Re: the imprecision of 4 color mapping and why it should be 2 color mapping and why FLT is also imprecise and thus false Re: E.E.E. also claims he disproved or neutralized Wiles proof!

From: Tim Mellor (timm_at_amsta.leeds.ac.uk)
Date: 09/11/04 

Date: 11 Sep 2004 10:43:15 -0700

> "Archimedes Plutonium" <a_plutonium@iw.net> wrote in message
> news:4142AACC.76116DC0@iw.net...
> <snip horrifically formatted introduction>
>
> > It is sad that mathematicians today act more like politicians rather
> > than scientists. I say this because if I ask *** Winter or Andrew Wiles
> > or most every other mathematician, ask them why they cannot sit down and
> > state that the formulation of 4 Color Mapping is wrong and imprecise
> > because it assumes the borderline is no color at all when we all know
> > the borderline is a Black color and thus 5 colors are involved.

I'm sure I've done this before for you, but here we go again. Below, I
preceed the terms I am defining with \em

Firstly however, note that the following setup is equivalent to the
rather more convenient setup of planar graphs that you would find in
the litterature. If you looked...

Defn.
1. Let n be a natural number. Let {X_i:i<n} be a set of mutually
disjoint open connected two dimentional subsets of R^2. Suppose that
R^2 \ union of {X_i:i<n} is of dimention 1. Then we say the pair
(n,{X_i,i<n}) is a \em map.

(intuativela, a map...)

2. Given a map (n,{X_i,i<n}), and i<j<n we say that X_i is \em
adjacent to X_j iff there exists some Y_1,Y_2 connected one
dimentional subsets of R^2 with the following properties:

Firstly, for each k in {1,2}
a) Y_k intersects both X_i and X_j, and
b) |Y_k \ (X_i union X_j)|=1

Also Y_k \ (X_i union X_j) differs from Y_2 \ (X_i union X_j)

(intuatively, two areas of the map are adjacent iff they share a
boundry of more than one point)

3. Given a map (n,{X_i,i<n}), and a natural number m, a \em nice
colouring of (n,{X_i,i<n}) to {1,2,...,m} is a function f:{X_i:i<n} ->
{1,2,...,m} with the following property:

For each i<j<n, if X_i is adjacent to X_j, then f(X_i) does not equal
f(X_j).

(we have to colour adjacent maps differently)

Theorem [Appel and Haken 1977]
Given any map (n,{X_i,i<n}), there exists a nice colouring to
{1,2,3,4}.