# Dividing lines in unstable theories and subclasses of Baire 1 functions

## TL;DR

This paper introduces new characterizations of the strict order property in model theory, linking dividing lines in theories to subclasses of Baire 1 functions and refining Shelah's theorem on order and independence properties.

## Contribution

It provides a novel functional analytic perspective on dividing lines in first order theories and introduces new classes of theories based on these characterizations.

## Key findings

- Characterization of SOP via formula behavior in models
- Refinement of Shelah's theorem relating OP, IP, and SOP
- Connections between dividing lines and subclasses of Baire 1 functions

## Abstract

We give a new characterization of $SOP$ (the strict order property) in terms of the behaviour of formulas in any model of the theory as opposed to having to look at the behaviour of indiscernible sequences inside saturated ones. We refine a theorem of Shelah, namely a theory has $OP$ (the order property) if and only if it has $IP$ (the independence property) or $SOP$, in several ways by characterizing various notions in functional analytic style. We point out some connections between dividing lines in first order theories and subclasses of Baire 1 functions, and give new characterizations of some classes and new classes of first order theories.