Submission date: 19 April 2022

Abstract:

This paper concerns the study of Bi-colored expansions of geometric theories in the light of the Fraïssé-Hrushovski construction method. Substructures of models of a geometric theory T are expanded by a color predicate p, and the dimension function associated with the pre-geometry of the T-algebraic closure operator together with a real number 0<α ≤ 1 is used to define a pre-dimension function δ_{α}. The pair (K_{α}^{+},≤_{α}) consisting of all such expansions with a hereditary positive pre-dimension along with the notion of substructure ≤_{α} associated to δ_{α} is then used as a natural setting for the study of generic bi-colored expansions in the style of Fraïssé-Hrushovski construction. Imposing certain natural conditions on T, enables us to introduce a complete axiomatization 𝕋_{α} for the class of rich structures in this class. We will show that if T is a dependent theory (NIP) then so is 𝕋_{α}. We further prove that whenever α is rational the strong dependence transfers to 𝕋_{α}. We conclude by showing that if T defines a linear order and α is irrational then 𝕋_{α} is not strongly dependent.

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Full text arXiv 2204.09142: pdf, ps.