Definition:Big Model

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Let $\mathcal{M}$ be an $\mathcal{L}$-structure with universe $M$.

Let $\kappa$ be a cardinal.

$\mathcal{M}$ is $\kappa$-big if for every subset $A\subset M$ with cardinality $|A| < \kappa$, the following holds:

if $\mathcal{L}_A$ is the language obtained from $\mathcal{L}$ by adding new constant symbols for each $a\in A$, then
for every language $\mathcal{L}_A^*$ obtained by adding a new relation symbol $R$ to $\mathcal{L}_A$, and
for every $\mathcal{L}_A^*$-structure $\mathcal{N}$ such that $\mathcal{M}$ and $\mathcal{N}$ are elementary equivalent as $\mathcal{L}_A$-structures,
there is a relation $R^\mathcal{M}$ on $M$ such that $(\mathcal{M},R^\mathcal{M})$ is elementary equivalent to $\mathcal{N}$ as an $\mathcal{L}_A^*$-structure.


Note that any function symbol or constant symbol can be replaced by a relation symbol along with suitable sentences mentioning only that symbol. So, the focus on relation symbols in the definition is just for convenience.

Informally, being $\kappa$-big means that $\mathcal{M}$ already has all of the structural features that are consistent with the behavior of $\mathcal{M}$ and the parameters in $A$.