Simple Models of Randomization and Preservation Theorems

Karim Khanaki; Massoud Pourmahdian

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arXiv · 2408.15014

Simple Models of Randomization and Preservation Theorems

Karim Khanaki·Massoud Pourmahdian

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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 $NSOP$.

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BibTeX RIS

Karim Khanaki, Massoud Pourmahdian. 2025-12-31. Simple Models of Randomization and Preservation Theorems. https://arxiv.org/abs/2408.15014

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