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

The chromatic number $\chi \left({\mathcal{E}}^{\mathcal{2}}\right)$ 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 $\chi \left({\mathcal{E}}^{\mathcal{2}}\right)$ by pasting Moser Spindles together, even countably many.