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Topic: 4 colors problem
Replies: 86   Last Post: Mar 13, 2014 4:36 PM

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magidin@math.berkeley.edu

Posts: 11,116
Registered: 12/4/04
Re: 4 colors problem
Posted: Mar 7, 2014 11:19 AM
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On Friday, March 7, 2014 12:04:09 AM UTC-6, swtch...@gmail.com wrote:
> That's a very good point, thank you.
>
> The theorem does not guarantee a pattern or an algorithm.
>
> Only that there are 4 colors that exist to fill the map.
>
> Then what are the approaches or methods that are used to
>
> actually prove the theorem via supercomputers and mathematical
>
> equations? How do they do it?


On the assumption that you will actually listen and on the hope that you will stop your incessant blathering on this:

The proof of the 4-color map theorem was done by contradiction, as follows:

1) Assume that there is at least one (finite) map that cannot be 4-colored. Then there must exist a *smallest* (finite) map that cannot be 4-colored. We will show that given such a map, it is possible to find a strictly smaller map that can also not be 4-colored, giving a contradiction.

2) Prove that there is a finite family of "unavoidable configurations" that any (finite) map must have; these are a family of maps (in fact, graphs since the map can be represented by a planar graph) such that any given map will contain this map "inside it."

3) Construct this finite family. Then show that each element of the family is "reducible": given a map M (graph) that contains an element A of the family, we can replace A with a strictly smaller submap with the property that the resulting smaller map M' can be 4-colored if and only if the original map M can be 4-colored.

4) Conclude that no "smallest counterexample" can exist, thereby proving the theorem.

Now, please stop your nonsensical endless blathering.

--
Arturo Magidin


Date Subject Author
3/3/14
Read 4 colors problem
stumblin' in
3/3/14
Read Re: 4 colors problem
Brian Q. Hutchings
3/3/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
g.resta@iit.cnr.it
3/3/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
g.resta@iit.cnr.it
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
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3/4/14
Read Re: 4 colors problem
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3/4/14
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3/4/14
Read Re: 4 colors problem
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3/4/14
Read Re: 4 colors problem
g.resta@iit.cnr.it
3/4/14
Read Re: 4 colors problem
magidin@math.berkeley.edu
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
Port563
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
Brian Q. Hutchings
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
Port563
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
Brian Q. Hutchings
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read there should be a noncomputer proof of 4CT, I agree
Brian Q. Hutchings
3/4/14
Read Re: 4 colors problem
g.resta@iit.cnr.it
3/4/14
Read Re: 4 colors problem
stumblin' in
3/4/14
Read Re: 4 colors problem
Port563
3/4/14
Read Re: 4 colors problem
stumblin' in
3/5/14
Read Re: 4 colors problem
stumblin' in
3/5/14
Read Re: 4 colors problem
magidin@math.berkeley.edu
3/5/14
Read Re: 4 colors problem
stumblin' in
3/5/14
Read Re: 4 colors problem
magidin@math.berkeley.edu
3/5/14
Read Re: 4 colors problem
quasi
3/5/14
Read Re: 4 colors problem
stumblin' in
3/5/14
Read Re: 4 colors problem
magidin@math.berkeley.edu
3/5/14
Read Re: 4 colors problem
stumblin' in
3/5/14
Read Re: 4 colors problem
quasi
3/5/14
Read Re: 4 colors problem
magidin@math.berkeley.edu
3/5/14
Read also, what is the spatial analog
Brian Q. Hutchings
3/5/14
Read Re: 4 colors problem
stumblin' in
3/5/14
Read Re: 4 colors problem
magidin@math.berkeley.edu
3/5/14
Read Re: 4 colors problem
stumblin' in
3/5/14
Read Re: 4 colors problem
Brian Q. Hutchings
3/5/14
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Virgil
3/5/14
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3/5/14
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3/5/14
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3/5/14
Read Re: 4 colors problem
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3/5/14
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3/5/14
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3/5/14
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Virgil
3/5/14
Read Re: 4 colors problem
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3/6/14
Read Re: 4 colors problem
Virgil
3/6/14
Read Re: 4 colors problem
Virgil
3/6/14
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Brian Q. Hutchings
3/6/14
Read Re: 4 colors problem
stumblin' in
3/6/14
Read Re: 4 colors problem
ross.finlayson@gmail.com
3/7/14
Read Re: 4 colors problem
Brian Q. Hutchings
3/7/14
Read Re: 4 colours problem
Robin Chapman
3/6/14
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3/6/14
Read Re: 4 colors problem
stumblin' in
3/7/14
Read Re: 4 colors problem
magidin@math.berkeley.edu
3/7/14
Read Re: 4 colors problem
Peter Percival
3/7/14
Read Re: 4 colors problem
Peter Percival
3/6/14
Read Re: 4 colors problem
stumblin' in
3/7/14
Read Re: 4 colors problem
stumblin' in
3/7/14
Read Re: 4 colors problem
magidin@math.berkeley.edu
3/7/14
Read Re: 4 colors problem
Peter Percival
3/7/14
Read Re: 4 colors problem
Peter Percival
3/7/14
Read Re: 4 colors problem
stumblin' in
3/7/14
Read Re: 4 colors problem
stumblin' in
3/9/14
Read Re: 4 colors problem
stumblin' in
3/9/14
Read Re: 4 colors problem
Peter Percival
3/9/14
Read Re: 4 colors problem
stumblin' in
3/9/14
Read Re: 4 colors problem
magidin@math.berkeley.edu
3/9/14
Read Re: 4 colors problem
Brian Q. Hutchings
3/9/14
Read Re: 4 colors problem
stumblin' in
3/9/14
Read Re: 4 colors problem
stumblin' in
3/11/14
Read Re: 4 colors problem
stumblin' in
3/11/14
Read Re: 4 colors problem
Brian Q. Hutchings
3/13/14
Read Re: 4 colors problem
stumblin' in
3/13/14
Read Re: 4 colors problem
Brian Q. Hutchings
3/13/14
Read Re: 4 colors problem
stumblin' in

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