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From: eklhad@ihnet.UUCP (K. A. Dahlke)
Newsgroups: net.math
Subject: coloring the 3-dimensional map
Message-ID: <151@ihnet.UUCP>
Date: Mon, 20-Aug-84 09:31:28 EDT
Article-I.D.: ihnet.151
Posted: Mon Aug 20 09:31:28 1984
Date-Received: Tue, 21-Aug-84 00:14:27 EDT
Organization: AT&T Bell Labs, Naperville, IL
Lines: 47

Thanks for the many responses to the 3-D map coloring problem.
Everyone responded correctly, I guess it was easy.
Some people were more formal than others, ranging from
"globs can grow tenticles that reach around touching all other globs"  to
"the general point-line graph is isomorphic to 3-dimensional regional maps"
Above quotes are not exact.
Here is my solution.

There is no limit to the number of colors that might be required by a
3-dimensional map.
I mentioned n dimensions just to confuse you.
I shall construct a map which requires an infinite number of colors.
Note: i mean countably infinite, since n-dimensional space cannot be
partitioned into an uncountably infinite number of disjoint non-zero
volume regions.
Each glob consists of two connected rectangular solids.
These rectangular bars are one square unit in cross-section,
and infinitely long.
Thinking in an x-y-z coordinant system,  piece J (J=-inf,+inf) is:
0<=z<=1,J-1<=x<=J,-inf= n+2.
Certainly for 3 dimensions we can go much higher than 5 colors.
I believe i have created a 12 color map.
Can anyone beat this, or prove 12 is maximum.
I will be on vacation for a while, but send responses,
I will get to them.
enjoy!!
-- 

Karl Dahlke    ihnp4!ihnet!eklhad