This a specific question about Ramsey type colorings.
The arrow notation: If $\kappa$, $\lambda$, $\mu$ are cardinals and $n<\omega$, then $\kappa\rightarrow(\lambda)^n_\mu$ if for any function $f:[\kappa]^n\longrightarrow\mu$ (where $[\kappa]^n$ is the set of subsets of $\kappa$ of size $n$), there is a subset $A\subseteq\kappa$ of size $\lambda$ such that $f$ is constant on $[A]^n$.
I am aware of the traditional Ramsey's Theorem, which talks about this arrow notation for countable cardinals; as well as the Erdős-Rado theorem, which says $(\beth_n(\kappa))^+\rightarrow(\kappa^+)^{n+1}_{\kappa}$ for any infinite $\kappa$.
My questions are:
For what infinite cardinals $\kappa$ is it true that $\kappa^+\rightarrow(\omega)^2_\kappa$
For what infinite cardinals $\kappa$ is it true that $2^\kappa\rightarrow(\omega)^2_\kappa$