From: cet1@cus.cam.ac.uk (Chris Thompson)
Newsgroups: sci.math
Subject: Re: On coloring the real plane.
Date: 27 Aug 1995 00:30:16 GMT
In article <1995Aug26.144241.13777@jarvis.cs.toronto.edu>, fpitt@cs.toronto.edu
(Francois Pitt) writes:
|> This is something that I posted to `rec.puzzles'. Someone there
|> (sorry, I can't remember who) told me it was an unsolved problem, and
|> someone else suggested that I post it here. I already checked in the
|> FAQ and I didn't see this, so here goes!
|>
|> Say you want to assign a color to every point in the real plane (i.e.,
|> every pair (x,y) where x and y are real numbers), in such a way that the
|> following property holds:
|>
|> For every two points a and b, if the distance between a and b
|> is equal to 1, then a and b are _not_ the same color.
|>
|> There is an elegant proof that this cannot be done with 3 colors or
|> less, and I have found a simple coloring using 7 colors. The question
|> now is: what is the mininum number of colors required? (In other words,
|> can this be done with 4, 5, 6 colors or not?)
For a review of the status (as of 1990) of this and several related problems,
check section G10 in [1]. Specifically
Nothing better than 4 <= #colours <= 7 is known unless some
restrictions are placed on the sets of points with a fixed colour.
If these sets are required to be Lesbesgue measurable, then
Falconer has proved #colours >= 5
If the sets are unions of regions bounded by rectifiable Jordan curves
(with the points on the boundaries assigned to some neighbouring
region) then Woodall has proved #colours >= 6
[1] Hallard T. Croft, Kenneth J. Falconer, Richard K. Guy
Unsolved Problem In Geometry
Springer-Verlag (1991)
ISBN 0-387-97506-3
Chris Thompson
Email: cet1@cus.cam.ac.uk