TITLE: Using a forcing extension $V[G]$ to determine properties of $V$.
QUESTION [5 upvotes]: It is well known that you can use forcing to change the truth value of various sentences ($CH$, $\Diamond$, et cetera). However, often when performing such a construction over a model $V$, the action is generally in $V[G]$ (or some inner model thereof - as when we violate $AC$ in $HOD(x)^{V[G]}$); we want to figure out what goes on in $V[G]$ for the purposes of, say, a relative consistency proof.
There are constructions, however, that also tell us about what goes on in $V$ (possibly in relation to $V[G]$), as when we consider embeddings of the form $j: V \longrightarrow M \subseteq V[G]$. Such embeddings can have quite small critical points (even as small as $\omega_1$), giving us a new perspective on the small uncountable sets of our original model $V$.
My question:
Are there other forcing constructions (i.e. not generic embeddings) that tell us what the structure of our original model $V$ is like with respect to uncountable sets (so not in the trivial sense of showing the consistency of proof-codes in $V_\omega$) and how $V$ relates to $V[G]$? To put it another way, are there other constructions that allow us to see the structure of $V$ more clearly from $V[G]$?

REPLY [4 votes]: My favorite family of examples come from the partition calculus. The original proof of the Baumgartner-Hajnal theorem $\omega_1\to (\alpha)^2_n$ for all finite $ n $ and all countable $\alpha $ appealed to the absoluteness of well-foundedness. The homogeneous set was found in an extension where $\mathsf{MA} $ holds, and that means that a certain ground-model tree is illfounded in the extension.
This idea has been used in a variety of similar results: In Schipperus's proof of the topological version of the Baumgartner-Hajnal theorem, in Todorcevic's proof of the extension of the BH result to non-special posets, in several of the partial results towards the conjecture $\omega_1\to (\alpha, n)^3$ for all countable $ \alpha $ and finite $ n $,  etc.