A lower bound on the number of colours needed to nicely colour a sphere

TL;DR

This paper establishes a new lower bound of 8 colours for properly colouring large spheres under certain restrictions, extending known bounds from the plane to higher dimensions.

Contribution

It proves that large spheres require at least 8 colours for proper colouring under specific natural restrictions, improving previous bounds and extending results to higher dimensions.

Findings

Large spheres need at least 8 colours for proper colouring.

The result extends the lower bounds from the plane to spheres of large radius.

Provides a new lower bound for 3-dimensional space colouring.

Abstract

The Hadwiger--Nelson problem is about determining the chromatic number of the plane (CNP), defined as the minimum number of colours needed to colour the plane so that no two points of distance 1 have the same colour. In this paper we investigate a related problem for spheres and we use a few natural restrictions on the colouring. Thomassen showed that with these restrictions, the chromatic number of all manifolds satisfying certain properties (including the plane and all spheres with a large enough radius) is at least 7. We prove that with these restrictions, the chromatic number of any sphere with a large enough radius is at least 8. This also gives a new lower bound for the minimum colours needed for colouring the 3-dimensional space with the same restrictions.