\begin{definition}[Definition:Homogeneous (Model Theory)]
Let $T$ be an $\LL$-theory.
Let $\kappa$ be an infinite cardinal.
A model $\MM$ of $T$ is $\kappa$-'''homogeneous''' {{iff}} for every subset $A$ and element $b$ in the universe of $\MM$ with the cardinality of $A$ strictly less than $\kappa$, if $f: A \to \MM$ is partial elementary, then $f$ extends to an elementary map $f^*: A \cup \set b \to \MM$.
That is, $\MM$ is $\kappa$-homogeneous {{iff}} for all $A \subseteq \MM$ with  $\card A < \kappa$ and all $b \in \MM$, every elementary $f: A \to \MM$ extends to an elementary $f^*: A \cup \set b \to \MM$.
We say $\MM$ is '''homogeneous''' {{iff}} it is $\kappa$-homogeneous where $\kappa$ is the cardinality of the universe of $\MM$.
\end{definition}