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

TL;DR

This paper introduces a coloring method for points on large spheres in three-dimensional space, using at most seven colors to ensure no two points at unit distance share the same color, highlighting a chromatic number bound in 3D.

Contribution

It provides a new construction demonstrating that large spheres in three dimensions can be colored with at most seven colors to avoid monochromatic unit distances.

Findings

For radii greater than about 12.44, seven colors suffice.

Contrasts with previous results requiring more colors under certain restrictions.

Shows a difference between 3D sphere coloring and planar cases.

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 $\varepsilon \leq\frac{\sqrt{7}}{2} - 1$.