Ramsey degrees: Big v. small

Mašulović, Dragan

doi:10.1016/j.ejc.2021.103323

Cited by 7 publications

(6 citation statements)

References 20 publications

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“…After proving a general transport principle in [47], he applied it to prove finite big Ramsey degrees for many universal structures and also for homogenous metric spaces with finite distance sets with a certain property which he calls compact with one nontrivial block. Mašulović proved in [48] that in categories satisfying certain mild conditions, small Ramsey degrees are minima of big Ramsey degrees. In the paper [49] with Šobot (not using category theory), finite big Ramsey degrees for finite chains in countable ordinals were shown to exist if and only if the ordinal is smaller than ω ω .…”

Section: Historical Highlights Recent Results and Methodsmentioning

confidence: 99%

Ramsey theory of homogeneous structures: current trends and open problems

Dobrinen¹

2021

Preprint

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This article highlights historical achievements in the partition theory of countable homogeneous relational structures, and presents recent work, current trends, and open problems. Exciting recent developments include new methods involving logic, topological Ramsey spaces, and category theory. The paper concentrates on big Ramsey degrees, presenting their essential structure where known and outlining areas for further development. Cognate areas, including infinite dimensional Ramsey theory of homogeneous structures and partition theory of uncountable structures, are also discussed.This article will appear in the 2022 ICM Proceedings. It is dedicated to Norbert Sauer for his seminal works on the partition theory of homogeneous structures, and for his mathematical and personal generosity.

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Section: Historical Highlights Recent Results and Methodsmentioning

confidence: 99%

Ramsey theory of homogeneous structures: current trends and open problems

Dobrinen¹

2021

Preprint

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“…The following result was first proved for categories of structures in [3], and for general categories in [26]. Theorem 3.9.…”

Section: Ramsey Properties In a Categorymentioning

confidence: 99%

“…(cf. [3,26]) Let C and C * be locally small categories such that all the morphisms in C and C * are mono. Let U : C * → C be an expansion with unique restrictions.…”

Section: Ramsey Properties In a Categorymentioning

confidence: 99%

Dual Ramsey properties for classes of algebras

Mašulović¹

2021

Preprint

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Almost any reasonable class of finite relational structures has the Ramsey property or a Ramsey expansion. In contrast to that, the list of classes of finite algebras with the Ramsey expansion is surprisingly short. In this paper we show that any nontrivial variety (that is, equationally defined class of algebras) enjoys various dual Ramsey properties. We develop a completely new set of strategies that rely on the fact that right adjoints preserve the Ramsey property while left adjoints preserve the dual Ramsey property, and then treat classes of algebras as Eilenberg-Moore categories for a (co)monad. We first show that for any group G (finite or infinite) finite G-sets have finite small Ramsey degrees, and that every finite G-set has a finite big Ramsey degree in the cofree G-set on countably many cofree generators. We then show that finite algebras in any nontrivial variety have finite dual small Ramsey degrees, and that every finite algebra has finite dual big Ramsey degree in the free algebra on countably many free generators. As usual, these come as consequences of ordered versions of the statements. To the best of our knowledge, this is the first calculation of dual big Ramsey degrees after the Infinite Dual Ramsey Theorem of Carlson and Simpson.

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“…The following result was first proved for categories of structures in [2], and for general categories in [12]. Theorem 3.6.…”

Section: Ramsey Properties In a Categorymentioning

confidence: 99%

“…(cf. [2,12]) Let C and C * be locally small categories such that all the morphisms in C and C * are mono. Let U : C * → C be an expansion with unique restrictions.…”

Section: Ramsey Properties In a Categorymentioning

confidence: 99%

Coalgebraic methods for Ramsey degrees of unary algebras

Mašulović¹

2021

Preprint

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In this paper we are interested in the existence of small and big Ramsey degrees of classes of finite unary algebras in arbitrary (not necessarily finite) algebraic language Ω. We think of unary algebras as M -sets where M = Ω * is the free monoid of words over the alphabet Ω and show that for an arbitrary monoid M (finite or infinite) the class of all finite M -sets has finite small Ramsey degrees. This immediately implies that the class of all finite G-sets, where G is an arbitrary group (finite or infinite), has finite small Ramsey degrees, and that the class of all finite unary algebras over an arbitrary (finite or infinite) algebraic language Ω has finite small Ramsey degrees. This generalizes some Ramsey-type results of M. Sokić concerning finite unary algebras over finite languages and finite G-sets for finite groups G. To do so we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat M -sets as Eilenberg-Moore coalgebras for "half a comonad" and using pre-adjunctions transport the Ramsey properties we are interested in from the category of finite or countably infinite chains of order type ω. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators.

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Ramsey degrees: Big v. small

Cited by 7 publications

References 20 publications

Ramsey theory of homogeneous structures: current trends and open problems

Ramsey theory of homogeneous structures: current trends and open problems

Dual Ramsey properties for classes of algebras

Coalgebraic methods for Ramsey degrees of unary algebras

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