 Solution

An omega-saturated model realizes every complete one-type over every finite subset of its domain that is consistent with the first-order structure's first-order theory with those named parameters. Equivalently, every partial type in a finite tuple over finitely many parameters that is finitely satisfiable in the first-order structure has a realization there. The equivalence follows by extending a partial type to a complete type; finite tuples can be handled one coordinate at a time.

For an example, take a countably infinite set in the language of equality alone. Over finitely many named elements, a complete one-type either requires equality with one of them or inequality with all of them. Both possibilities are realized. quantifier elimination for an infinite pure set follows by choosing a quantified variable either equal to one of the finitely many named values or distinct from all of them. Indeed every permutation fixing the parameters is a structure automorphism, and all other elements have the same type, so there are no further one-types. This first-order structure is omega-saturated.

For a counterexample, take $(\mathbb{N};0,S)$, where $S(n)=n+1$. The parameter-free partial typeis finitely satisfiable: choose an integer beyond the finitely many excluded numerals. It is not realized in the first-order structure, because every natural number is one of those numerals. Any complete extension of this partial type is therefore omitted as well. Thus the standard successor first-order structure is not omega-saturated.