# Finite Ramsey degrees and Fra\"iss\'e expansions with the Ramsey property

## TL;DR

This paper offers a new proof, using classical Fra"iss"e theory, that structures with finite Ramsey degrees can be expanded to have the Ramsey property, simplifying previous dynamic-based methods.

## Contribution

It provides an alternative proof for the existence of Fra"iss"e expansions with the Ramsey property, avoiding ultrafilter dynamics.

## Key findings

- New proof based on classical Fra"iss"e tools
- Establishes existence of expansions with Ramsey property
- Simplifies previous dynamic-based approach

## Abstract

By a result of Zucker, every Fra\"iss\'e structure $\bf F$ for which the elements of $\mathrm{Age}(\bf F)$ have finite Ramsey degrees admits a Fra\"iss\'e precompact expansion $\bf F^{*}$ whose age $\mathrm{Age}(\bf F^{*})$ has the Ramsey property. While the original method uses dynamics in spaces of ultrafilters, the purpose of the present short note is to provide a different proof, based on classical tools from Fra\"iss\'e theory.