The chromatic number of Euclidean space with dense color classes

Abstract

In this note we construct colorings of Euclidean space ${\mathbb{R}}^n$ with finitely many colors such that any two points at unit distance have different colors and, in addition, each color class is dense in ${\mathbb{R}}^n$. In particular, 12 dense colors suffice to color ${\mathbb{R}}^2$. In arbitrary dimension, we show that $n\chi ({\mathbb{R}}^n)+1$ colors suffice, where $\chi ({\mathbb{R}}^n)$ denotes the chromatic number of ${\mathbb{R}}^n$ in the standard formulation.

10 pages, 1 figure