On the cromatic number of infinitesimal plane layer

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