Multi-Dimensional Four-color Theorem

Date: 08/08/97 at 10:46:53
From: Kenny Jensen
Subject: Multi-dimensional Four-color Theorem

I was wondering if any work had been done on theorems like the four-
color theorem for different dimensions.  It seems to me that a more 
general theorem could be made such as:

The number of colors needed to color an n-dimensional map is 2^n.

This seems to work for points, lines, and planes.

If there has been any work on this subject could you please point me 
to the sources?

Date: 08/11/97 at 14:24:25
From: Doctor Ceeks
Subject: Re: Multi-dimensional Four-color Theorem

Hi,

This is a very nice question. It's a very good idea to try to
generalize theorems in ways as natural as the one you are suggesting.

In this case, however, once you go to three dimensions, you can make 
partitions of space into regions for which you need N colors
to color the regions in order that no two adjacent regions will have 
the same color for any N.

You can make an example by starting with one ball.

Now, add a ball to the picture and connect it with a thin tube to
the first ball.

Now, add a third ball to the picture and connect this ball with two
thin tubes to the two balls already in the picture.

You can keep adding balls and connecting them to all the other balls
like this because there is enough space in three dimensions to work 
with.

If the balls represent regions, since each ball is touching every 
other ball, you need at least as many colors as there are balls to 
color them.

Do you understand what I'm saying?

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 08/11/97 at 18:41:45
From: Anonymous
Subject: Re: Multi-dimensional Four-color Theorem

Hello,

Thanks for the explanation. It helped me visualize the question better 
than I had previously.