On the chromatic numbers of 3-dimensional slices

Abstract

We prove that for an arbitrary $\epsilon >0$ holds \[ χ(\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10, \] 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.