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Topic: Problem understanding Rado's proof of the canonical Ramsey theorem
Replies: 39   Last Post: Nov 4, 2013 4:15 AM

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 Paul Posts: 780 Registered: 7/12/10
Problem understanding Rado's proof of the canonical Ramsey theorem
Posted: Oct 31, 2013 6:41 PM

I am having trouble understanding the paper with the URL: http://www.cs.umd.edu/~gasarch/TOPICS/canramsey/Rado.pdf

I get stuck around the middle of page 2 where it says:
f(z0, ..., z_r-1) = f(y0,..., y_r-1)

This assertion doesn't seem to follow from the quantifiers defining L.
I do see that there exists _some_ y0, y1,... and _some_ y0', y1' , ...

to make the above equality true but that's not enough because here the yi and yi' are arbitrary.

We are given that rho_0 does not belong to L. However, L is defined by a "for all" statement. So, for rho_0, the for-all statement is false and we can find some yi and yi' to make f(z0, ..., z_r-1) = f(y0,..., y_r-1) true.

But the author is stating something much stronger -- that we can deduce the equality for an arbitrary yi and yi'.

My ultimate goal is to understand _any_ proof of the Canonical Ramsey theorem. (I'm limiting my search to free web sources for now). The original Erdos/Rado paper does seem somewhat convoluted, which is presumably why Rado felt a need to rewrite the proof. Imre Leader proves it too. However, he only details a simple case, and leaves the rest to the reader.

Many thanks for any help or insights.

Paul Epstein

Date Subject Author
10/31/13 Paul
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