Measurable Chromatic Number of Spheres

TL;DR

This paper investigates the measurable chromatic number of spheres' surfaces, demonstrating that lower bounds similar to those in the plane apply broadly, and revealing the non-monotonic relationship with curvature.

Contribution

It extends lower bound techniques for chromatic numbers from the plane to spheres of various radii, showing non-monotonic behavior with respect to curvature.

Findings

Lower bounds similar to the plane apply to most sphere radii.

Measurable chromatic number varies non-monotonically with curvature.

The relationship between chromatic number and radius is complex and not straightforward.

Abstract

We examine the measurable chromatic number of distance colorings on the surface of 2-dimensional spheres of varying radii, showing in particular that similar arguments to those used to raise lower bounds in the plane work for all but a countable set of radii. Furthermore, we show that measurable chromatic number as a function of the radius, or more generally the curvature, is not monotonic.