Saturated Implies Universal

Theorem
Let $\kappa$ be an infinite cardinal. Let $\mathcal{M}$ be a model of the $\mathcal{L}$-theory $T$.

If $\mathcal{M}$ is $\kappa$-saturated, then it is $\kappa^+$-universal, where $\kappa^+$ is the successor cardinal of $\kappa$.

Proof
The idea of the proof is that $\mathcal{M}$ being saturated means that when we want to define an elementary map $\mathcal{N}\to\mathcal{M}$, we can find an image $y\in \mathcal{M}$ for an element $x\in \mathcal{N}$ by realizing the type made up of the formulas that such a $y$ would need to satisfy.

Let $\mathcal{N}$ be a model of $T$ with universe of cardinality strictly less than $\kappa$.

We will construct an elementary embedding of $\mathcal{N}$ into $\mathcal{M}$ by transfinite recursion.

Since $|\mathcal{N}|<\kappa$, we can write its elements as $n_\alpha$ for ordinals $\alpha < \kappa$.

For each ordinal $\alpha < \kappa$, let $A_\alpha$ be the subset $\{n_\beta : \beta < \alpha\}$ of the universe of $\mathcal{N}$.

Note for clarity that $n_\alpha \in A_{\alpha+1}$ but $n_\alpha \notin A_\alpha$.

Define $f_0 = \emptyset$.
 * Base case $\alpha = 0$:

Note that $f_0$ is trivially an elementary embedding from $A_0 = \emptyset$ into $\mathcal{M}$

Let $\displaystyle f_\alpha = \bigcup_{\beta < \alpha} f_\beta$.
 * Limit ordinals $\alpha$, assuming $f_\beta$ is defined and elementary $A_\beta \to \mathcal{M}$ for all $\beta < \alpha$:

If $\phi$ is an $\mathcal{L}$-sentence with parameters from $A_\alpha$, then since it involves only finitely many such parameters, they must all be contained in some $A_\beta$ for $\beta < \alpha$. But $f_\alpha \restriction A_\beta = f_\beta$ is elementary, so $f_\alpha$ must be as well.

We need to extend $f_\beta$ to $A_\alpha = A_\beta \cup \{n_\beta\}$ so that truth of $\mathcal{L}$-sentences with parameters from $A_\alpha$ is preserved.
 * Successor ordinals $\alpha = \beta + 1$, assuming $f_\beta$ is defined and elementary $A_\beta \to \mathcal{M}$:

Consider the subset $p = \{\phi(v, f_\beta(\bar{a})):\bar{a}\text{ is a tuple from }A_\beta \text{ and }\mathcal{N}\models\phi(n_\beta,\bar{a})\}$ of the set of $\mathcal{L}$-formulas with one free variable and parameters from the image $f_\beta (A_\beta)$ of $A_\beta$ under $f_\beta$.

The set $p$ is a $1$-type over the image $f_\beta (A_\beta)$ in $\mathcal{M}$.

Since $|A_\beta| < \kappa$ and by assumption $\mathcal{M}$ is $\kappa$-saturated, this means that $p$ is realized in $\mathcal{M}$ by some element $b$.

Thus, defining $f_\alpha$ to be $f_\beta \cup \{(n_\beta, b)\}$ makes it an elementary embedding $A_\alpha \to \mathcal{M}$.

Now, define $\displaystyle f = \bigcup_{\alpha < \kappa} f_\alpha$.

Then $f$ is an elementary embedding from $\mathcal{N}$ to $\mathcal{M}$ since $\displaystyle \bigcup_{\alpha < \kappa} A_\alpha = \mathcal{N}$, any finite set of parameters from $\mathcal{N}$ must belong to one $A_\alpha$, and $f \restriction A_\alpha = f_\alpha$ is elementary.