The surface of a sufficiently large sphere has chromatic number at most 7

Abstract

We present a method to assign, for any radius $r$ greater than about 12.44, one of seven colors to each point in $\mathbb{R}^3$ lying at distance $r$ from the origin, such that no two points at unit distance from each other are assigned the same color. The existence of such a construction contrasts with the recent demonstration that, for any positive value $\varepsilon$, if no two points assigned the same color lie at any distance in $[1,1+\varepsilon]$ (and with certain other restrictions that are also satisfied with our coloring), then eight colors are needed for any finite $r\ge18$, even though seven colors suffice in the plane when $\varepsilon \leq\frac{\sqrt{7}}{2} - 1$.