Categorical equivalence and the Ramsey property for finite powers of a primal algebra

Pantic

2019

Mathematica Slovaca

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In contrast to the abundance of "direct" Ramsey results for classes of finite structures (such as finite ordered graphs, finite ordered metric spaces and finite posets with a linear extension), in only a handful of cases we have a meaningful dual Ramsey result. In this paper we prove a dual Ramsey theorem for finite ordered oriented graphs. Instead of embeddings, which are crucial for "direct" Ramsey results, we consider a special class of surjective homomorphisms between finite ordered oriented graphs. Since the setting we are interested in involves both structures and morphisms, all our results are spelled out using the reinterpretation of the (dual) Ramsey property in the language of category theory.

Section: Category Theory and The Ramsey Propertymentioning

confidence: 99%

“…In Section 3 we provide basics of category theory and give a categorical reinterpretation of the Ramsey property as proposed in [9]. We define the Ramsey property and the dual Ramsey property for a category and illustrate these notions using some well-known examples.…”

Section: Introductionmentioning

confidence: 99%

A dual Ramsey theorem for finite ordered oriented graphs

Pantic

2019

Mathematica Slovaca

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“…In this sense it can be considered as a dual of the Nešetřil-Rödl Theorem (without forbidden substructures): objects are combinatorial cubes with selected combinatorial subspaces and morphisms preserve the types of the selected subspaces. In this paper, however, we consider a dual of the Nešetřil-Rödl Theorem spelled out in the language of relational structures and base our approach on [9] which can be thought of as a simplified version of the approach taken in [16].…”

Section: Introductionmentioning

confidence: 99%

On dual Ramsey theorems for relational structures

2020

Czech.Math.J.

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In this paper we provide explicit dual Ramsey statements for several classes of finite relational structures (such as finite linearly ordered graphs, finite linearly ordered metric spaces and finite posets with a linear extension) and conclude the paper with an explicit dual of the Nešetřil-Rödl Theorem for relational structures. Instead of embeddings which are crucial for "direct" Ramsey results, for each class of structures under consideration we propose a special class of surjective maps and prove a dual Ramsey theorem in such a setting. In contrast to on-going Ramsey classification projects where the research is focused on fine-tuning the objects, in this paper we advocate the idea that fine-tuning the morphisms is the key to proving dual Ramsey results. Since the setting we are interested in involves both structures and morphisms, all our results are spelled out using the reinterpretation of the (dual) Ramsey property in the language of category theory.

“…The purpose of this paper is to show the following result which clearly generalizes each of the results listed in Theorem 1.1: It was Leeb who pointed out in 1970 [11] that the use of category theory can be quite helpful both in the formulation and in the proofs of results pertaining to structural Ramsey theory. We pursued this line of thought in several papers [12,13,14] and demonstrated that reinterpreting the Ramsey property in the context of category theory and using the machinery of category theory can lead to essentially new proving strategies. The proof of Theorem 1.2 will represent another demonstration of these new strategies.…”

Section: Introductionmentioning

confidence: 99%

A Ramsey theorem for multiposets

Draganić

European Journal of Combinatorics

2019

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In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result from the 1980's claims that the class of all finite posets with a linear extension has the Ramsey property. In 2010 Sokić proved that the class of all finite structures consisting of several linear orders has the Ramsey property. This was followed by a 2017 result of Solecki and Zhao that the class of all finite posets with several linear extensions has the Ramsey property.Using the categorical reinterpretation of the Ramsey property in this paper we prove a common generalization of all these results. We consider multiposets to be structures consisting of several partial orders and several linear orders. We allow partial orders to extend each other in an arbitrary but fixed way, and require that every partial order is extended by at least one of the linear orders. We then show that the class of all finite multiposets conforming to a fixed template has the Ramsey property.