Date: Tue, 19 Apr 2005 10:41:01 -0400

From: Bill Thurston <wpt4@cornell.edu> Subject: Re: [math-fun] Density question .........

What's known about the chromatic number of the plane? I'm curious whether enough is known to be able to prove there can't be a measurable best coloring.

A google search came up with http://planetmath.org/encyclopedia/ChromaticNumberOfASpace.html which has a reference to the paper below, which proves that if a 4-colouring exists then one of the monochromatic sets is not Lebesgue measurable. K. J. Falconer. The realization of distances in measurable subsets covering R^n J Comb. Theory A 31, 184-189, 1981. Gary McGuire