2026/07/22 by Maxim Didin, Vsevolod Voronov
#math.CO
In this note we construct colorings of Euclidean space ℝn with finitely many colors such that any two points at unit distance have different colors and, in addition, each color class is dense in ℝn. In particular, 12 dense colors suffice to color ℝ2. In arbitrary dimension, we show that nχ(ℝn)+1 colors suffice, where χ(ℝn) denotes the chromatic number of ℝn in the standard formulation.