Let $M$ be an $\aleph_0$-saturated structure (in a possibly uncountable language). I have a set $A \subseteq M$. Let $F$ be the family of $M$-definable subsets of $M$ containing $A$. I have the following question:
Is there a partial type $\Sigma(x)$ over a finite subset of $M$ such that every $M$-definable subset $X$ of $M$ is a superset of $\Sigma(M)$ iff $X \in F$?
That is, I want to find a set like $\bigcap F$ that is type-definable by fewer (finite) number of parameters.
When is can I find such a set, and when can I not? (For instance, it seems that I can find such a partial type for any subset $A$ in $(\Bbb Q, <)$, but that structure has very tame definable sets.)