On the chromatic numbers of 3-dimensional slices

TL;DR

This paper establishes a lower bound of 10 on the chromatic number of certain 3-dimensional geometric graphs, extending understanding of coloring problems in high-dimensional Euclidean spaces.

Contribution

It proves a new lower bound for the chromatic number of graphs defined on 3D slices with a small thickness, advancing the theory of geometric graph coloring.

Findings

Chromatic number of 3D slices is at least 10.

Lower bounds hold for arbitrary small thickness .

Results contribute to high-dimensional geometric coloring theory.

Abstract

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