Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2010 / B / 5 / d

Suppose $T$ is an omega-categorical theory. By the Ryll-Nardzewski theorem, every finite-arity complete type over the empty set is isolated. Let $p(x,a)$ be a complete type over a finite tuple in the unique countable model $M$, and take an elementary extension realizing it at $b$. The joint type of $(b,a)$ is isolated by a formula $\theta (x,y)$.

The complete type of $a$ contains $\exists x\theta (x,y)$, since this formula is true of $a$ in the extension. Hence it is true of $a$ in $M$, and some $b^{\prime }\in M$ satisfies $\theta (b^{\prime },a)$. Isolation of the joint type implies that $b^{\prime }$ realizes $p$. Thus the countable model is an omega-saturated model.

The converse fails. The complete successor theory from question 4 has a countable modelwith countably infinitely many two-sided successor components. It is omega-saturated. Over finitely many parameters, a complete one-variable type either fixes a finite successor/predecessor translate of a parameter or zero, or asks for a point in no component represented by those parameters and zero. The first kind is realized in the named component, and the second in an unused integer component. The quantifier-elimination description of classification of models of the successor theory shows there are no other cases.

Nevertheless this theory is not omega-categorical: $M$ and its standard model $\mathcal{N}$ are countable and not isomorphic. Thus existence of a countable saturated model is strictly weaker than countable categoricity.