Pre-adjunctions and the Ramsey property

TL;DR

This paper introduces a category-theoretic strategy using pre-adjunctions to establish the Ramsey property for various classes of finite structures, simplifying proofs of known results.

Contribution

It proposes a new, simpler method based on category theory to prove the Ramsey property, demonstrated through multiple classical examples.

Findings

The strategy simplifies proofs of the Ramsey property for finite linearly ordered posets.

It applies to convexly ordered ultrametric spaces, confirming their Ramsey property.

The method also proves the Ramsey property for finite rational metric spaces.

Abstract

Showing that the Ramsey property holds for a class of finite structures can be an extremely challenging task and a slew of sophisticated methods have been proposed in literature. In this paper we propose a new strategy to show that a class of structures has the Ramsey property. The strategy is based on a relatively simple result in category theory and consists of establishing a pre-adjunction between the category of structures which is known to have the Ramsey property, and the category of structures we are interested in. This strategy was implicitly used already in 1981 by H.J. Pr\"omel and B. Voigt in their proof of the Ramsey property for the class of finite linearly ordered graphs. We demonstrate the applicability of this strategy by providing short proofs of three important well known results: we show the Ramsey property for the category of all finite linearly ordered posets with embeddings, for the category of finite convexly ordered ultrametric spaces with embeddings, and for the category of all finite linearly ordered metric spaces (rational metric spaces) with embeddings.