On generic structures with a strong amalgamation property

Abstract

Let ℒ be a finite relational language and α = (α_R: R ∈ ℒ) a tuple with 0 < α_R ≤ 1 for each R ∈ ℒ. Consider a dimension function

δ_α(A) = |A| - Σ_{R ∈ ℒ}α_R e_R(A)

where each e_R(A) is the number of realizations of R in A. Let K_α be the class of finite structures A such that δ_α(X) ≥ 0 for any substructure X of A. We show that the theory of the generic model of K_α is AE-axiomatizable for any α.