2021/07/25 by Tomas Sirgedas, Sirgedas, Tomas
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2107.11900
openalex publication_date 2021/07/25 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We present a method to assign, for any radius r greater than about 12.44,\none of seven colors to each point in \ℝ3 lying at distance r from\nthe origin, such that no two points at unit distance from each other are\nassigned the same color. The existence of such a construction contrasts with\nthe recent demonstration that, for any positive value \ε, if no two\npoints assigned the same color lie at any distance in [1,1+\ε] (and\nwith certain other restrictions that are also satisfied with our coloring),\nthen eight colors are needed for any finite r\≥18, even though seven colors\nsuffice in the plane when \ε \≤\(\√(7))/(2) - 1.\n