Let ℒ be a finite relational language and α = (αR: R ∈ ℒ) a tuple with 0 < αR ≤ 1 for each R ∈ ℒ. Consider a dimension function
δα(A) = |A| - ΣR ∈ ℒα_R e_R(A)
where each eR(A) is the number of realizations of R in A. Let Kα be the class of finite structures A such that δα(X) ≥ 0 for any substructure X of A. We show that the theory of the generic model of Kα is AE-axiomatizable for any α.
Koichiro Ikeda. Hirotaka Kikyo. Akito Tsuboi. "On generic structures with a strong amalgamation property." J. Symbolic Logic 74 (3) 721 - 733, September 2009. https://doi.org/10.2178/jsl/1245158082