Geodesic
Countable infinite existentially closed models of universally axiomatizable theories
Matematičeskie trudy, Tome 18 (2015) no. 1, pp. 48-97.

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In the present article, we obtain a new criterion for a model of a universally axiomatizable theory to be existentially closed. The notion of a maximal existential type is used in the proof and for investigating properties of countable infinite existentially closed structures. The notions of a prime and a homogeneous model, which are classical for the general model theory, are introduced for such structures. We study universal theories with the joint embedding property admitting a single countable infinite existentially closed model. We also construct, for every natural $n$, an example of a complete inductive theory with a countable infinite family of countable infinite models such that $n$ of them are existentially closed and exactly two are homogeneous. 
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A. T. Nurtazin. Countable infinite existentially closed models of universally axiomatizable theories. Matematičeskie trudy, Tome 18 (2015) no. 1, pp. 48-97. http://geodesic.mathdoc.fr/item/MT_2015_18_1_a3/

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