A new proof of the Larman-Rogers upper bound for the chromatic number of\n the Euclidean space

2016/10/10 by Роман Просанов, Prosanov, Roman · 2 citations
Mathematics · #52C10 (Primary) #52C17 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1610.02846

openalex publication_date 2016/10/10 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

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

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