Finite Ramsey theorem: $ \def\nn{\mathbb{N}} $
For any $e,k,r \in \nn$, there exists a least natural number $m=R(e,r,k)$ so that, for any set $M$ with cardinality at least $m$, with each of the $e$-sets of $M$ coloured with one of $r$ colours, there exists a subset $H$ of $M$ with cardinality $k$ so that all $e$-sets of $H$ are coloured with the same colour. (Here an $e$-set of $M$ is a subset of $M$ of size $e$.)
Infinite Ramsey theorem:
For any $e,r \in \nn$, and any infinite set $M$, with each of the $e$-sets of $M$ coloured with one of $r$ colours, there exists an infinite subset $H$ of $M$ so that all $e$-sets of $H$ are coloured with the same colour.
I read that the finite Ramsey theorem can be expressed in PA. I don't understand why, because the universal and existence quantifier are used over sets, not just over variables (not appropriate for a first-order theory). I have read somewhere that $R$ as above is a recursive function and those functions are representable in PA with a formula -- maybe that has something to do with it (but my professor thinks otherwise).
I am very sorry for not being precise, because I read about this a long time ago and I don't remember the source, but I have also read (if I remember correctly) that the infinite Ramsey theorem cannot even be expressed in PA (not just that it cannot be proved).
Is it that the notion of an 'arbitrary infinite set' is problematic? I don't know generally how the notion of an 'arbitrary finite set' can be expressed in PA. I know about coding of the finite sets of natural numbers but I am not sure that enables the fact that the finite Ramsey theorem can be expressed in the language of PA.