Simple Models of Randomization and Preservation Theorems

The main purpose of this paper is to present new and more uniform model-theoretic/combinatorial proofs of the theorems (in [5] and [4]): The randomization \(T^{R}\) of a complete first-order theory \(T\) with \(NIP\)/stability is a (complete) first-order continuous theory with \(NIP\)/stability. The proof method for both theorems is based on the significant use of a particular type of models of \(T^{R}\), namely simple models, and certain indiscernible arrays. Using simple models of \(T^R\) gives the advantage of re-proving these theorems in a simpler and quantitative manner. We finally turn our attention to \(NSOP\) in randomization. We show that based on the definition of \(NSOP\) given [11], \(T^R\) is stable if and only if it is \(NIP\) and \(NSOP\).