Ramsey Degrees Research Articles

Coalgebraic methods for Ramsey degrees of unary algebras

Abstract In this paper, we prove the existence of small and big Ramsey degrees of classes of finite unary algebras in an arbitrary (not necessarily finite) algebraic language Ω. Our results generalize some Ramsey-type results of M. Sokić concerning finite unary algebras over finite languages. To do so, we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat unary algebras as Eilenberg-Moore coalgebras for a functor with comultiplication, 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.

Big Ramsey degrees and infinite languages

Big Ramsey degrees and infinite languages

Exact big Ramsey degrees for finitely constrained binary free amalgamation classes

We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recent upper bounds given by Zucker. Using this characterization, we show that the Fraïssé limit of each such class admits a big Ramsey structure satisfying the infinite Ramsey theorem, implying that the automorphism group of the Fraïssé limit has a metrizable universal completion flow.

Big Ramsey degrees in ultraproducts of finite structures

We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct L⁎ has, as a spine, η1, an uncountable analogue of the order type of rationals η. Finite big Ramsey degrees for η were exactly calculated by Devlin in [5]. It is immediate from [39] that η1 fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to η1 to show that it witnesses big Ramsey degrees of finite tuples in η on every copy of η in η1, and consequently in L⁎. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.

A NEW PERSPECTIVE ON SEMI-RETRACTIONS AND THE RAMSEY PROPERTY

Abstract We investigate the notion of a semi-retraction between two first-order structures (in typically different signatures) that was introduced by the second author as a link between the Ramsey property and generalized indiscernible sequences. We look at semi-retractions through a new lens establishing transfers of the Ramsey property and finite Ramsey degrees under quite general conditions that are optimal as demonstrated by counterexamples. Finally, we compare semi-retractions to the category theoretic notion of a pre-adjunction.

Carlson-Simpson's lemma and applications in reverse mathematics

We study the reverse mathematics of infinitary extensions of the Hales-Jewett theorem, due to Carlson and Simpson. These theorems have multiple applications in Ramsey's theory, such as the existence of finite big Ramsey degrees for the triangle-free graph, or the Dual Ramsey theorem. We show in particular that the Open Dual Ramsey theorem holds in ACA0+.

Dual Ramsey properties for classes of algebras

Almost any reasonable class of finite relational structures has the Ramsey property or a precompact Ramsey expansion. In contrast to that, the list of classes of finite algebras with the precompact 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 left adjoints preserve the dual Ramsey property, and then treat classes of algebras as Eilenberg–Moore categories for a monad. We 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.

Type-respecting amalgamation and big Ramsey degrees

We give an infinitary extension of the Nešetřil-Rödl theorem for category of relational structures with special type-respecting embeddings.

Ramsey Properties of Products and Pullbacks of Categories and the Grothendieck Construction

In this paper we provide purely categorical proofs of two important results of structural Ramsey theory: the result of M. Sokić that the free product of Ramsey classes is a Ramsey class, and the result of M. Bodirsky that adding constants to the language of a Ramsey class preserves the Ramsey property. The proofs that we present here ignore the model-theoretic background of these statements. Instead, they focus on categorical constructions by which the classes can be constructed generalizing the original statements along the way. It turns out that the restriction to classes of relational structures, although fundamental for the original proof strategies, is not relevant for the statements themselves. The categorical proofs we present here remove all restrictions on the signature of first-order structures and provide the information not only about the Ramsey property but also about the Ramsey degrees.

The Ramsey theory of Henson graphs

Analogues of Ramsey’s Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author’s recent result for the triangle-free Henson graph, we prove that for each [Formula: see text], the k-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey’s Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker.

Big Ramsey degrees in universal inverse limit structures

We build a collection of topological Ramsey spaces of trees giving rise to universal inverse limit structures, extending Zheng’s work for the profinite graph to the setting of Fraïssé classes of finite ordered binary relational structures with the Ramsey property. This work is based on the Halpern-Läuchli theorem, but different from the Milliken space of strong subtrees. Based on these topological Ramsey spaces and the work of Huber-Geschke-Kojman on inverse limits of finite ordered graphs, we prove that for each such Fraïssé class, its universal inverse limit structure has finite big Ramsey degrees under finite Baire-measurable colorings. For such Fraïssé classes satisfying free amalgamation as well as finite ordered tournaments and finite partial orders with a linear extension, we characterize the exact big Ramsey degrees.

Big Ramsey Spectra of Countable Chains

A big Ramsey spectrum of a countable chain (i.e. strict linear order) C is a sequence of big Ramsey degrees of finite chains computed in C. In this paper we consider big Ramsey spectra of countable scattered chains. We prove that countable scattered chains of infinite Hausdorff rank do not have finite big Ramsey spectra, and that countable scattered chains of finite Hausdorff rank with bounded finite sums have finite big Ramsey spectra. Since big Ramsey spectra of all non-scattered countable chains are finite by results of Galvin, Laver and Devlin, in order to complete the characterization of countable chains with finite big Ramsey spectra (or degrees) one still has to resolve the remaining case of countable scattered chains of finite Hausdorff rank whose finite sums are not bounded.

On big Ramsey degrees for binary free amalgamation classes

Generalizing and simplifying recent work of Dobrinen, we show that if L is a finite binary relational language and F is a finite set of finite irreducible L-structures, then the class K=Forb(F) has finite big Ramsey degrees.

Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces

This paper investigates properties of \(\sigma \)-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter \({\mathcal {U}}\) for n-tuples, denoted \(t({\mathcal {U}},n)\), is the smallest number t such that given any \(l\ge 2\) and coloring \(c:[\omega ]^n\rightarrow l\), there is a member \(X\in {\mathcal {U}}\) such that the restriction of c to \([X]^n\) has no more than t colors. Many well-known \(\sigma \)-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at best. In this paper, we utilize methods of topological Ramsey spaces to calculate Ramsey degrees of several classes of ultrafilters generated by \(\sigma \)-closed forcings. These include a hierarchy of forcings due to Laflamme which generate weakly Ramsey and weaker rapid p-points, forcings of Baumgartner and Taylor and of Blass and generalizations, and the collection of non-p-points generated by the forcings \({\mathcal {P}}(\omega ^k)/\mathrm {Fin}^{\otimes k}\). We provide a general approach to calculating the Ramsey degrees of these ultrafilters, obtaining new results as well as streamlined proofs of previously known results. In the second half of the paper, we calculate pseudointersection and tower numbers for these \(\sigma \)-closed forcings and their relationships with the classical pseudointersection number \({\mathfrak {p}}\).

Big Ramsey Degrees of 3-Uniform Hypergraphs Are Finite

We prove that the universal homogeneous 3-uniform hypergraph has finite big Ramsey degrees. This is the first case where big Ramsey degrees are known to be finite for structures in a non-binary language.Our proof is based on the vector (or product) form of Milliken’s Tree Theorem and demonstrates a general method to carry existing results on structures in binary relational languages to higher arities.

Ramsey degrees: Big v. small

In this paper we investigate algebraic properties of big Ramsey degrees in categories satisfying some mild conditions. As the first nontrivial consequence of the generalization we advocate in this paper we prove that small Ramsey degrees are the minima of the corresponding big ones. We also prove that big Ramsey degrees are subadditive and show that equality is enforced by an abstract property of objects we refer to as self-similarity. Finally, we apply the abstract machinery developed in the paper to show that if a countable relational structure has finite big Ramsey degrees, then so do its quantifier-free reducts. In particular, it follows that the reducts of (Q,<), the random graph, the random tournament and (Q,<,0) all have finite big Ramsey degrees.

Countable Ordinals and Big Ramsey Degrees

In this paper we consider big Ramsey degrees of finite chains in countable ordinals. We prove that a countable ordinal has finite big Ramsey degrees if and only if it is smaller than ωω. Big Ramsey degrees of finite chains in all other countable ordinals are infinite.

Finite big Ramsey degrees in universal structures

Big Ramsey degrees of finite structures are usually considered with respect to a Fraïssé limit. The only available strategy to prove that a Fraïssé limit supports finite big Ramsey degrees was suggested by Sauer in 2006 and relies on representing structures by binary trees and then invoking Milliken's Theorem.In this paper we consider structures which are not Fraïssé limits, and still have the property that their finite substructures have finite big Ramsey degrees in them. For example, the class of all finite acyclic oriented graphs is not a Fraïssé age, and yet we show that there is a countably infinite acyclic oriented graph in which every finite acyclic oriented graph has finite big Ramsey degree. Our approach to proving this and a few more statements of similar kind is based on a new strategy of transporting the property from one category of structures to another using the machinery of category theory.

Big Ramsey degrees and topological dynamics

We consider Fraïssé structures whose objects have finite big Ramsey degree and ask what consequences this has for the dynamics of the automorphism group. Motivated by a theorem of D. Devlin about the partition properties of the rationals, we define the notion of a big Ramsey structure , a single structure which codes the big Ramsey degrees of a given Fraïssé structure. This in turn leads to the definition of a completion flow ; we show that if a Fraïssé structure admits a big Ramsey structure, then the automorphism group admits a unique universal completion flow. We also discuss the problem of when big Ramsey structures exist and explore connections to the notion of oscillation stability defined by Kechris, Pestov, and Todorčević.

Ramsey degrees, labeled and unlabeled partitions

Let K be a class of finite relational structures. Let Kn be the class obtained by adding labeled partition with at most n parts to all structures in K in all possible ways. Let EK be the class obtained by adding unlabeled partition, i.e. an equivalence relation, to structures in K. Assuming that K has an ordered Ramsey expansion and imposing additional properties on K we calculate Ramsey degrees for the structures in Kn and EK.