A new proof of the Larman–Rogers upper bound for the chromatic number of the Euclidean space

A new proof of the Larman–Rogers upper bound for the chromatic number of the Euclidean space

The chromatic numberχ(Rn) of the Euclidean space Rn is the smallest number of colors sufficient for coloring all points of the space in such a way that any two points at the distance 1 have different colors. In 1972 Larman–Rogers proved that χ(Rn)⩽(3+o(1))n. We give a new proof of this bound.

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Veröffentlicht in:Discrete Applied Mathematics 2020-04, Vol.276, p.115-120
1. Verfasser: Prosanov, Roman
Format: Artikel
Sprache:eng
Schlagworte:
Chromatic number
Euclidean geometry
Euclidean space
Geometric covering
Upper bounds
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Zusammenfassung:The chromatic numberχ(Rn) of the Euclidean space Rn is the smallest number of colors sufficient for coloring all points of the space in such a way that any two points at the distance 1 have different colors. In 1972 Larman–Rogers proved that χ(Rn)⩽(3+o(1))n. We give a new proof of this bound.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2019.05.020