Model-theoretic dividing lines via posets

TL;DR

This paper establishes a unified framework linking various model-theoretic dividing lines to embeddings of specific posets, introducing a new property that bridges existing classifications.

Contribution

It constructs posets characterizing key model-theoretic properties and introduces a novel property SUP that connects and extends these classifications.

Findings

Each property OP, IP, TP1, TP2, ATP, SOP3 corresponds to a specific poset   extbf{}      embedding.

The property SUP is consistent with NIP2 and implies ATP and SOP.

The framework provides a poset-based classification of model-theoretic dividing lines.

Abstract

We show that for each property $\mathsf{P}\in \{\mathsf{OP}, \mathsf{IP}, \mathsf{TP}_1, \mathsf{TP}_2, \mathsf{ATP}, \mathsf{SOP}_3\}$ there is a poset $\Sigma_{\mathsf{P}}$ such that a theory has property $\mathsf{P}$ if and only if some model interprets a poset in which $\Sigma_{\mathsf{P}}$ can be embedded. We also introduce a new property $\mathsf{SUP}$, consistent with $\mathsf{NIP}_2$ and implying $\mathsf{ATP}$ and $\mathsf{SOP}$.