Let $L$ be a first order language, and let $R$ be an elementary class of $L$ -structures. Let $\kappa$ be a cardinal. $R_{\kappa}$ be the set of structures from $R$ with cardinality less than or equal to $\kappa$ .

Let $M \in R_{\kappa}$ . Suppose that for every $N \in R_{\kappa}$ there is an embedding of $N$ into $M$ . Then we say $M$ is universal.