Extending the Continuum of Six-Colorings

2024/04/08 by Konrad Mundinger, Mundinger, Konrad, Sebastian Pokutta +5
Engineering · #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2404.05509

openalex publication_date 2024/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance d in the sixth color. Such colorings have previously been known to exist for 0.41 < √(2) - 1 ≤ d ≤ 1 / √(5) < 0.45. Our results significantly expand that range to 0.354 ≤ d ≤ 0.657, the first improvement in 30 years. Notably, the constructions underlying this were derived by formalizing colorings suggested by a custom machine learning approach.

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