On the chromatic number of slices without monochromatic unit arithmetic progressions

For $h,n\geq 1$ and $e>0$ we consider a chromatic number of the spaces $\mathbb{R}^n\times[0, e]^h$ and general results in this problem. Also we consider the chromatic number of normed spaces with forbidden monochromatic arithmetic progressions. We show that for any $n$ there exists a two-coloring of $\mathbb{R}^n$ such that all long unit arithmetic progressions contain points of both colors and this coloring covers spaces of the form $\mathbb{R}^n\times[0, e]^h$.