Definition:Saturated Model

Definition
Let $T$ be an $\LL$-theory.

Let $\kappa$ be an infinite cardinal.

A model $\MM$ of $T$ is $\kappa$-saturated for every subset $A$ of the universe of $\MM$ of cardinality strictly less than $\kappa$, and for every $n \in \N$, every complete $n$-type $p$ over $A$ is realized in $\MM$.

That is, $\MM$ is $\kappa$-saturated for all $A \subseteq \MM$ with $\card A < \kappa$, and for all $n \in \N$, each $p \in \map {S_n^\MM} A$ is realized in $\MM$.

We say $\MM$ is saturated it is $\kappa$-saturated where $\kappa$ is the cardinality of the universe of $\MM$.