Upper bounds on chromatic number of \(\mathbb{E}^n\) in low dimensions

Let \(\chi(\mathbb{E}^n)\) denote the chromatic number of the Euclidean space \(\mathbb{E}^n\), i.e., the smallest number of colors that can be used to color \(\mathbb{E}^n\) so that no two points unit distance apart are of the same color. We present explicit constructions of colorings of \(\mathbb{E}^n\) based on sublattice coloring schemes that establish the following new bounds: \(\chi(\mathbb{E}^5)\le 140\), \(\chi(\mathbb{E}^n)\le 7^{n/2}\) for \(n\in\{6,8,24\}\), \(\chi(\mathbb{E}^7)\le 1372\), \(\chi(\mathbb{E}^{9})\leq 17253\), and \(\chi(\mathbb{E}^n)\le 3^n\) for all \(n\le 38\) and \(n=48,49\).