Geodesic
Definable sets in generic structures and their cardinalities
Matematičeskie trudy, Tome 20 (2017) no. 2, pp. 52-79

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Analyzing diagrams forming generative classes, we describe definable sets and their links in generic structures as well as cardinality bounds for these definable sets, finite or infinite. Introducing basic characteristics for definable sets in generic structures, we compare them each others and with cardinalities of these sets.We introduce calculi for (type-)definable sets allowing to compare their cardinalities. In terms of these calculi, Trichotomy Theorem for possibilities comparing cardinalities of definable sets is proved. Using these calculi, we characterize the possibility to construct a generic structure of a given generative class. 
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     author = {I. Kiouvrekis and P. Stefaneas and S. V. Sudoplatov},
     title = {Definable sets in generic structures and their cardinalities},
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I. Kiouvrekis; P. Stefaneas; S. V. Sudoplatov. Definable sets in generic structures and their cardinalities. Matematičeskie trudy, Tome 20 (2017) no. 2, pp. 52-79. http://geodesic.mathdoc.fr/item/MT_2017_20_2_a2/