Pre-adjunctions and the Ramsey property

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.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.

Canonizing structural Ramsey theorems

“…One useful strategy for proving the Ramsey property for categories consists of establishing a pre-adjunction between two categories (see [5]). As the canonical Ramsey property is much stronger than the "usual" Ramsey property, we shall need a stronger version which we refer to as a canonical pre-adjunction.…”

“…As a demonstration of this strategy we shall show that the class of all finite linearly ordered metric spaces has the canonical Ramsey property. The proof is a modification of the proof of [5,Theorem 4.4] and the technical results that we inherit from [5] shall not be repeated here.…”

“…< t ℓ } ⊆ R be a finite set of nonnegative reals. We say that T is tight [5] if t i+j t i + t j for all 0 i j i + j ℓ. It is easy to show (see [5]) that (M × {0, 1, .…”

“…In this paper we build on the results of [13] to provide several new structural canonical Ramsey results. In contrast to [13] where the authors prove canonical Ramsey statements using the "classical" Ramsey-theoretic approach, in this paper we modify the ideas from [5] and using the appropriate "transfer techniques" formulated in the language of category theory we prove the canonical Ramsey theorem for the class of all finite linearly ordered tournaments, the class of all finite posets with linear extensions, the class of all finite linearly ordered metric spaces and the class of all finite linearly ordered oriented graphs. We conclude the paper with the canonical version of the celebrated Nešetřil-Rödl Theorem.…”

“…In Section 4 we present a technical result which enables us to transfer the canonical Ramsey property from a category to its hereditary subcategory, and as an immediate consequence prove the canonical Ramsey property for the class of all finite posets with linear extensions. In Section 5 we introduce canonical pre-adjunctions between two categories (see [5] for the motivation) and use them to prove the canonical Ramsey property for the class of all finite linearly ordered metric spaces, as well as some standard subclasses thereof. The paper concludes with Section 6 in which we prove the canonical version of the Nešetřil-Rödl Theorem (Theorem 1.3) and from it easily derive the canonical Ramsey property for the class of all finite linearly ordered oriented graphs.…”