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

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.

To appear in Australasian Journal of Combinatorics