As a follow-up to my previous post, I wanted to discuss an overview of de Grey’s proof that the chromatic number of the plane is at least 5, as well as some recent progress that has been made on the polymath proposal page and elsewhere.

The core idea of de Grey’s proof is to select a finite point set H in the plane and a coloring property P, and then demonstrate two incongruous statements:

(a) For every proper k-coloring of the plane, there exists a copy of H whose inherited coloring satisfies P.

(b) For every proper k-coloring of the plane, every copy of H inherits a coloring that doesn’t satisfy P.

Indeed, if both statements hold, then there is no proper k-coloring of the plane. To prove (a) and (b), we pass to finite unit-distance graphs in the plane. Explicitly, to prove (a), one finds a finite point set L such that for every proper k-coloring of L, there exists a copy of H in L whose inherited coloring satisfies P. Similarly, to prove (b), one finds a finite superset M of H such that every proper k-coloring of M forces the inherited coloring of H to not satisfy P.

Continue reading The chromatic number of the plane is at least 5, part II