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 TR 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 TR, namely simple models, and certain indiscernible arrays. Using simple models of TR 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], TR is stable if and only if it is $NIP$ and $NSOP$.