Hi Warren, 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. Cris On Apr 3, 2013, at 6:51 PM, Warren D Smith wrote:
Here's an interesting possible attack on this problem. Let N be an integer with a ton of representations of the form a^2+b^2-ab (equivalent form: u^2+3*v^2). For example N=7*13*19*31*37 works well, (product of K primes that are each 1 mod 3 will have a number T of reps exponential in K).
Now within the set of vertices of the equilateral triangle lattice of sidelength=1, consider the distance=sqrt(N) graph. This graph is countably infinite and regular with (arbitrarily large) valency T. Now ask your computer: what is the chromatic number of some finite subgraph of it? That will be a lower bound on the chromatic number of the plane distance=1 graph.
The analogous attack can be done using any underlying 2-variable integer quadratic form (not necessarily a^2+b^2-ab), and which quadratic form and which N would yield the best results, I do not know. Because of the large valency it seems plausible you might get a large chromatic number. Unfortunately the specific N just suggested would yield a rather large graph... your poor computer would be laboring trying to find its chromatic number...
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