Coloring Some Finite Sets in $ \mathbb{R}^n $

This note relates to bounds on the chromatic number $ \chi (\mathbb{R}^n)$ of the Euclidean space, which is the minimum number of colors needed to color all the points in $ \mathbb{R}^n$ so that any two points at the distance 1 receive different colors. In [6] a sequence of graphs $ G_n $ in $ \mathbb{R}_n $ was introduced showing that $ \chi(\mathbb{R}^n) \ge \chi(G_n) \ge (1+ o(1))\frac{n^2}{6} $. For many years, this bound has been remaining the best known bound for the chromatic numbers of some lowdimensional spaces. Here we prove that $ \chi(G_n) \sim \frac{n^2}{6} $ and find an exact formula for the chromatic number in the case of $ n = 2^k $ and $ n = 2^k − 1 $.