On the cromatic number of infinitesimal plane layer

We consider natural generalization of plane chromatic number problem. We consider chromatic numbers $\chi$ of spaces $$\mathbb{R}$^n \times [0,\varepsilon]^k$forarbitrarysmall$\varepsilon$. We prove that $5\le \chi ({\mathbb{R}}^2\times [0,\epsilon ])\le 7$ and $6\le \chi ({\mathbb{R}}^2\times [0,\epsilon {]}^2)\le 7$. Also we consider natural questions, arising from this considerations.