Ramsey theorems for the naturals and for general infinite sets In reverse mathematics and in recursion theory, the infinite Ramsey theorems are usually stated in terms of coloring of $[\Bbb N]^n$. How do these (not) imply the Ramsey theorems for general infinite sets $X \subseteq \Bbb N$, in the metatheory, over RCA$_0$, or computability-theoretically? A first try is to extend a coloring of $[X]^n$ to a coloring of $[\Bbb N]^n$ so that one can get an infinite homogeneous set for the former from the one for the latter. I believe that, in order to do this, I should be careful not to introduce additional structures in the coloring of $\Bbb N \setminus X$. But this is precisely what the Ramsey's theorem state is impossible, if $X$ is coinfinite!

Just use the order-preserving bijection $f:X\to\mathbb N$ to define a coloring of $[\mathbb N]^n$ which is isomorphic to the given coloring of $[X]^n.$ Namely, for $s\in[\mathbb N]^n,$ give $s$ the color originally assigned to $f^{-1}(s).$