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?




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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?



Separation of almost disjoint families by ground model almost disjoint families
Perhaps I am misunderstanding your question, but I think you intend that $A\_n'\in V$ and also that the family $\{A\_n'\mid n\in\omega\}$ is in $V$. Is that right?


In this case, the answer is no. Let $A\_0$ consist of every other element of $c$, and $A\_1$ consist of every other element of the remaining elements of $c$ and so on with $A\_n$. These are fully disjoint. Now, suppose that we have $A\_n\subset A\_n'\in V$ and the family $\{A\_n'\mid n\in\omega\}$ is in $V$ and almost disjoint. So $A\_0'$ is a specific infinite co-infinite set in $V$. Let $p$ force that $\dot A\_0\subset \check A\_0'$, using a name $\dot A\_0$ for this definition of $A\_0$. But this is impossible, since we can extend $p$ so as to control the later elements of $c$ in such a way that would violate $A\_0\subset A\_0'$, by inserting the next two elements into $c$ from outside of $A\_0'$. So this is an almost disjoint family in $V[c]$ that cannot be covered by a ground model family $A\_n'$ in the way you described.


The same argument seems to work with higher cardinals and larger families. One simply extends the every-other-element set $A\_0$ to a maximal almost disjoint family of any kind, and runs the same argument. Basically, the obstacle is that there is no ground model co-infinite set $A$ that you can force to cover of the every-other-element-set of the generic set.


