Andreas Blass
We survey some connections between topological dynamics, semigroups of ultrafilters, and combinatorics. As an application, we give a proof, based on ideas of Bergelson and Hindman, of the Hales-Jewett partition theorem.
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We are interested in generalizing part of the theory of ultrafilters on omega to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal.
We investigate the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. In particular, this paper provides several techniques to construct ultrafilters which are not Tukey maximal. Furthermore, we connect this analysis with a cardinal invariant of Boolean algebras, the ultrafilter number, and prove consistency results concerning its possible values on Cohen and random algebras.
In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers.
Combinatorics of Schur ultrafilters
本文研究了可数交换群上Schur ultrafilters的元素的组合特征，构造了一个自由Schur ultrafilter，它不是无限Schur的，并在连续假设下证明了存在一个自由Schur P-point。
Ultrafilters are a tool, originating in mathematical logic and general topology, that has steadily found more and more uses in multiple areas of mathematics, such as combinatorics, dynamics, and algebra, among others. The purpose of this article is to introduce ultrafilters in a friendly manner and present some applications to the branch of combinatorics known as Ramsey theory, culminating with a new ultrafilter-based proof of van der Waerden's theorem.
We generalize to the relations $(λ, μ) \stackrelκ{\Rightarrow} (λ', μ')$ and $\alm (λ, μ) \stackrelκ{\Rightarrow} \alm (λ', μ')$ some results obtained in Parts II and IV. We also present a multi-cardinal version.
We extend to singular cardinals the model-theoretical relation $λ\stackrelκ{\Rightarrow} μ$ introduced in P. Lipparini, The compactness spectrum of abstract logics, large cardinals and combinatorial principles, Boll. Unione Matematica Italiana ser. VII, {\bf 4-B} 875--903 (1990). We extend some results obtained in Part II, finding equivalent conditions involving uniformity of ultrafilters and the existence of certain infinite matrices. Our present definition suggests a new compactness property for abstract logics.
We introduce the relation of "almost-reduction" in an arbitrary topological Ramsey space R, as a generalization of the relation of "almost-inclusion" on the space of infinite sets of natural numbers (the Ellentuck space). This leads us to a type of ultrafilter U on the set of first approximations of the elements of R which corresponds to the well-known notion of "selective ultrafilter" on N, the set of natural numbers. The relationship turns out to be rather exact in the sense that it permits us to lift several well-known facts about selective ultrafilters on N and the Ellentuck space to the u…
This paper investigates properties of $σ$-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 $c:[ω]^n\rightarrow l$, 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 $σ$-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at …
We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters. We also list some problems, and furnish applications to topological spaces and to extended logics.
Topological Ramsey spaces are spaces which support infinite dimensional Ramsey theory similarly to the Ellentuck space. Each topological Ramsey space is endowed with a partial ordering which can be modified to a $σ$-closed `almost reduction' relation analogously to the partial ordering of `mod finite' on $[ω]^ω$. Such forcings add new ultrafilters satisfying weak partition relations and have complete combinatorics. In cases where a forcing turned out to be equivalent to a topological Ramsey space, the strong Ramsey-theoretic techniques have aided in a fine-tuned analysis of the Rudin-Keisler a…
We find many conditions equivalent to the model-theoretical property $λ\stackrelκ{\Rightarrow} μ$ introduced in [L1]. Our conditions involve uniformity of ultrafilters, compactness properties of products of topological spaces and the existence of certain infinite matrices.
We generalize the results from "P. Lipparini, Productive $[λ,μ]$-compactness and regular ultrafilters, Topology Proceedings, 21 (1996), 161--171"; in particular the present results apply to singular cardinals, too.
We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces.
We discuss the connection between various orders on the class of all the ultrafilters and certain compactness properties of abstract logics and of topological spaces. We present a model theoretical characterization of Comfort order. We introduce a new order motivated by considerations in abstract model theory. For each of the above orders, we show that if $E$ is a $(λ, λ)$-regular ultrafilter, and $D$ is not $(λ, λ)$-regular, then $E \not \leq D$. Many problems are stated.
Schur ultrafilters and Bohr compactifications of topological groups
本文利用Schur超滤子与Stone-Čech紧化的代数结构，给出了拓扑群Bohr紧化的新描述，并证明了chart群为拓扑群等价于其每个Schur超滤子收敛到单位元。
We discuss the existence of complete accumulation points of sequences in products of topological spaces. Then we collect and generalize many of the results proved in Parts I, II and IV. The present Part VI is complementary to Part V to the effect that here we deal, say, with uniformity, complete accumulation points and $ κ$-$(λ)$-compactness, rather than with regularity, $[ λ, μ]$-compactness and $ κ$-$ (λ, μ)$-compactness. Of course, if we restrict ourselves to regular cardinals, Parts V (for $ λ= μ$) and Part VI essentially coincide.
We introduce the notion of a coherent $P$-ultrafilter on a complete ccc Boolean algebra, strenghtening the notion of a $P$-point on $ω$, and show that these ultrafilters exist generically under ${\mathfrak c} = {\mathfrak d}$. This improves the known existence result of Ketonen. Similarly, the existence theorem of Canjar can be extended to show that coherently selective ultrafilters exist generically under ${\mathfrak c} = {cov(M)}$. We use these ultrafilters in a topological application: a coherent $P$-ultrafilter on an algebra $B$ is an untouchable point in the Stone space of $B$, witnessing…
There exist two distinct types of ultrafilter extensions of binary relations, one discovered in universal algebra and modal logic, and another, in model theory and algebra of ultrafilters. We show that the extension of the latter type is properly included in the extension of the former type, and describe their interaction with the relation algebra operations. Then we provide topological characterizations of both extensions and show that the larger extension continuously maps the space of ultrafilters into the space of filters endowed with the Vietoris topology.
We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space $\mathcal{R}$, are the notions of selective for $\mathcal{R}$ and Ramsey for $\mathcal{R}$ equivalent? Every topological Ramsey space $\mathcal{R}$ has an associated notion of Ramsey ultrafilter for $\mathcal{R}$ and selective ultrafilter for $\mathcal{R}$ (see \cite{MijaresSelective}). If $\mathcal{R}$ is taken to be the Ellentuck space then the two concepts reduce to the familiar notions of Ramsey and selective ultrafilters on $ω$; so by a well-known result of Kunen the two are equivalent. We…
We use Shelah's theory of possible cofinalities in order to solve a problem about ultrafilters. THEOREM. Suppose that $ λ$ is a singular cardinal, $ λ' < λ$, and the ultrafilter $D$ is $ κ$-decomposable for all regular cardinals $ κ$ with $ λ' < κ< λ$. Then $D$ is either $ λ$-decomposable, or $ λ^+$-decomposable. We give applications to topological spaces and to abstract logics.
Finitely additive measures on $\mathbb Z$ and additive combinatorics
该研究以$\text{ba}(\boldsymbol{\text{Z}})$为工具，将超滤子理论方法拓展至测度论场景，证明满足测度阈值的整数子集为$\text{IP}_n$集，还给出示例并讨论开放问题。
Every subset of a topological Ramsey space is Ramsey in the Solovay model
本文在Solovay模型下，无需超滤子和几乎归约，证明了拓扑Ramsey空间的每个子集都具有Ramsey性质。
Contents: 1. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, I; 2. Frechet-Urysohn fans in free topological groups; 3. Packing index of subsets in Polish groups; 4. Symmetric monochromatic subsets in colorings of the Lobachevsky plane; 5. Structural Ramsey theory of metric spaces and topological dynamics of isometry groups; 6. Distinguishing Number of Countable Homogeneous Relational Structures; 7. Indestructible colourings and rainbow Ramsey theorems; 8. Products of Borel subgroups; 9. Selection theorems and treeabili…
If $S$ is a discrete semigroup, then $βS$ has a natural, left-topological semigroup structure extending $S$. Under some very mild conditions, $U(S)$, the set of uniform ultrafilters on $S$, is a two-sided ideal of $βS$, and therefore contains all of its minimal left ideals and minimal idempotents. We find some very general conditions under which $U(S)$ contains prime minimal left ideals and left-maximal idempotents. If $S$ is countable, then $U(S) = S^*$, and a special case of our main theorem is that if a countable discrete semigroup $S$ is a weakly cancellative and left-cancellative, then $S…
A classical result of topological algebra states that any compact left topological semigroup has an idempotent. We refine this by showing that any compact left topological left semiring has a common, i.e. additive and multiplicative simultaneously, idempotent. As an application, we partially answer a question related to algebraic properties of ultrafilters over natural numbers. Finally, we observe that similar arguments establish the existence of common idempotents in much more general, non-associative universal algebras.
We prove that if an ultrafilter L is not coherent to a Q-point, then each analytic non-sigma-bounded topological group G admits an increasing chain <G_a : a < b(L)> of its proper subgroups such that: (i) U_{a in b(L)} G_a=G; and $(ii)$ For every sigma-bounded subgroup H of G there exists a such that H is a subset of G_a. In case of the group Sym(w) of all permutations of w with the topology inherited from w^w this improves upon earlier results of S. Thomas.
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Borel versions of greedy algorithms and augmenting procedures, local rules, Borel transversals, etc. Also, we present the construction of Andrew Marks of acyclic Borel graphs for which …
Denote the integers by Z and the positive integers by N. The groups Z^k (k a natural number) are discrete, and the classification up to isomorphism of their (topological) subgroups is trivial. But already for the countably infinite power Z^N of Z, the situation is different. Here the product topology is nontrivial, and the subgroups of Z^N make a rich source of examples of non-isomorphic topological groups. Z^N is the Baer-Specker group. We study subgroups of the Baer-Specker group which possess group theoretic properties analogous to properties introduced by Menger (1924), Hurewicz (1925), Ro…
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