The Ramsey number $R(m,n)$ is easy to describe. It's the smallest positive integer such that any graph with at least $R(m,n)$ vertices has at least a clique of size $m$ or an independent set of size $n$. Generalized Ramsey numbers are defined like
Let $G_1,G_2,\dots,G_m$ be simple graphs. The generalised Ramsey Number $r(G_1,G_2,\dots,G_m)$ is the smallest integer $n$ such that every $m$-edge colouring $(E_1,E_2,\dots,E_m)$ of $K_n$ contains, for some $i$, a subgraph isomorphic to $G_i$ in colour $i$.
As far as I'm aware, probabilistc arguments have been used to show the existence of these numbers. But what happens if we take this concept a step further?
Let $\mathcal G = \{G_\gamma\}_{\gamma<\alpha}$ be a family of (possibly infinite multi)graphs induced by a cardinal $\alpha$. Let $r(\mathcal G)$ be the smallest cardinal $\beta$ such that every $\alpha$-edge coloring $\{E_\gamma\}_{\gamma<\alpha}$ of $K_\beta$ contains, for some $\gamma$, a subgraph isomorphic to $G_\gamma$ in color $\gamma$.
Does $r(\mathcal G)$ always exist and has it been studied in such a generalized setting? Searches return work more or less exclusively related to cardinals, removing the graph theory. But maybe this is just a natural generalization I cannot yet grasp?