Definition 15.3 (Partial elementary / homogeneous). Let $M,N$ be $L$-structures, $A\subseteq M$, $B\subseteq N$. A function $f:A\to B$ is partial elementary if for every $L$-formulas $\phi (x_1,\dots ,x_n)$ and $a_1,\dots ,a_n\in A$ we have

$$M\models \phi (\overline{a})⟺N\models \phi (f(\overline{a})).$$

Given $\kappa \ge |L|+{\aleph }_0$, $M$ is $\kappa$-homogeneous if for any $A\subseteq M$ with $|A|<\kappa$, any partial elementary map $fA\to M$ and any $c\in M$ there is some $d\in M$ with $f\cup \{(c,a)\}$ partial elementary. In other words, “partial elementary maps can be extended”.