The main result of the present article is the following: Let N be an infinite subset of,, and let be a matrix with infinitely many rows of completely Ramsey subsets of such that for every n,. Then there exist, a sequence of nonempty finite subsets of N, and an infinite subset T of such that for every infinite subset I of. We also give an application of this result to partitions of an uncountable analytic subset of a Polish space X (...) into sets belonging to the σ‐algebra generated by the analytic subsets of X. (shrink)

Hindman’s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. Ramsey algebras are structures that satisfy an analogue of Hindman’s Theorem. This paper introduces Ramsey algebras and presents some elementary results. Furthermore, their connection to Ramsey spaces will be addressed.

Hindman’s Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman’s Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper develops a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable (...) field of sets. This amounts to a positive result that addresses Carlson’s question in some way. (shrink)

Hindman’s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. A Ramsey algebra is a structure that satisfies an analogue of Hindman’s theorem. In this paper, we present the basic notions of Ramsey algebras by using terminology from mathematical logic. We also present some results regarding classification of Ramsey algebras.

We use games of Kastanas to obtain a new characterization of the classC ℱ of all sets that are completely Ramsey with respect to a given happy family ℱ. We then combine this with ideas of Plewik to give a uniform proof of various results of Ellentuck, Louveau, Mathias and Milliken concerning the extent ofC ℱ. We also study some cardinals that can be associated with the ideal ℐℱ of nowhere ℱ-Ramsey sets.

We present a version for κ-distributive ideals over a regular infinite cardinal κ of some of the combinatorial results of Mathias on happy families. We also study an associated notion of forcing, which is a generalization of Mathias forcing and of Prikry forcing.

On the occasion of his 70th birthday, the work of Adrian Mathias in set theory is surveyed in its full range and extent.

We improve a theorem of Raisonnier by showing that Cons(ZFC+every Σ 2 1 -set of reals in Lebesgue measurable+every Π 2 1 -set of reals isK σ-regular) implies Cons(ZFC+there exists an inaccessible cardinal). We construct, fromL, a model where every Δ 3 1 -sets of reals is Lebesgue measurable, has the property of Baire, and every Σ 2 1 -set of reals isK σ-regular. We prove that if there exists a Σ n+1 1 unbounded filter on ω, then there exists (...) a nonK σ-regular Π 2 1 -subset. (shrink)

We show that a theory of autonomous iterated Ramseyness based on second order arithmetic (SOA) is proof-theoretically equivalent to ${\Pi^1_2}$ -comprehension. The property of Ramsey is defined as follows. Let X be a set of real numbers, i.e. a set of infinite sets of natural numbers. We call a set H of natural numbers homogeneous for X if either all infinite subsets of H are in X or all infinite subsets of H are not in X. X has the property (...) of Ramsey if there exists a set which is homogeneous for X. The property of Ramsey is considered in reverse mathematics to compare the strength of subsystems of SOA. To characterize the system of ${\Pi^1_2}$ -comprehension in terms of Ramseyness we introduce a system of autonomous iterated Ramseyness, called R-calculus. We augment the language of SOA with additional set terms (called R-terms) ${R\vec{x}X\phi(\vec{x},X)}$ for each first order formula ${\phi(\vec{x},X)}$ (where φ may contain further R-terms). The R-calculus is a system which comprises comprehension for all first order formulas (which may contain R-terms or other set parameters) and defining axioms for the R-terms which claim that for each ${\vec{x}}$ , we can remove finitely many elements from the set ${R\vec{x}X\phi(\vec{x},X)}$ such that the remaining set is homogeneous for ${\{{X}{\phi(\vec{x},X)\}}}$ . We show that the R-calculus proves the same ${\Pi^1_1}$ -sentences as the system of ${\Pi^1_2}$ -comprehension. (shrink)

For a ⊆ b ⊆ ω with b\ a infinite, the set D = {x ∈ [ω]ω : a ⊆ x ⊆ b} is called a doughnut. A set S ⊆ [ω]ω has the doughnut property [MATHEMATICAL SCRIPT CAPITAL D] if it contains or is disjoint from a doughnut. It is known that not every set S ⊆ [ω]ω has the doughnut property, but S has the doughnut property if it has the Baire property ℬ or the Ramsey property ℛ. (...) In this paper it is shown that a finite support iteration of length ω1 of Cohen forcing, starting from L, yields a model for CH + equation image + equation image + equation image. (shrink)

A set-theoretical proof of Gowers’ Dichotomy Theorem is presented together with its application to another dichotomy concerning asymptotic l 2 basic sequences.

A set-theoretical proof of Gowers’ Dichotomy Theorem is presented together with its application to another dichotomy concerning asymptotic l2 basic sequences.

We present and analyze \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_\sigma}$$\end{document}-Mathias forcing, which is similar but tamer than Mathias forcing. In particular, we show that this forcing preserves certain weak subsystems of second-order arithmetic such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{ACA}_0}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{WKL}_0 + \mathsf{I}\Sigma^0_2}$$\end{document}, whereas Mathias forcing does not. We also show that the needed reals for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_\sigma}$$\end{document}-Mathias forcing (in the sense of Blass in Ann Pure Appl Logic 109(1–2):77–88, 2001) are just the computable reals, as opposed to the hyperarithmetic reals for Mathias forcing. (shrink)

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the (...) Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor generating p-points which are k-arrow but not \-arrow, and in a partial order of Blass producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \\). If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \. In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template. (shrink)

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. (shrink)

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 \({\mathfrak {p}}\). (shrink)

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for barriers on the Ellentuck (...) space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text]. (shrink)

Given a topological Ramsey space math formula, we extend the notion of semiselective coideal to sets math formula and study conditions for math formula that will enable us to make the structure math formula a Ramsey space and also study forcing notions related to math formula which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck's space. This extends results from to the most general context of topological Ramsey spaces. As applications, we (...) prove that for every topological Ramsey space math formula, under suitable large cardinal hypotheses every semiselective ultrafilter math formula is generic over math formula; and that given a semiselective coideal math formula, every definable subset of math formula is math formula-Ramsey. This generalizes the corresponding results for the case when math formula is equal to Ellentuck's space. (shrink)

Solovay has shown that if $\cal{O}$ is an open subset of $P(\omega)$ with code $S$ and no infinite set avoids $\cal{O}$ , then there is an infinite set hyperarithmetic in $S$ that lands in $\cal{O}$ . We provide a direct proof of this theorem that is easily formalizable in $ATR_0$.