Definition:Homogeneous (Model Theory)

Definition
Let $T$ be an $\mathcal{L}$-theory.

Let $\kappa$ be an infinite cardinal.

A model $\mathcal{M}$ of $T$ is $\kappa$-homogeneous if for every subset $A$ and element $b$ in the universe of $\mathcal{M}$ with the cardinality of $A$ strictly less than $\kappa$, if $f:A\to \mathcal{M}$ is elementary, then $f$ extends to an elementary map $f^*: A\cup\{b\} \to \mathcal{M}$.

That is, $\mathcal{M}$ is $\kappa$-homogeneous if for all $A  \subseteq \mathcal{M}$ with  $|A|<\kappa$ and all $b\in \mathcal{M}$, every elementary $f:A\to \mathcal{M}$ extends to an elementary $f^*:A\cup\{b\}\to\mathcal{M}$.

We say $\mathcal{M}$ is homogeneous if it is $\kappa$-homogeneous where $\kappa$ is the cardinality of the universe of $\mathcal{M}$.

Equivalent Definition
$\mathcal{M}$ is homogeneous if and only if it has some infinite cardinality $\kappa$ and for every $A\subseteq \mathcal{M}$ with $|A|<\kappa$, each partial elementary map $f:A\to \mathcal{M}$ extends to an automorphism.

The equivalence of this definition is proved in Homogeneous iff Partial Elementary Maps Extend to Automorphisms.