From: Cris Moore <moore@santafe.edu> I was thinking along similar lines. Consider the triangular lattice, and consider all the distances you can get between pairs of lattice points, but not between two points in the same sublattice (corresponding to the same color in a 3-coloring of the lattice). These distances separate pairs of points that we can force to be different colors. Now, suppose there are two distances in this set with ratio phi (the golden ratio). I think actually there are no such pairs, which is a nice problem in algebra... but if there were, then you could build a K_5, which would give a lower bound of 5 on the chromatic number of the plane.
--well, in spirit you may have a good idea (I do not know and feel confused)... but your idea in detail is a bust: Every distance between two points of the eq.tri. lattice is sqrt(integer) and obviously the ratio of two such, cannot be phi, since it must be sqrt(rational) and phi^2=phi+1=irrational.