In message <BWpz$uCS15fSFw6Q@212648.invalid>, David Hartley <firstname.lastname@example.org> writes >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.
That was a bit silly of me. For any A, if Ramsey's theorem applies to an infinite subset B of A then it applies to A. So, if you're worrying about choice, it is certainly true for at least any set with a denumerable subset.
The statement and proof of Ramsey's theorem in the Erdos-Rado paper apply only to sets of natural numbers. They can easily be extended to any countably infinite set but not obviously to an infinite set with no denumerable subset. So if you want the Canonical theorem to apply to all infinite sets, neither proof works without at least a weak form of choice.
I suspect that Rado's proof can be modified to show that if Ramsey's theorem extended to all infinite sets holds in ZF, then so does the extended canonical theorem, but I haven't checked it through carefully. -- David Hartley