Measurable Chromatic Number of Spheres
Greg Malen

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComputational Geometry and Mesh Generation · Digital Image Processing Techniques · Quasicrystal Structures and Properties
