In message <email@example.com>, Paul <firstname.lastname@example.org> writes >No, B' is a subset of A which is not countable, in general.
Third line of Rado's paper: "Let A = (0,1,2,...}"
As far as I know (which isn't far) Ramsey's theorem only applies to countable sets so Rado's proof will only work with countable A.
>Your initial point about Erdos-Rado stating that they avoid AC is a >repetition of my point. In summary, the Erdos-Rado result proves >Canonical Ramsey without AC whereas Rado deduces Canonical Ramsey from >AC.
I don't agree. Both papers prove the Canonical Theorem from the standard Ramsey theorem without any further use of AC. As long as Ramsey's theorem can be proved without AC, so can the Canonical theorem.