# Big Ramsey degrees and topological dynamics

## TL;DR

This paper explores the relationship between big Ramsey degrees in Fra"iss"e structures and the topological dynamics of their automorphism groups, introducing the concept of big Ramsey structures and their implications for universal flows.

## Contribution

It introduces the notion of big Ramsey structures and demonstrates their role in establishing unique universal completion flows for automorphism groups.

## Key findings

- Existence of big Ramsey structures implies a unique universal completion flow.
- Connections established between big Ramsey degrees and topological dynamics.
- Discussion on conditions for the existence of big Ramsey structures.

## Abstract

We consider Fra\"iss\'e structures whose objects have finite big Ramsey degree and ask what consequences this has for the dynamics of the automorphism group. Motivated by a theorem of D. Devlin about the partition properties of the rationals, we define the notion of a big Ramsey structure, a single structure which codes the big Ramsey degrees of a given Fra\"iss\'e structure. This in turn leads to the definition of a completion flow; we show that if a Fra\"iss\'e structure admits a big Ramsey structure, then the automorphism group admits a unique universal completion flow. We also discuss the problem of when big Ramsey structures exist and explore connections to the notion of oscillation stability defined by Kechris, Pestov, and Todor\v{c}evi\'c