We define the interpolative fusion [Formula: see text] of a family [Formula: see text] of first-order theories over a common reduct [Formula: see text], a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each [Formula: see text] is model-complete, [Formula: see text] coincides with the model companion of [Formula: see text]. By obtaining sufficient conditions for the existence of [Formula: see text], we develop new tools to show that theories of interest have model companions.

We consider the structures $(\mathbb {Z}; \mathrm {SF}^{\mathbb {Z}})$ , $(\mathbb {Z}; <, \mathrm {SF}^{\mathbb {Z}})$ , $(\mathbb {Q}; \mathrm {SF}^{\mathbb {Q}})$ , and $(\mathbb {Q}; <, \mathrm {SF}^{\mathbb {Q}})$ where $\mathbb {Z}$ is the additive group of integers, $\mathrm {SF}^{\mathbb {Z}}$ is the set of $a \in \mathbb {Z}$ such that $v_{p}(a) < 2$ for  every prime p and corresponding p-adic valuation $v_{p}$ , $\mathbb {Q}$ and $\mathrm {SF}^{\mathbb {Q}}$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

We study interpolative fusion, a method of combining theories T 1 and T 2 in distinct languages in a "generic" way over a common reduct T∩, to obtain a theory T * ∪ . When each T i is model-complete, T * ∪ is the model companion of the union T 1 ∪ T 2 . Our goal is to prove preservation results, i.e., to find sufficient conditions under which model-theoretic properties of T 1 and T 2 are inherited by T * ∪ . We first prove preservation results for quantifier elimination, modelcompleteness, and related properties. We then apply these tools to show that, under mild hypotheses, including stability of T∩, the property NSOP 1 is preserved. We also show that simplicity is preserved under stronger hypotheses on algebraic closure in T 1 and T 2 . This generalizes many previous results; for example, simplicity of ACFA and the random n-hypergraph are both non-obvious corollaries. We also address preservation of stability, NIP, and ℵ 0 -categoricity, and we describe examples which witness that these results are sharp.