The Hadwiger-Nelson problem over certain fields

TL;DR

This paper determines the minimum number of colors needed to color points in certain algebraic number fields' planes so that no two points at distance one share the same color, extending the classical Hadwiger-Nelson problem.

Contribution

It computes the Hadwiger-Nelson numbers for specific quadratic fields, providing new exact values and bounds, and discusses implications over various fields and the axiom of choice.

Findings

(\u211a(\u221a{2})^2) = 2

(\u211a({3})^2) = 3

({7})^2) = 3 despite being triangle-free in the graph representation

Abstract

We compute the Hadwiger-Nelson numbers $\chi(E^2)$ for certain number fields $E$, that is, the smallest number of colors required to color the points in the plane with coordinates in~$E$ so that no two points at distance $1$ from one another have the same color. Specifically, we show that $\chi(\mathbb{Q}(\sqrt{2})^2) = 2$, that $\chi(\mathbb{Q}(\sqrt{3})^2) = 3$, that $\chi(\mathbb{Q}(\sqrt{7})^2) = 3$ despite the fact that the graph $\Gamma(\mathbb{Q}(\sqrt{7})^2)$ is triangle-free, and that $4 \leq \chi(\mathbb{Q}(\sqrt{3}, \sqrt{11})^2) \leq 5$. We also discuss some results over other fields, for other quadratic fields. We conclude with some comments on the use of the axiom of choice.