
A Simple Visual Proof of a Powerful Idea in Graph Theory - dnetesn
http://nautil.us/blog/a-simple-visual-proof-of-a-powerful-idea-in-graph-theory
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kleiba
Strictly speaking this "visual proof" is not a proof at all, it's just an
illustration of one simple case. But what's good about is that it helps give
you an intuition about why Ramsey's theorem is true: namely, that you'd
eventually run into a dead-end when trying to color edges, in the sense that
you're bound to create a sub-graph where all edges are either colored red or
blue.

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air7
I remember hearing about a social scientist "discovering" that in any
classroom he visited, there always seemed to be 4 kids that were either all
friends, or were all not friends. He wrote a paper that attempted to explain
this using social theory/group dynamics...

(This was told by my Discrete math professor when learning about Ramsey's
Theorem )

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DennisP
The article says this is a useful tool, but doesn't say why. What's an example
of something this is used for?

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nandushines
"Ramsey’s theorem states that in any graph where all points are connected by
either red lines or blue lines, you’re guaranteed to have a _large subset_ of
the graph that is completely uniform—that is, either all red or all blue."

Can someone please explain how a large subset is defined? I am new to Graph
Theory.

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Sniffnoy
"Large subset" is not a technical term. The right question here is not "how is
large subset" defined; it's "What does Ramsey's theorem actually state?"

A totally abstract version of the theorem would be (note: I'm only considering
finite versions of Ramsey's theorem here), Ramsey's theorem states that given
any n and m, there's a number R(n,m) such that if you have a complete graph on
R(n,m) vertices, and you color each edge red or blue, there's either a
uniformly red subset of n vertices or a uniformly blue subset of m vertices.

A more useful version would give an example of a number you could pick for
R(n,m). (Normally we define R(n,m) to be the smallest such number, so you're
giving an upper bound on it.) For instance, Erdos and Szekeres proved that
R(n,m)<=(r+s-2 choose r-1).

