On the cromatic number of infinitesimal plane layer

TL;DR

This paper investigates the chromatic number of infinitesimal layers of the plane, establishing bounds for spaces like  imes [0, ] and  imes [0, ]^2, and explores related questions.

Contribution

It provides new bounds for the chromatic number of infinitesimal plane layers and discusses related open questions.

Findings

 imes [0, ] has chromatic number between 5 and 7.

 imes [0, ]^2 has chromatic number between 6 and 7.

The paper raises natural questions from these results.

Abstract

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.