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The chromatic number $\chi(\mathbb{R}^n)$ of the Euclidean space$\mathbb{R}^n$ is the smallest number of colors sufficient for coloring allpoints of the space in such a way that any two points at the distance 1 havedifferent colors. In 1972 Larman--Rogers proved that $\chi(\mathbb{R}^n) \leq(3 + o(1))^n$. We give a new proof of this bound.

A new proof of the Larman-Rogers upper bound for the chromatic number of the Euclidean space