Prime model

A prime model of a complete theory elementarily embeds into every model of that theory.

A model is atomic when the complete type over the empty set of every finite tuple is isolated. For a complete theory in a countable language, a model is prime exactly when it is countable and atomic.

Take a countable structure with countably many independent unary predicates, every finite predicate pattern realized infinitely often, and a constant naming every element. Its complete theory has an atomic model: tuples of named elements have types isolated by equations to their names. But every infinite binary predicate pattern, together with inequalities to every named element, is consistent. There are uncountably many such one-types, so no countable omega-saturated model exists. Thus existence of a countable atomic model does not imply existence of a countable saturated one.

A countable atomic model of a complete theory elementarily embeds into every model. Enumerate it and recursively extend a finite partial elementary map; an isolating formula for the next joint type guarantees a realization in the target.

If $M,N$ are infinite atomic models of a complete theory and a finite partial elementary map sends $\bar{a}$ to $\bar{b}$, then for every $c\in M$ there is $d\in N$ extending it by $c\mapsto d$. An isolating formula for $\mathrm{tp}(\bar{a},c)$ supplies $d$. Back-and-forth therefore makes any two countable atomic models isomorphic.