Saturated model

Summary

A saturated model is a model that realizes as many types as its size allows. More precisely, $M$ is $\kappa$-saturated when every complete type over fewer than $\kappa$ parameters of $M$ is realized in $M$. It is saturated when it is $\abs{M}$-saturated. Saturated models of a complete theory are unique up to isomorphism in each cardinality. Every model of the theory of at most the same size embeds elementarily into a saturated one. They serve as large "universal" models in which most model-theoretic work is done.