
Ramsey's Theorem shows complete disorder is impossible - ColinWright
http://www.math.grin.edu/~miletijo/museum/ramsey.html
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dmvaldman
This is a sensationalist headline. Ramsey's theorem concerns patterns in a
simple system, the coloring of edges in a graph. It says regardless of how the
graph is colored you can always find a triangle of vertices with certain
properties.

This does not mean complete disorder is impossible. It's just a combinatorial
argument that says this simple game of coloring edges on a graph has some
patterns to it.

But then maybe you're of the mindset that the universe is a complete graph and
physics is a color by number game...in which case....oh god, we're all
screwed...

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ColinWright
There is an infinite version of Ramsey's theorem.

Suppose you consider the collection of pairs of positive whole numbers, and
each pair is colored either red or blue. Then there is an infinite set S of
positive whole numbers such that any pair of items from S are the same color.

In symbols:

P = { (a,b) : a in N, b in N }

C : P -> {0,1}

There exists S an infinite subset of N and c in {0,1} such that for all a in S
and b in S, C(a,b)=c.

The proof is quite simple, I use it regularly to boggle 13 and 14 year olds.

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btilly
I figured out the proof on my drive in to work. It is simple.

At all points we're going to have an infinite set of integers to process, a
clique of integers whose connections to each other and all of the integers
left to process is red, and a clique of integers whose connections to each
other and all of the integers left to process is blue. (red clique, blue
clique)

We start with all of N as our set, and our cliques are empty sets.

At each point we take the first thing in our set, and remove it from the set.
If there are an infinite number of things left in our set that it has a red
connection to, then put it into the red clique and remove all things from the
set that its connection is blue to. Else put it into the blue clique and
remove all things from the set that its connection is red to. This step can
always be done and leaves us with 2 cliques (one of which grew by one) and an
infinite set of integers to process.

After we do this an infinite number of times at least one of the red and blue
cliques must now be infinite in size. And we're done.

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ColinWright
That's an elegant way to express it - thank you. My method has more of the "TA
DA!" about it, and tends to generate a sense of "Gosh!", but I like the quiet
elegance of yours.

Nice one.

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jgorham
I don't know if Ramsey's theorem really says anything as grand about disorder
as Maxwell's demon and the like, but it does express something quantitative
about the existence of subsets containing some specific properties as the size
of the graph increases.

For me, Ramsey's theory is particularly interesting its rare that an open
problem in mathematics can be explained to anyone with relatively little
background in mathematics or logic for that matter. Mathematicians have
discovered some bounds on R(n), but an exact solution looks like its no where
in sight.

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rupak
Rainbow Ramsey's theorem shows that complete disorder is unavoidable as well.
:)

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powertower
In other words, when complete disorder is achieved, complete order is formed
(since that disorder becomes completly homogeneous).

~~~
saurabh
“You know the thing about chaos? It's fair.” - The Joker

