Simple Models of Randomization and Preservation Theorems

TL;DR

This paper introduces a uniform model-theoretic proof that the randomization of a complete first-order theory with NIP retains NIP, and explores the stability properties of such randomizations.

Contribution

It provides a simpler, more uniform proof of the preservation of NIP under randomization and analyzes the stability conditions of the randomized theories.

Findings

Randomization $T^R$ of a complete NIP theory is itself NIP.

Simple models and indiscernible arrays are key tools in the proof.

$T^R$ is stable if and only if it is both NIP and $NSOP$.

Abstract

The main purpose of this paper is to present a new and more uniform model-theoretic/combinatorial proof of the theorem ([5]): The randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$ is a (complete) first-order continuous theory with $NIP$. The proof method is based on the significant use of a particular type of models of $T^{R}$, namely simple models, certain indiscernible arrays, and Rademacher mean width. Using simple models of $T^R$ gives the advantage of re-proving this theorem 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 [13], $T^R$ is stable if and only if it is $NIP$ and $NSOP$.