а


DOI: https://doi.org/10.4213/sm8316

Full text: PDF file (731 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2014, 205:8, 1160–1200

Bibliographic databases:

UDC: 514.174
MSC: 52C17
Received: 16.12.2013

Citation: V. P. Filimonov, “Covering sets in $\mathbb{R}^m$”, Mat. Sb., 205:8 (2014), 95–138; Sb. Math., 205:8 (2014), 1160–1200

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