# Remarks on generic stability in independent theories

## TL;DR

This paper explores various notions of generic stability in different classes of theories, establishing equivalences and providing examples that distinguish stable from unstable behaviors, with connections to Ramsey theory.

## Contribution

It shows that the standard definition of generic stability aligns with frequency interpretation measures and provides combinatorial examples illustrating differences in stability notions.

## Key findings

- Generic stability for types coincides with frequency interpretation measures.
- Examples of types in NSOP theories that are finitely approximated but not generically stable.
- Connections between stability notions and classical Ramsey theory results.

## Abstract

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of "generic stability" in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as $\phi$-types in simple theories that are definable and finitely satisfiable in a small model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs.