New upper bounds for the chromatic numbers of Euclidean spaces

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. W