TITLE: Separation of almost disjoint families by ground model almost disjoint families
QUESTION [6 upvotes]: Suppose that $V$ is a model of $\sf ZFC$, and for concreteness I should point that at this point I am interested in $V=L$ as a ground model.
Suppose that $V[c]$ is a Cohen extension of $V$ where $c$ is a real number, and $\{A_n\mid n\in\omega\}$ is an almost disjoint family in $V[c]$ of subsets of $\omega$. Can we always find an almost disjoint family $\{A'_n\mid n\in\omega\}$ such that $A_n\subseteq A'_n$?
What about larger cardinals? Suppose that $\{A_\alpha\mid\alpha<\kappa\}$ is an almost disjoint family of subsets of a regular $\kappa$ (almost disjoint means that the intersection of any distinct two is of size ${<}\kappa$). Can we find in $V$ a separation family?

While this is not the case I care about, I am curious about the case where the almost disjoint family is maximal, or at least of size $\kappa^+$. Can we still find such family?

REPLY [9 votes]: Here's a possibly simpler example than Joel's, in the case of $\omega$.  Let $A_0$ be $c$ (where I think of the Cohen real as a subset of $\omega$), and let the rest of the $A_n$'s be any partition of $\omega-c$.  (In fact, you could let $A_1=\omega-c$ and let all the later $A_n$'s be empty, if you're willing to let the empty set into your almost disjoint family.) An easy genericity argument shows that the only ground-model subsets $A_0'$ of $\omega$ that satisfy $c\subseteq A_0'$ are cofinite.  So $A_0'$ would have to meet $A_1'$ infinitely.
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