From Ramsey degrees to Ramsey expansions via weak amalgamation

TL;DR

This paper develops a method to create Ramsey expansions for categories of finite objects with finite small Ramsey degrees, using weak amalgamation and weak Fra"iss"e categories, without requiring additional assumptions.

Contribution

It introduces a new construction of Ramsey expansions based on weak amalgamation, generalizing previous results that depended on stronger structural assumptions.

Findings

Every class with finite Ramsey degrees has weak amalgamation.

The construction leverages weak Fra"iss"e categories.

Improves upon prior methods by removing the need for a larger ambient class.

Abstract

Under no additional assumptions, in this paper we construct a Ramsey expansion for every category of finite objects with finite small Ramsey degrees. Our construction is based on the relationship between small Ramsey degrees, weak amalgamation and recent results about weak Fra\"iss\'e categories. Namely, generalizing the fact that every Ramsey class has amalgamation, we show that classes with finite Ramsey degrees have weak amalgamation. We then invoke the machinery of weak Fra\"iss\'e categories to perform the construction. This improves previous similar results where an analogous construction was carried out under the assumption that everything sits comfortably in a bigger class with enough infrastructure, and that in this wider context there is an ultrahomogeneous structure under whose umbrella the construction takes place.