Saturated model

Remarks

A $\kappa$-saturated structure also realizes every complete $n$-type over fewer than $\kappa$ parameters, for every $n \ge 1$.

Example: $(\Q, <)$ is saturated. A complete $1$-type over a finite set $A \subseteq \Q$ says where $x$ lies relative to the elements of $A$. Either $x$ equals some $a \in A$, or $x$ lies in one of the open intervals cut out by $A$, including the two unbounded ones. Each of these intervals contains a rational number, so the type is realized.

Non-example: the field $\overline{\Q}$ of algebraic numbers is a model of the theory of algebraically closed fields of characteristic $0$, but it is not $\aleph_0$-saturated. The set $\set{f(x) \ne 0 : f \in \Z[x], f \ne 0}$ is consistent with its theory over $A = \emptyset$, as $\C$ shows, and it extends to a complete type. No element of $\overline{\Q}$ realizes it, since every algebraic number is a root of a nonzero integer polynomial.

Every structure has a $\kappa$-saturated elementary extension for each $\kappa$. Saturated models of the same complete theory and the same cardinality are isomorphic.