Can a model $\mathcal M$ be $|\mathcal M|^+$-saturated? If $\kappa$ is an infinite cardinal, we say a model $\mathcal M$ is $\kappa$-saturated when for all $A\subset M$ with $|A|<\kappa$ all complete $1$-types $p(x)$ in $\mathcal L_A$ are realised in $\mathcal M$. Furthermore $\mathcal M$ is just called saturated when it is $|\mathcal M|$-saturated. Is it also possible for a model to be $|\mathcal M|^+$-saturated?

Consider the (partial) type $$p_\mathcal{M}=\\{x\
ot=a: a\in\mathcal{M}\\}.$$ Since $\mathcal{M}$ is infinite, $p$ is indeed finitely consistent over $\mathcal{M}$, but cannot be realized.

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In fact, in some sense this is the canonical example of an "interesting" partial type: given a parameter set $A$, there are two kinds of type over $A$ \- those which have a realization in $A$ and those which _(regardless of whether they're realized in the ambient model)_ don't. Every type of the latter kind extends the analogous partial type $$p_A=\\{x\
ot=a:a\in A\\}.$$