Let $T$ be a complete $\omega$-categorical theory which have infinite models, and $C$ a $\omega$-saturated model of $T$. Let $A\subseteq C$ and $T_A$ be the theory of the model $C_A$, the structure with symbols to name elements of $A$. I have to prove that $T_A$ is $\omega$-categorical iff $A$ is finite.
For the forward direction, I use Ryll-Nardzewski theorem in the following manner: $T_A$ being $\omega$-categorical, it has only a finite number of complete 1-types. Hence all the different 1-types $tp^{C_A}(a)$ for $a\in A$ must form a finite set, i.e $A$ is finite.
I'm not sure how to prove the converse. Could you provide me a hint?
Hint: A theory is $\omega$-categorical if and only if it has finitely many $n$-types for each $n < \omega$. But consistent $1$-types in $T_A$ can be associated naturally to consistent $(k + 1)$-types in $T$, for some fixed $k$. And, of course, there's really only one obvious choice for $k$...