Type is Realized in some Elementary Extension

Theorem
Let $\mathcal{M}$ be an $\mathcal{L}$-structure, let $A$ be a subset of the universe of $\mathcal{M}$, and let $p$ be an $n$-type over $A$.

There exists an elementary extension of $\mathcal{M}$ which realizes $p$.

Proof
The idea is to work in a language with constant symbols for all elements of $\mathcal{M}$ and show that the union of $p$ and the elementary diagram of $\mathcal{M}$ is satisfiable. Since $\mathcal{M}$ naturally embeds into any model of such a theory, this will prove the theorem.

Let $\mathcal{L}_\mathcal{M}$ be the language obtained by adding to $\mathcal{L}$ constant symbols for each element of $\mathcal{M}$.

Denote by $\operatorname{Diag}_{\mathrm{el}} (\mathcal{M})$ the elementary diagram of $\mathcal{M}$.

Let $T$ be $p \cup \operatorname{Diag}_{\mathrm{el}} (\mathcal{M})$.

We will show that $T$ is finitely satisfiable. It will follow by the Compactness Theorem that $T$ is satisfiable.

To this end, let $\Delta$ be a finite subset of $T$.

Since $\Delta$ is finite, it consists of finitely many $\mathcal{L}_A$-sentences $\phi_0,\dots,\phi_n$ from $p$ (which are $\mathcal{L}_\mathcal{M}$ sentences since $A\subseteq \mathcal{M}$) along with finitely many $\mathcal{L}_\mathcal{M}$-sentences $\psi_0,\dots,\psi_k$ from $\operatorname{Diag}_{\mathrm{el}} (\mathcal{M})$.

By definition, $p$ is satisfiable by some $\mathcal{L}_A$-structure $\mathcal{N}$ such that $\mathcal{N}\models p\cup \operatorname{Th}_A (\mathcal{M})$.

Thus, since $\phi_0,\dots,\phi_n \in p$, we have that $\mathcal{N}$ satisfies $\phi_0,\dots,\phi_n$.

We will show that the same $\mathcal{N}$ also satisfies $\psi_0,\dots,\psi_k$.

The obstacle to overcome is that the $\psi_i$ are $\mathcal{L}_\mathcal{M}$-formulas, and we only know $\mathcal{N}$ as an $\mathcal{L}_A$-structure which satisfies sentences with parameters from $A$. The $\psi_i$ may have parameters from $\mathcal{M}$ outside of $A$. The idea is to quantify away the excess parameters and appropriately select the interpretation of new symbols so that $\mathcal{N}$ is a good $\mathcal{L}_\mathcal{M}$-structure. Explicitly:

Let $\psi$ be the conjunction $\psi_0 \wedge \cdots \wedge \psi_k$. Note that since $\psi$ is an $\mathcal{L}_\mathcal{M}$-sentence, it can be written as an $\mathcal{L}_A$-formula $\psi(\bar{b})$, where $\bar{b}$ is a tuple of parameters from $\mathcal{M}$ not in $A$.

By existentially quantifying away the tuple $\bar{b}$, we obtain an $\mathcal{L}_A$-sentence $\exists \bar{x} \psi(\bar{x})$.

Now, since $\mathcal{M}\models\psi(\bar{b})$, we have $\mathcal{M}\models\exists \bar{x} \psi(\bar{x})$, and hence $\exists \bar{x} \psi(\bar{x})$ is in $\operatorname{Th}_\mathcal{A} (\mathcal{M})$.

By choice of $\mathcal{N}$, it follows that $\mathcal{N}\models \exists \bar{x} \psi(\bar{x})$, and thus there must be some tuple $\bar{c}$ of elements from $\mathcal{N}$ such that $\mathcal{N}\models \psi(\bar{c})$.

Now, by interpreting the $\mathcal{L}_\mathcal{M}$-symbols $\bar{b}$ as the elements $\bar{c}$, we can view $\mathcal{N}$ as an $\mathcal{L}_\mathcal{M}$-structure which satisfies $\phi_0 \wedge \cdots \wedge \phi_n \wedge \psi_0 \wedge \cdots \wedge \psi_k$.

Thus, $\mathcal{N}$ satisfies all of $\Delta$.

This demonstrates that $T$ is finitely satisfiable and hence satisfiable by the Compactness Theorem.

This means that there is an $\mathcal{L}_\mathcal{M}$-structure $\mathcal{M}^*$ which satisfies $p\cup \operatorname{Diag}_{\mathrm{el}} (\mathcal{M})$.

Since $\mathcal{M}^*$ interprets a symbol for each element of $\mathcal{M}$, there is an obvious embedding of $\mathcal{M}$ into $\mathcal{M}^*$. This embedding is elementary since $\mathcal{M}^*$ satisfies the elementary diagram of $\mathcal{M}$.

Thus $\mathcal{M}^*$ is an elementary extension of $\mathcal{M}$.

Finally, since $\mathcal{M}^*$ satisfies $p$, there must be a tuple of elements $\bar{d}$ such that $\mathcal{M}^* \models \phi(d)$ for each $\phi(\bar{v}) \in p$.

Thus $\mathcal{M}^*$ realizes $p$.