Let $L$ be the a language containing countably many unary predicates $\{P_i : i\in \mathbb{N} \}$ and countable many constants $\{a_{i,j} : (i,j)\in \mathbb{N}^2 \}$. Let $T$ be the theory which says $a_{i,j}\neq a_{i,k}$ for any $i$ and any pair $j<k$, $\forall x(\neg P_i(x) \vee \neg P_j(x))$ for any pair $i<j$, and $P_i(a_{i,j})$ for any pair $i,j$. Show the following:
In any $\omega$-saturated model $M$ of $T$ the following are infinite: $M - \bigcup_{i\in \mathbb{N}}P_i^M$ and $P_i^M - \{ a_{i,j}^M : j\in \mathbb{N} \}$ for each $i$.
$T$ is complete and eliminates quatifiers.
The converse of 1. holds.
$T$ has a prime model and a countable saturated model. Describe them.
$T$ has $2^{\aleph_0}$ countable models.
What are the isolated types in $S_n^T$ ?
$I({\aleph_\alpha},T)=(2+|\alpha|)^{\aleph_0}$.
I do have the skeleton of (almost) every item; so the help would consist on stating the technical details.
For 1.: since in each saturated model every type with a countable number of variables is realized, we can select countably-many variables expressing "we are outside of $P_i$ for all $i$ and we are all different" which will show the first and, using the same idea, the second follows as well.
For 2.: Regarding completeness it is sufficient to show that this theory has an algebraically closed model (with which we'd have the first part of 4.); this model will correspond to the "minimal" posible model, i.e., the model containing just the predicates $P_i$ and the constants $a_{i,j}$.
As for the Q.E. we use the following criteria: T.F.A.E.:
(a) $T$ has Q.E.
(b) Let $M, M^*$ be models of $T$ where $M^*$ is $\omega$-saturated, and let $a \in M$ and $f : N \to M^*$ embedding where $N$ is a finitely generated substructure of $M$. Then $f$ can be extended to an embedding from $(N,a)_M$ to $M^*$.
For 4. We described the prime model before; as for the countable saturated - following the same idea as above - consider now the model for which we add all the individuals.
For 5. Count all possible added individuals to each $P_i$.
Any help would be very appreciated.