Why Archimedes Plutonium will never proof the 4CT


Archimedes Plutonium [AP] has posted two alleged proofs of the 4CT
which are short but not correct. There are two basic facts that he is
repeatedly ignoring, and since the threads containing those proofs have
turned into many-headed hydrae, and since AP doesn't like reading long
posts, my points are lost. So I decided to start a new thread.

Here are the two basic facts that AP is ignoring:

(1) If you can color every triangulated map with 4 colors, that does
not mean you can color any map with 4 colors.

The proof of this fact should be obvious. AP claims that if there is a
country with more than 4 boundary points, then that country will be
triangulated. Now, when the new map is colored, the country that was
triangulated will be colored with at least 2 colors. This is not a
coloring of the original map, where that country must be colored the
same throughout.

In short, AP is not solving the problem for the original map, which is
what the 4CT requires.

(2) If the 4CT is false, that does not immediately imply that there are
5 countries which are adjacent to each other.

(The discussion here has been muddied by several posters who say that
since the 4CT is true, that any statement of the form "If the 4CT is
false, then ..." is true. Here, we are not assuming that the 4CT is
true or false.)

The proof of this fact is simply the exhibition of another
configuration (a set of countries and how they're adjacent) which
requires 5 colors. I have referred AP to the Mycielski Graph page at
MathWorld, which is nice in that it actually has a picture.

http://mathworld.wolfram.com/MycielskiGraph.html

Now, in this particular picture, the red dots correspond to countries,
one dot per country, and a blue (black?) line between two dots means
the two countries are adjacent.

For example, if the map was a pie chart which consisted of 5 slices,
each slice would be represented by a single dot, and each dot would be
connected to 2 others. The resulting graph is called a pentagon, and is
in the middle of the image on the Mycielski Graph page.

BTW, the error in ignoring (2) is called "arguing from ignorance",
which is basically, "I can't imagine any other possibility, so my
possibility must be true". The same error is present, in the same form,
in the following "proof":

"THEOREM" There is no real number x such that x^2 = 9.

"Proof". If there is a real number x such that x^2 = 9, then x = 1. But
1^2 is not 9. Therefore, the theorem holds.

***

BTW, a list of countries and a list of which countries are adjacent to
which is all that is needed in order to color a map. The following
scenario should make it clear, even to AP:

"The leaders of the various countries got together and decided they
would like to have high-speed roads ("highways") connecting the capital
cities of their countries. To reduce the cost and eliminate routing
conflicts, it was decided that two capitals would have a highway
between them only if those two countries were adjacent (but not at a
point), and that the route between the two capitals should pass through
their common boundary."

Now the capitals together with these roads is called a graph, and is
sometimes also called a network. If the "nodes" of the networks are
colored (which are the capitals) so that two capitals with a road
between them receive different colors, then the map can be colored so
that adjacent countries get different colors. It should be obvious how
to do this: Just color a country with the same color that its capital
is colored.

This tranforms map coloring into "vertex coloring". After all, a
coloring of the USA only requires that you assign colors to Alabama,
Arkansas, Arizona, etc., so that

California and Arizona receive different colors, and
California and Nevada receive different colors, and
Arizona and Nevada receive different colors, and
.... (more conditions)

which is what the edges in the graph are representing.

Now the complaint may be made that some graphs cannot come from maps.
There actually is a property of graphs that can be checked, which will
determine whether this is so. It is called Kuratowski's Theorem, and
says that a graph comes from some map if and only if it does not
"contain" (in a well-defined way) K5 or K(3,3). So the 4CT can, and is,
studied without needing a map.

--- Christopher Heckman

.

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