On the chromatic numbers of 3-dimensional slices

We prove that for an arbitrary $\epsilon >0$ holds $$\chi ({\mathbb{R}}^3\times [0,\epsilon {]}^6)\ge 10,$$ where $\chi (M)$ stands for the chromatic number of an (infinite) graph with the vertex set $M$ and the edge set consists of pairs of monochromatic points at the distance 1 apart.