Extending the Continuum of Six-Colorings
arxiv(2024)
摘要
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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