Ages of expansion of \(\omega\)-categorical structures (Q2469432)

Recall that the age of a structure is the set of (isomorphism types of) its finite substructures. The authors define an expansion \((M,\bar r)\) of a countable \(\omega\)-categorical structure to be \(\mathcal J\)-generic if for all locally finite structures \(N\) of the same age as \(M\) there is an expansion \((N,\bar s)\) of the same age as \((M,\bar r)\). Recall that if \(M\) is a structure which eliminates quantifiers and the infinite quantfier \(\exists^\infty\), the theory of \(M\) with a new predicate has a model-completion Th\((M)_P\), the theory of a generic predicate (which the authors call mc-generic. They show that mc-generic predicates on \(\omega\)-stable \(\omega\)-categorical structures yield \(\mathcal J\)-generic expansions. The authors also define a topology on the set of all \(\bar r\)-expansions of \(M\), and define a particular expansion to be generic if its orbit under the automorphism group of \(M\) is comeagre. They show that if a generic expansion exists and satisfies a certain joint embedding property, then it is \({\mathcal J}\)-generic. The proofs show something stronger, namely \({\mathcal J}\)-smoothness: Any two finite subsets of the expansion have isomorphic copies which are independent (in the sense of the original structure).