\begin{definition}[Definition:Big Model]
Let $\MM$ be an $\LL$-structure with universe $M$.
Let $\kappa$ be a cardinal.
$\MM$ is '''$\kappa$-big''' if for every subset $A \subset M$ with cardinality $\card A < \kappa$, the following holds:
:if $\LL_A$ is the language obtained from $\LL$ by adding new constant symbols for each $a \in A$, then
:for every language $\LL_A^*$ obtained by adding a new relation symbol $R$ to $\LL_A$, and
:for every $\LL_A^*$-structure $\NN$ such that $\MM$ and $\NN$ are elementary equivalent as $\LL_A$-structures,
:there is a relation $R^\MM$ on $M$ such that $\struct {\MM, R^\MM}$ is elementary equivalent to $\NN$ as an $\LL_A^*$-structure.
\end{definition}