Fix logic $L$ with equality and a binary relation symbol $E$.

The class of graphs can be identified with the class of models of the universal first-order Horn $L$-sentences $\forall x,y\; E(x,y) \rightarrow E(y,x)$ and $\forall x\; \lnot E(x,x)$. Note that this class includes infinite graphs.

An odd hole is a cycle with an odd number of vertices, at least five. An odd antihole is the complement of an odd hole. A Berge graph contains no induced odd holes or antiholes. By the Strong Perfect Graph theorem, a graph is perfect iff it is Berge. It is therefore possible to define the class of perfect graphs as the finite models of an infinite set of universal first-order sentences, each expressing a property "this graph does not contain an odd (anti)hole of order $2k+3$", for $k=1,2,\dots$. Call the class of all models of this set of sentences the $\omega$-perfect graphs (to highlight the fact that it may include infinite structures).

My primary question:

Is the class of $\omega$-perfect graphs finitely axiomatizable?

One could conclude this if the class of all $L$-structures that are not $\omega$-perfect were axiomatizable, but if this is the case then it is not obvious to me.

~~Further, the direct product of two perfect graphs is again perfect (for instance, this is an immediate corollary of a theorem of Ravindra and Parthasarathy). Since the class of $\omega$-perfect graphs can be axiomatized by a set of universal sentences, it is closed under ultraproducts. Hence the class of $\omega$-perfect graphs is closed under isomorphism, substructures (induced subgraphs), direct products, and ultraproducts, and therefore is a quasivariety definable by a set of universal (first-order) Horn sentences of $L$. (Although their existence seems guaranteed, justifying the title of this question, I don't actually know an explicit set of universal Horn axioms for the $\omega$-perfect graphs.) If the answer to the main question is affirmative, then this leads to the follow-on question:~~

~~Is the class of $\omega$-perfect graphs finitely axiomatizable by universal Horn sentences?~~

*Edit:* Thanks to bof for pointing out that if $G$ is $C_5$ with an edge added, then $G \times G$ is not perfect! I misinterpreted the result, the correct one is (in modern notation):

(Ravindra and Parthasarathy 1977, Theorem 3.2) $G_1 \times G_2$ is perfect iff either

1. $G_1$ or $G_2$ is bipartite, or

2. $G_1$ and $G_2$ are both (odd-hole,paw)-free.

The quasivariety generated by the $\omega$-perfect graphs therefore includes many non-perfect finite graphs. To match the question title, my second question should therefore be:

Is the quasivariety generated by all perfect graphs finitely axiomatizable?