Theorem: Let $\kappa$ be a regular cardinal in $M$. Let $\mathbb{P} \in M$ be a separative partial order of size $\kappa$, and let $G$ be $\mathbb{P}$-generic over $M$. Then the following are equivalent:
(1) There is some $p \in G$ such that in $M$, $\mathbb{P} \restriction p$ has a dense subset of size $< \kappa$.
(2) For all sets of ordinals $A \in M[G]$ of size $\kappa$, there is $B \subseteq A$ such that $B \in M$ and $|B| = \kappa$.