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Filter games on ω and the dual ideal

Claude Laflamme, Christopher C. Leary (2002)

Fundamenta Mathematicae

We continue the efforts to characterize winning strategies in various infinite games involving filters on the natural numbers in terms of combinatorial or structural properties of the given filter. Previous results in the literature included those games where player II responded with natural numbers, or finite subsets of natural numbers. In this paper we concentrate on games where player II responds with members of the dual ideal. We also give a summary of known results on filter games.

Finite Embeddability of Sets and Ultrafilters

Andreas Blass, Mauro Di Nasso (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper we study it in its own right. We also study a related notion of finite embeddability of ultrafilters on the natural numbers. Among other results, we obtain connections between finite embeddability and the algebraic and topological structure of the Stone-Čech...

Forcing tightness in products of fans

Jörg Brendle, Tim La Berge (1996)

Fundamenta Mathematicae

We prove two theorems that characterize tightness in certain products of fans in terms of families of integer-valued functions. We also define several notions of forcing that allow us to manipulate the structure of the set of functions from some cardinal θ to ω, and hence, the tightness of these products. These results give new constructions of first countable <θ-cwH spaces that are not ≤θ-cwH.

Fragments of strong compactness, families of partitions and ideal extensions

Laura Fontanella, Pierre Matet (2016)

Fundamenta Mathematicae

We investigate some natural combinatorial principles related to the notion of mild ineffability, and use them to obtain new characterizations of mild ineffable and weakly compact cardinals. We also show that one of these principles may be satisfied by a successor cardinal. Finally, we establish a version for κ ( λ ) of the canonical Ramsey theorem for pairs.

Fraïssé structures and a conjecture of Furstenberg

Dana Bartošová, Andy Zucker (2019)

Commentationes Mathematicae Universitatis Carolinae

We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between S ( G ) , the Samuel compactification, and E ( M ( G ) ) , the enveloping semigroup of the universal minimal flow. We resolve Furstenberg’s problem for several automorphism groups and give a detailed study in the case of G = S ∞ , leading us to define and investigate several new types...

Generating countable sets of surjective functions

J. D. Mitchell, Y. Péresse (2011)

Fundamenta Mathematicae

We prove that any countable set of surjective functions on an infinite set of cardinality ℵₙ with n ∈ ℕ can be generated by at most n²/2 + 9n/2 + 7 surjective functions of the same set; and there exist n²/2 + 9n/2 + 7 surjective functions that cannot be generated by any smaller number of surjections. We also present several analogous results for other classical infinite transformation semigroups such as the injective functions, the Baer-Levi semigroups, and the Schützenberger monoids.

Guessing clubs in the generalized club filter

Bernhard König, Paul Larson, Yasuo Yoshinobu (2007)

Fundamenta Mathematicae

We present principles for guessing clubs in the generalized club filter on κ λ . These principles are shown to be weaker than classical diamond principles but often serve as sufficient substitutes. One application is a new construction of a λ⁺-Suslin-tree using assumptions different from previous constructions. The other application partly solves open problems regarding the cofinality of reflection points for stationary subsets of [ λ ] ℵ ₀ .

Hausdorff gaps and towers in 𝓟(ω)/Fin

Piotr Borodulin-Nadzieja, David Chodounský (2015)

Fundamenta Mathematicae

We define and study two classes of uncountable ⊆*-chains: Hausdorff towers and Suslin towers. We discuss their existence in various models of set theory. Some of the results and methods are used to provide examples of indestructible gaps not equivalent to a Hausdorff gap. We also indicate possible ways of developing a structure theory for towers based on classification of their Tukey types.

Hurewicz scheme

Michal Staš (2008)

Acta Universitatis Carolinae. Mathematica et Physica

Ideals which generalize (v 0)

Piotr Kalemba, Szymon Plewik (2010)

Open Mathematics

Countable products of finite discrete spaces with more than one point and ideals generated by Marczewski-Burstin bases (assigned to trimmed trees) are examined, using machinery of base tree in the sense of B. Balcar and P. Simon. Applying Kulpa-Szymanski Theorem, we prove that the covering number equals to the additivity or the additivity plus for each of the ideals considered.

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