Is there a relationship between the recent result on Keller's Conjecture;
The conjecture states that any tiling of identical hypercubes that fills space must contain a pair of neighbors that share an entire face.
and the Hadwiger-Nelson Problem?
I have only just heard about Keller's Conjecture and thought that it was awfully similar to the Chromatic number of the plane, and was wondering if it was possible to extend the results from it to the Hadwiger-Nelson problem.
i.e., both problems are concerned with these n-dimensional euclidean spaces, they both have their proofs rooted in dense packings, Keller's being the dense packing of dimensional analogues of the cube in this space, and Hadwiger-Nelson's being graphs with dense packings of the moser-spindle. Also both of their usage of tiling as a means proof, with:
The upper bound of seven on the chromatic number follows from the existence of a tessellation of the plane by regular hexagons, with diameter slightly less than one, that can be assigned seven colors in a repeating pattern to form a 7-coloring of the plane.