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

TL;DR

The paper investigates sphere colorings under Thomassen-like tiling restrictions for a Hadwiger--Nelson-type problem. It proves that for sufficiently large spheres, at least eight colours are needed under these restrictions, by converting a hypothetical seven-colouring tiling into a fully triangulated planar graph with twelve irregular vertices and employing curvature arguments and Isbell-type colorings on the infinite triangular grid. The key technical move is a graph-theoretic reduction and a case analysis that rules out the existence of a nice $7$-colouring when $r\ge 46.5/\pi$. As a consequence, the work improves the sphere lower bound and, via a geometric-to-graph translation, strengthens the lower bound for nicely tiling $\mathbb{R}^3$ under the same restrictions.

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.

Figures (28)

• Figure 1:
• Figure 2:
• Figure 3: Tiles with distance $1$
• Figure 4: Siamese tiles
• Figure 5: Case 1a, Case 1b and Case 2

Theorems & Definitions (61)

• Definition 1
• Theorem 2: Thomassen t
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Definition 6
• Lemma 7
• proof