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.