Existentially closed models of bowtie-free graphs

Question

Setting

Definition(1). $\mathcal{M} \models T$ is an existentially closed (e.c.) model of $T$ if whenever $\mathcal{N} \models T$, $\mathcal{N} \supseteq \mathcal{M}$, and $\mathcal{N}\models \exists \bar{v} \phi(\bar{v},\bar{a})$, where $\bar{a} \in M$ and $\phi$ is a quantifier free formula, then $\mathcal{M} \models \exists \bar{v} \phi(\bar{v},\bar{a})$.

Definition(2). Let $T$ be a theory. A theory $T^*$ is called the model companion of $T$ if $$\mathcal{M}\models T^* \text{ iff } \mathcal{M} \text{ is an e.c. model of } T$$

So $T^*= \bigcap_{\mathcal{M} \text{ is e.c.} } Th(\mathcal{M})$, where $Th(\mathcal{M})$ is the full theory of $\mathcal{M}$.

Case of study

Let $T$ be the theory of bowtie-free graphs (i.e. $T$ is the theory of graphs plus the axiom that expresses every five points does not form a bowtie), and $mod(T)$ be the class of all models of $T$.

Question(1). Is $mod(T)$ closed under the union of chain? (equivalently we may ask, does $T$ have $\forall\exists$-axiomatization?)

If the answer of the first question is positive, we can ask the following questions.

Question(2). What are the $e.c.$ models of $T$?

Question(3). Does $T$ have model-companion?

Answer 1

The answer to question (3) is yes. This is a theorem of Cherlin, Shelah and Shi in the paper Universal graphs with forbidden subgraphs and algebraic closure. More generally, they prove (Theorem 1) that for any finite set $\mathcal{F}$ of finite graphs, the theory of $\mathcal{F}$-free graphs has a model companion.

The proof of Theorem 1 is fairly constructive, so you can extract an axiomatization of the model companion from the proof. This axiomatization will answer question (2), but maybe not in a way you find satisfying. The best you can hope for in an answer to question (2) is a reduction of the general definition of existentially closed to a more concrete set of instances of existential closure, in the form of "extension axioms".