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 $0.41<\sqrt{2}-1\le d\le 1\mathrm{/}\sqrt{5}<0.45$. Our results significantly expand that range to $0.354\le d\le 0.657$, the first improvement in 30 years. Notably, the constructions underlying this were derived by formalizing colorings suggested by a custom machine learning approach.