The vector graph and the chromatic number of the plane, or how NOT to prove that $\chi(\mathbb{E}^2)>4$

TL;DR

This paper investigates the chromatic number of the plane, demonstrating limitations on increasing its lower bound through the method of combining Moser Spindles, thus clarifying the constraints of current proof techniques.

Contribution

It introduces a limiting result showing that pasting Moser Spindles cannot raise the lower bound of the plane's chromatic number, highlighting the boundaries of existing approaches.

Findings

Cannot surpass the lower bound of 4 using Moser Spindles

Limits of current methods in proving ch(2) > 4

Provides a new perspective on the chromatic number problem

Abstract

The chromatic number $\chi\left(\mathcal{E^2}\right)$ of the plane is known to be some integer between 4 and 7, inclusive. We prove a limiting result that says, roughly, that one cannot increase the lower bound on $\chi\left(\mathcal{E^2}\right)$ by pasting Moser Spindles together, even countably many.