Dual Ramsey Theorem Research Articles

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 Theorem for Trees

Dual Ramsey Theorem for Trees

Borel combinatorics fail in HYP

We characterize the completely determined Borel subsets of HYP as exactly the [Formula: see text] subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author.

Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals

We define a collection of topological Ramsey spaces consisting of equivalence relations on ω with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of ω. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) in order to obtain a dual Ramsey theorem for such ordinals.

The Ramsey Properties for Grassmannians Over ℝ, ℂ

In this note we study and obtain factorization theorems for colorings of matrices and Grassmannians over ℝ and ℂ, which can be considered metric versions of the Dual Ramsey Theorem for Boolean matrices and of the Graham-Leeb-Rothschild Theorem for Grassmannians over a finite field.

Effectiveness for the Dual Ramsey Theorem

We analyze the dual Ramsey theorem for k partitions and ℓ colors (DRTℓk) in the context of reverse math, effective analysis, and strong reductions. Over RCA0, the dual Ramsey theorem stated for Baire colorings Baire-DRTℓk is equivalent to the statement for clopen colorings ODRTℓk and to a purely combinatorial theorem CDRTℓk. When the theorem is stated for Borel colorings and k≥3, the resulting principles are essentially relativizations of CDRTℓk. For each α, there is a computable Borel code for a Δα0-coloring such that any partition homogeneous for it computes ∅(α) or ∅(α−1) depending on whether α is infinite or finite. For k=2, we present partial results giving bounds on the effective content of the principle. A weaker version for Δn0-reduced colorings is equivalent to D2n over RCA0+IΣn−10 and in the sense of strong Weihrauch reductions.

The Ramsey property for Banach spaces and Choquet simplices

We show that the Gurarij space \mathbb{G} has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. We also compute the universal minimal flow of the automorphism group of the Poulsen simplex \mathbb{P} and we prove that it consists of the canonical action on \mathbb{P} itself. This answers a question of Conley and Törnquist. We show that the pointwise stabilizer of any closed proper face of \mathbb{P} is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simplex) is extremely amenable. These results are obtained via several Kechris–Pestov–Todorcevic correspondences, by establishing the approximate Ramsey property for several classes of finite-dimensional Banach spaces and function systems and their versions with distinguished contractions. This is the first direct application of the Kechris–Pestov–Todorcevic correspondence in the setting of metric structures. The fundamental combinatorial principle that underpins the proofs is the Dual Ramsey Theorem of Graham and Rothschild.

The Ramsey property for operator spaces and noncommutative Choquet simplices

The noncommutative Gurarij space NG, initially defined by Oikhberg, is a canonical object in the theory of operator spaces. As the Fraïssé limit of the class of finite-dimensional nuclear operator spaces, it can be seen as the noncommutative analogue of the classical Gurarij Banach space. In this paper, we prove that the automorphism group of NG is extremely amenable, i.e. any of its actions on compact spaces has a fixed point. The proof relies on the Dual Ramsey Theorem, and a version of the Kechris–Pestov–Todorcevic correspondence in the setting of operator spaces.Recent work of Davidson and Kennedy, building on previous work of Arveson, Effros, Farenick, Webster, and Winkler, among others, shows that nuclear operator systems can be seen as the noncommutative analogue of Choquet simplices. The analogue of the Poulsen simplex in this context is the matrix state space NP of the Fraïssé limit A(NP) of the class of finite-dimensional nuclear operator systems. We show that the canonical action of the automorphism group of NP on the compact set NP1 of unital linear functionals on A(NP) is minimal and it factors onto any minimal action, whence providing a description of the universal minimal flow of Aut(NP).

Amalgamation and Ramsey properties of Lp spaces

We study the dynamics of the group of isometries of Lp-spaces. In particular, we study the canonical actions of these groups on the space of δ-isometric embeddings of finite dimensional subspaces of Lp(0,1) into itself, and we show that for every real number 1≤p<∞ with p≠4,6,8,… they are ε-transitive provided that δ is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fraïssé Banach space which underlies these results and of which the known separable examples are the spaces Lp(0,1), p≠4,6,8,… and the Gurarij space. We also give a proof of the Ramsey property of the classes {ℓpn}n, p≠2,∞, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matoušek and Rödl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space X with a Ramsey property of the collection of finite dimensional subspaces of X.

English

We discuss 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 another rendering of the Nesetřil-Rodl 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 quotient maps and prove a dual Ramsey theorem in such a setting. Although our methods are based on reinterpreting the (dual) Ramsey property in the language of category theory, all our results are about classes of finite structures.

A dual Ramsey theorem for finite ordered oriented graphs

Abstract 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.

A computable analysis of variable words theorems

The Carlson-Simpson lemma is a combinatorial statement occurring in the proof of the Dual Ramsey theorem. Formulated in terms of variable words, it informally asserts that given any finite coloring of the strings, there is an infinite sequence with infinitely many variables such that for every valuation, some specific set of initial segments is homogeneous. Friedman, Simpson, and Montalban asked about its reverse mathematical strength. We study the computability-theoretic properties and the reverse mathematics of this statement, and relate it to the finite union theorem. In particular, we prove the Ordered Variable word for binary strings in ACA0.

A Dual Ramsey Theorem for Permutations

In 2012 M. Sokić proved that the that the class of all finite permutations has the Ramsey property. Using different strategies the same result was then reproved in 2013 by J. Böttcher and J. Foniok, in 2014 by M. Bodirsky and in 2015 yet another proof was provided by M. Sokić.Using the categorical reinterpretation of the Ramsey property in this paper we prove that the class of all finite permutations has the dual Ramsey property as well. It was Leeb who pointed out in 1970 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. In this paper we argue that this is even more the case when dealing with the dual Ramsey property.

On the Dual Ramsey Property for Finite Distributive Lattices

The class of finite distributive lattices, as many other natural classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Sokic have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the class of finite distributive lattices does not have the dual Ramsey property either. However, we are able to derive a dual Ramsey theorem for finite distributive lattices endowed with a particular linear order. Both results are consequences of the recently observed fact that categorical equivalence preserves the Ramsey property.

Abstract approach to finite Ramsey theory and a self-dual Ramsey theorem

We give an abstract approach to finite Ramsey theory and prove a general Ramsey-type theorem. We deduce from it a self-dual Ramsey theorem, which is a new result naturally generalizing both the classical Ramsey theorem and the dual Ramsey theorem of Graham and Rothschild. In fact, we recover the pure finite Ramsey theory from our general Ramsey-type result in the sense that the classical Ramsey theorem, the Hales–Jewett theorem (with Shelahʼs bounds), the Graham–Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and the new self-dual Ramsey theorem are all obtained as iterative applications of the general result.

Minimal hyperspace actions of homeomorphism groups of h-homogeneous spaces

Suppose that X is an h-homogeneous zero-dimensional compact Hausdorff space, i.e., X is a Stone dual of a homogeneous Boolean algebra. Using the dual Ramsey theorem and a detailed combinatorial analysis of what we call stable collections of subsets of a finite set, we obtain a complete list of the minimal sub-systems of the compact dynamical system (Exp(Exp(X)), Homeo(X)), where Exp(X) denotes the hyperspace comprising the closed subsets of X equipped with the Vietoris topology. The importance of this dynamical system stems from Uspenskij’s characterization of the universal ambit of G = Homeo(X). The results apply to the Cantor set, the generalized Cantor sets X = {0,1}κ for noncountable cardinals κ, and to several other spaces. A particular interesting case is X = ω* = βω \ ω, where βω denotes the Stone- Čech compactification of the natural numbers. This space, called the corona or the remainder of ω, has been extensively studied in the fields of set theory and topology.

Effectiveness for infinite variable words and the Dual Ramsey Theorem

We examine the Dual Ramsey Theorem and two related combinatorial principles VW(k,l) and OVW(k,l) from the perspectives of reverse mathematics and effective mathematics. We give a statement of the Dual Ramsey Theorem for open colorings in second order arithmetic and formalize work of Carlson and Simpson [1] to show that this statement implies ACA 0 over RCA 0 . We show that neither VW(2,2) nor OVW(2,2) is provable in WKL 0 . These results give partial answers to questions posed by Friedman and Simpson [3].

A generalization of the Dual Ellentuck Theorem

We prove versions of the Dual Ramsey Theorem and the Dual Ellentuck Theorem for families of partitions which are defined in terms of games.

A Result in Dual Ramsey Theory

We present a result which is obtained by combining a result of Carlson with the Finitary Dual Ramsey Theorem of Graham–Rothschild.

A dual form of Ramsey's Theorem

Let k ϵ ω, where ϵ is the set of all natural numbers. Ramsey's Theorem deals with colorings of the k-element subsets of ω. Our dual form deals with colorings of the k-element partitions of ω. Let ( ω) k (respectively ( ω) ω ) be the set of all partitions of ω having exactly k (respectively infinitely many) blocks. Given X ϵ ( ω) ω let ( X) k be the set of all Y ϵ ( ω) k such that Y is coarser than X. Dual Ramsey Theorem. If ( ω) k = C 0 ∪ … ∪ C t−1 where each C i is Borel then there exists X ϵ ( ω) ω such that ( X) k ⊆ C i for some i < l. Dual Galvin-Prikry Theorem. Same as before with k replaced by ω. We also obtain dual forms of theorems of Ellentuck and Mathias. Our results also provide an infinitary generalization of the Graham-Rothschild “parameter set” theorem [ Trans. Amer. Math. Soc. 159 (1971), 257–292] and a new proof of the Halpern-Läuchli Theorem [ Trans. Amer. Math. Soc. 124 (1966), 360–367].