I'll ask my question and then explain the background. Thanks in advance.
Question: Is it the case that $$ \beth_{\omega}\rightarrow (\beth_{\omega})_{{\aleph_{0}}}^{n} $$ for finite $n$? If not, what more generally can be said about cardinals $\kappa$ such that $$ \kappa\rightarrow (\kappa)_{{\aleph_{0}}}^{n} $$ holds, for finite $n$?
Background:
Wikipedia succinctly states the Erdős–Rado theorem as follows: $$ \beth_{n}^{+}\rightarrow (\aleph_{1})_{{\aleph_{0}}}^{{n+1}} $$ In words and paraphrasing slightly this says: if we countably colour the $n{+}1$-size subsets of a set $X$ of cardinality greater than $\beth_{n}$, then there exists a homogeneous subset of uncountable cardinality.
Then it re-states a more general form: $$ \exp_{n}(\kappa )^{+}\longrightarrow (\kappa^{+})_{\kappa }^{{n+1}} $$ An $\aleph_0$-colouring is also a $\kappa$-colouring for $\kappa\geq\aleph_0$, and $\beth_\omega > \exp_n(\beth_i)$ for finite $i$, so from this it seems to me that we can derive for every finite $i$ that $$ \beth_{\omega}\longrightarrow (\beth_i^{+})_{\aleph_0}^{{n+1}} . $$ This suggests that we might try to take a limit as $i$ rises towards $\omega$ --- and we arrive at my question.
Advice welcome, thank you.