New upper bounds for the chromatic numbers of Euclidean spaces

Source abstract

A coloring of $\R^n$ is \emph{proper for the forbidden distance segment} $[1,\ell]$ if no two points of the same color are at a distance from $[1,\ell]$; the minimum number of colors is $χ(\R^n,[1,\ell])$, and $\ell=1$ gives the classical chromatic number $χ(\R^n)$ of the Nelson--Hadwiger problem. We prove the new upper bounds $χ(\R^4)\le43$, $χ(\R^5)\le132$, $χ(\R^7)\le1029$, $χ(\R^9)\le7203$, $χ(\R^{10})\le45619$, improving the previously known $49$, $140$, $1372$, $17253$ and $3^{10}$; in particular, this refutes the conjecture of Arman, Bondarenko, Prymak and Radchenko that $49$ and $140$ are optimal among all lattice colorings of $\R^4$ and $\R^5$. The first four bounds come from explicit rational lattices --- an Eisenstein lattice in $\R^4$, a lattice in general position in $\R^5$, and laminations of the Eisenstein colorings $E_6^*/343$ and $E_8/2401$ in $\R^7$ and $\R^9$ --- and each is reduced, by one verification protocol, to a finite list of inequalities between explicitly written rational numbers checked in exact arithmetic. The fifth bound is analytic: we prove that for every Eisenstein lattice $Λ$ the distance between same-colored cells of $(3+ω)Λ$ equals $\sqrt{7/3}\,λ_1(Λ)$, which gives the exact widths of all known colorings with $7^{n/2}$ colors, and a product rule $\sum_i1/d_i^2\le1$ for the widths of orthogonal products; together they yield $45619=2401\cdot19$, the first bound in $\R^{10}$ below $3^n$, as well as $χ(\R^{25})\le4\cdot7^{12}$ and $χ(\R^{26})\le19\cdot7^{12}$. We also show that no sublattice of $E_8$ of index below $2401$ defines a proper coloring. All code, exact certificates and data are open.