I am trying to solve the excercise in the title, here its text.
With $S_n(T)$ we denote complete $n$-types without parameters of the tehory $T$
Now $(a)\rightarrow (b)$ is clear to me. So I would try to prove $(a)\rightarrow (b)\rightarrow (c)\rightarrow(a)$.
Concerning $(c)\rightarrow (a)$, what I need to show is that there is a countable $A\in Mod(T)$ which is $\omega$-saturated, i.e.
For any $L$-structure $B$, if $\bar{a},\bar{b}$ are n-tuples such that $(A,\bar{a})\equiv (B,\bar{b})$ and $d$ is any element of $B$ then there is $c$ in $A$ such that $(A,\bar{a},c)\equiv (B,\bar{b},d)$.
By $(A,\bar{a})\equiv (B,\bar{b})$ we know that $\bar{a}$ and $\bar{b}$ realize the same complete $n$ type in $A$ and $B$ respectively. What I would try to show is that, chosen $d\in B$, there is a $c\in A$ such that $\bar{a}c$ realizes the same $n+1$ complete type of $\bar{b}d$.
Under $(a)$ we know that $T$ has a countable atomic model, and I was trying to use this fact in order to get an $\omega$-saturated one. Anyway the only direction to which the existence of such a model leads seems to be the existence of a countable prime model, which is in some sense dual to that $\omega$-saturated. So I am stuck: $(c)$ seems to suggest the application of the countable omitting types theorem, but again I cannot figure how. Maybe the fact $T$ is countable plays a role too.
Sadly I have no idea about $(b)\rightarrow(c)$. Any idea about how to close?
More in generale what can we say about the relation between being an atomic model of a theory and being an $\omega$ saturated structure?
Thanks in advance