Setting
Definition $\mathcal{M} \models T$ is existentially closed if whenever $\mathcal{N} \models T$, $\mathcal{N} \supseteq \mathcal{M}$, and $\mathcal{N}\models \exists \bar{v} \phi(\bar{v},\bar{a})$, where $\bar{a} \in \mathbb{M}$ and $\phi$ is quantifier free, then $\mathcal{M} \models \exists \bar{v} \phi(\bar{v},\bar{a})$.
I would like to show that if T is $\forall\exists$-axiomatizable,, then T has an existentially closed model.
From now on we assume that if $\mathcal M \models T$, then there is $\mathcal N \supseteq \mathcal M$ existentially closed with $|\mathbb{N}| = |\mathbb{M}| + |\mathcal L| + \aleph_o$. Suppose that T has an infinite nonexistentially closed model, then I want to prove T has a nonexistentially closed model of cardinality $\kappa$ for any infinite cardinal $\kappa \ge |\mathcal{L}|$.
Finally, I would like to show that if T is $\kappa$-catagorical for some infinite $\kappa \ge |\mathcal{L}|$ and axiomatized by $\forall\exists$-sentences, then all models of T are existentially closed.