On the chromatic number of an infinitesimal plane layer

This paper is devoted to a natural generalization of the problem on the chromatic number of the plane. The chromatic number of the spaces Rn×[0,ε]k\mathbb {R}^n \times [0,\varepsilon ]^k is considered. It is proved that 5≤χ(R2×[0,ε])≤75 \leq \chi (\mathbb {R}^2\times [0,\varepsilon ])\leq 7 and 6≤χ(R2×[0,ε]2)≤7{6\leq \chi (\mathbb {R}^2\times [0,\varepsilon ]^2) \leq 7} for ε>0\varepsilon >0 sufficiently small. Also, some natural questions arising from these considerations are posed.