About three days ago, I got stuck when reading a paper by the great combinatorist, Rado. So I posted the question on sci.math and on stackexchange, although it's only been on stack exchange about 36 hours. No one responded on stack exchange. On sci.math, fom was very kind and spent a lot of time and effort trying to resolve the issue. I am genuinely appreciative of fom's efforts. However, despite fom's help and support, I am absolutely no nearer to resolving this point than I was when I began this thread.
Is it possible for anyone else to help? It's a very elementary point and could be trivially resolved by David Ullrich, quasi, Fred etc. (Just naming 3 active and well-informed posters at random here.)
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.