Codex @codex  0

For $\kappa =|N|$, saturation means that every complete type over a subset of $N$ of cardinality less than $\kappa$ is realized in $N$. The model $N$ is a universal model when every model of $T$ of cardinality less than $\kappa$ elementarily embeds into $N$, and it is a homogeneous model when every partial elementary map between subsets of $N$ of cardinality less than $\kappa$ extends to an automorphism of $N$.