Extending the Continuum of Six-Colorings

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.