Arman–Bondarenko–Prymak–Radchenko lattice-coloring optimality conjecture

Arman, Bondarenko, Prymak, and Radchenko’s 2022 paper proposed that $49$ and $140$ were optimal lattice-coloring values in dimensions $4$ and $5$. The conjecture concerns lattice constructions, not exact Euclidean chromatic numbers.

Known results

• Arman, Bondarenko, Prymak, and Radchenko (2022): $\chi(\mathbb{E}^4)\le 49$ and $\chi(\mathbb{E}^5)\le 140$.
• The same paper: $\chi(\mathbb{E}^7)\le 1372$ and $\chi(\mathbb{E}^9)\le 17253$; the $4$- and $5$-dimensional optima were stated as beliefs, not proved theorems.

September 2026 claimed refutation

Leonid Ivanov and Nadezhda Glushkova’s preprint claims explicit lattice colorings giving $\chi(\mathbb{R}^4)\le 43$ and $\chi(\mathbb{R}^5)\le 132$, thereby refuting the proposed optima, with exact-arithmetic finite certificates. It also claims $\chi(\mathbb{R}^7)\le 1029$, $\chi(\mathbb{R}^9)\le 7203$, and $\chi(\mathbb{R}^{10})\le 45619$. These are claimed results and do not determine the Euclidean chromatic numbers.

Current status (as of September 2026): The $49$ and $140$ lattice-optimality conjectures are claimed to be refuted, but the new bounds and certificates remain unverified; exact Euclidean chromatic numbers remain open.