On the chromatic number of 2-dimensional spheres

TL;DR

This paper proves Simmons' 1976 conjecture that any three-coloring of a sufficiently large 2-sphere necessarily contains two monochromatic points exactly one unit apart.

Contribution

It confirms a long-standing conjecture in geometric coloring, establishing a fundamental property of colorings on 2-spheres of radius greater than 1/2.

Findings

Proves Simmons' conjecture from 1976

Shows existence of monochromatic pairs at distance 1 in 3-colorings

Advances understanding of geometric coloring problems on spheres

Abstract

In 1976 Simmons conjectured that every coloring of a 2-dimensional sphere of radius strictly greater than $1/2$ in three colors has a couple of monochromatic points at the distance 1 apart. We prove this conjecture.