Need help in proving any countably universal model contains a countably saturated model

Question

I try to prove that a countably universal model $\mathfrak{B}$ must contain a countably saturated model. I believe it can be done by proving $\mathfrak{B}$ contains a countably homogeneous model because a countably homogeneous and universal model is countably saturated. To do this, I think it needs to construct an elementary chain of countably homogeneous models in $\mathfrak{B}$. But I am not sure how to do it.

Answer 1

I claim that ${\rm Th}(\mathfrak{B})$ is small, that is, it has countably many parameters-free types with finitely many variables.

In fact, any such type has to be realized in $\mathfrak{B}$ which is countable (the OP does not say that, I doubt the claim is true otherwise).

Small theories have countable saturated models. Such model embeds elementarily in $\mathfrak{B}$.