This is an elaboration of a comment on Suvrit's answer.

Ramsey numbers can be defined for (infinite) ordinals, just as in the finite case: $r(\alpha,\beta)$ is the least $\gamma$ such that for any $2$-coloring of the edges of the complete graph on $\gamma$ vertices there is a set of vertices of type $\alpha$ whose induced graph is red, or a set of vertices of type $\beta$ whose induced graph is blue.

Ramsey's theorem gives that $r(\omega,\omega)=\omega$, but already $r(\omega+1,\omega)=\omega_1$. On the other hand, if $\alpha\lt\omega_1$ and $n$ is finite, then $r(\alpha,n)\lt\omega_1$, and for reasonably small infinite values of $\alpha$, one can attempt to compute $r(\alpha,n)$ explicitly. It turns out that this computation reduces to (Ramsey-theoretic) finite problems, which, just as with the classic computation of finite Ramsey numbers, quickly become unfeasible.

For example:

- $r(\omega+3,3)=\omega\cdot2 + 8$. In general, if $0\lt n,m\lt\omega$, then
$$ r(\omega+n,m)=\omega\cdot(m-1)+(g(n,m)-(m-1)), $$
where $g(n,m)$ is the least $k$ such that any $2$-coloring of the edges of the complete graph on set of vertices $\{1,\dots,k\}$ such that the induced graph on $C=\{1,\dots,m-1\}$ is blue, either admits a blue $K_m$, or a red $K_{n+1}$ with one of its vertices in $C$.

This was first established by Haddad and Sabbagh in 1969. One has $r(n+1,m)\le g(n,m)\lt\infty$, but typically the first inequality is strict. For example, $r(4,3)=9$ but $g(3,3)=10$. In general, computing $g(n,m)$ is similar to, but harder than computing $r(n+1,m)$.

- $r(\omega\cdot3,3)=\omega\cdot9$. In general, if $0\lt n,m\lt\omega$, then
$$ r(\omega\cdot n,m)=\omega\cdot l(n,m), $$
where $l(n,m)$ is the least $k$ such that any $2$-coloring of the edges of the complete digraph on $k$ vertices either contains a red complete digraph on $n$ vertices, or a blue transitive tournament on $m$ vertices.

Here, in complete digraphs we have two arrows (going in opposite directions) between any two distinct vertices. This was shown by Erdős and Rado in 1955. As with $g$, the computation of the values of $l(n,m)$ quickly becomes unfeasible.

- $r(\omega^2\cdot2,3)=\omega^2\cdot10$. In general, if $0\lt n,m\lt\omega$, then $r(\omega^2\cdot m,n)=\omega^2\cdot h(m,n)$ for a Ramsey-theoretic function $h$ related to $3$-colorings of the edges of digraphs, though its exact description is somewhat technical to include here. This was shown fairly recently by Thilo Weinert, see here.