# Trois couleurs: A new non-equational theory

## TL;DR

This paper introduces a new class of non-equational ω-stable theories by coloring the free pseudospace, expanding the known examples beyond the previously limited stable theories.

## Contribution

It constructs the first known non-equational ω-stable theories using a novel coloring method inspired by Hrushovski and Srour's example.

## Key findings

- Successfully constructs non-equational ω-stable theories
- Demonstrates the existence of non-equational stable theories beyond known examples
- Provides a new approach to studying stability in first-order theories

## Abstract

A first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability yet so far only two examples of non-equational stable theories are known. We construct non-equational $\omega$-stable theories by a suitable colouring of the free pseudospace, based on Hrushovski and Srour's original example.