A question regarding $ZFC^{-}$ Given $ZFC^{-}$, that is, ZFC-Powerset+Collection+Separation, is there a set of alternative axioms $X$ (other than the trivial one, namely, {Powerset}) that, when added to $ZFC^{-}$, allow one to derive Powerset as a theorem of $ZFC^{-}$+$X$ and recover full $ZFC$? (Thanks to Prof. Hamkins for setting me straight on the correct formulation of $ZFC^{-}$.) 
 A: By compactness, if a sentence $\sigma$ is equivalent to the powerset axiom over $\mathrm{ZFC}^-$, then this equivalence is provable in Kripke-Platek set theory ($\mathrm{KP}$) with infinity, $\Sigma_n$-separation and $\Sigma_n$-collection for some $n$. Due to the connection between $\mathrm{KP}$ and admissible ordinals, a good place to look for this is in the $\alpha$-recursion theory literature. The only thing that came to mind were results by Evangelos Kranakis on partition relations in $\alpha$-recursion theory. I don't recall the specifics of Kranakis's work offhand, but that thought did lead me to an example.
Consider the following abstract extenson of Ramsey's Theorem to infinite cardinals.

For every cardinal $\kappa$ there is a cardinal $\lambda$ such that $\lambda\to(3)^2_\kappa$.

In $\mathrm{ZFC}$, one can show that $\lambda = (2^\kappa)^+$ works. In $\mathrm{ZFC}^-$, one can show that if $\lambda \leq 2^\kappa$ then $\lambda\not\to(3)^2_\kappa$. Thus, if there is a $\lambda$ such that $\lambda\to(3)^2_\kappa$ then $\lambda$ must be so large that $2^\kappa \lt \lambda$.
