Universal model

Fix an infinite cardinal $\kappa$ and a complete theory $T$. A model $N\models T$ of cardinality $\kappa$ is $\kappa$-universal when every model of $T$ of cardinality at most $\kappa$ has an elementary embedding into $N$. The bound includes models of size exactly $\kappa$.

Some conventions use a separate weaker notion requiring embeddings only for models of size less than $\kappa$. That convention must be distinguished here: at $\kappa ={\aleph }_0$, it can be vacuous for a theory with no finite models. A countable dense linear order without endpoints is universal for the countable models of its complete theory, since each such model is isomorphic to the rational order.