Let us call a theory $T$ good if
- it is complete;
- it is formulated in a countable language $L$;
- it has infinite models.
Suppose $T$ is good and $M \models T$, the exercise is to prove
$M$ is $\omega$-saturated iff it is $\omega$-homogeneous and it realizes all types [in $\cup_n S_n(T)$].
Right to left is immediate. We also know that
[*] $M$ is $\omega$-saturated iff it is $\omega$-homogeneous and $\omega^+$-universal.
So it suffices to show that realizing all types implies $\omega^+$-universality (maybe using $\omega$-homogeneity). My idea is to utilize the technique of the $\Rightarrow$-direction proof of [*], but adept it to use that $M$ realizes $\cup_n S_n(T)$ instead of $\omega$-saturation. But I am stuck.
I don't know any theorems that describe/involve models that realize all types, except that saturated models realize all types (but that is not very useful here). What do we know about such models, and how can I use that?