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Cardinal invariants of the lattice of partitions

Barbara Majcher-Iwanow (2000)

Commentationes Mathematicae Universitatis Carolinae

We study cardinal coefficients related to combinatorial properties of partitions of ω with respect to the order of almost containedness.

Cardinal invariants of ultraproducts of Boolean algebras

Andrzej Rosłanowski, Saharon Shelah (1998)

Fundamenta Mathematicae

We deal with some problems posed by Monk [Mo 1], [Mo 3] and related to cardinal invariants of ultraproducts of Boolean algebras. We also introduce and investigate several new cardinal invariants.

Clones on regular cardinals

Martin Goldstern, Saharon Shelah (2002)

Fundamenta Mathematicae

We investigate the structure of the lattice of clones on an infinite set X. We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg’s theorem: there are 2 2 λ maximal (= “precomplete”) clones on a set of size λ. The clones we construct do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem from pcf theory we show that for cardinals λ (in particular, for all successors of...

Club-guessing and non-structure of trees

Tapani Hyttinen (2001)

Fundamenta Mathematicae

We study the possibilities of constructing, in ZFC without any additional assumptions, strongly equivalent non-isomorphic trees of regular power. For example, we show that there are non-isomorphic trees of power ω₂ and of height ω · ω such that for all α < ω₁· ω · ω, E has a winning strategy in the Ehrenfeucht-Fraïssé game of length α. The main tool is the notion of a club-guessing sequence.

Club-guessing, good points and diamond

Pierre Matet (2007)

Commentationes Mathematicae Universitatis Carolinae

Shelah’s club-guessing and good points are used to show that the two-cardinal diamond principle κ , λ holds for various values of κ and λ .

Cofinal types of topological directed orders

SŁawomir Solecki, Stevo Todorcevic (2004)

Annales de l’institut Fourier

We investigate the structure of the Tukey ordering among directed orders arising naturally in topology and measure theory.

Cohen real and disjoint refinement of perfect sets

Miroslav Repický (2000)

Commentationes Mathematicae Universitatis Carolinae

We prove that if there exists a Cohen real over a model, then the family of perfect sets coded in the model has a disjoint refinement by perfect sets.

Coherent adequate sets and forcing square

John Krueger (2014)

Fundamenta Mathematicae

We introduce the idea of a coherent adequate set of models, which can be used as side conditions in forcing. As an application we define a forcing poset which adds a square sequence on ω₂ using finite conditions.

Coloring grids

Ramiro de la Vega (2015)

Fundamenta Mathematicae

A structure = ( A ; E i ) i n where each E i is an equivalence relation on A is called an n-grid if any two equivalence classes coming from distinct E i ’s intersect in a finite set. A function χ: A → n is an acceptable coloring if for all i ∈ n, the χ - 1 ( i ) intersects each E i -equivalence class in a finite set. If B is a set, then the n-cube Bⁿ may be seen as an n-grid, where the equivalence classes of E i are the lines parallel to the ith coordinate axis. We use elementary submodels of the universe to characterize those n-grids...

Coloring ordinals by reals

Jörg Brendle, Sakaé Fuchino (2007)

Fundamenta Mathematicae

We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of C s ( κ ) and F s ( κ ) of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence also HP(ℵ₂))...

Combinatorics of ideals --- selectivity versus density

A. Kwela, P. Zakrzewski (2017)

Commentationes Mathematicae Universitatis Carolinae

This note is devoted to combinatorial properties of ideals on the set of natural numbers. By a result of Mathias, two such properties, selectivity and density, in the case of definable ideals, exclude each other. The purpose of this note is to measure the ``distance'' between them with the help of ultrafilter topologies of Louveau.

Compactness of Powers of ω

Paolo Lipparini (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary extensions. We also have results involving products of possibly uncountable regular cardinals.

Comparing the closed almost disjointness and dominating numbers

Dilip Raghavan, Saharon Shelah (2012)

Fundamenta Mathematicae

We prove that if there is a dominating family of size ℵ₁, then there are ℵ₁ many compact subsets of ω ω whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.

Comparison game on Borel ideals

Michael Hrušák, David Meza-Alcántara (2011)

Commentationes Mathematicae Universitatis Carolinae

We propose and study a “classification” of Borel ideals based on a natural infinite game involving a pair of ideals. The game induces a pre-order and the corresponding equivalence relation. The pre-order is well founded and “almost linear”. We concentrate on F σ and F σ δ ideals. In particular, we show that all F σ -ideals are -equivalent and form the least equivalence class. There is also a least class of non- F σ Borel ideals, and there are at least two distinct classes of F σ δ non- F σ ideals.

Constructions of thin-tall Boolean spaces.

Juan Carlos Martínez (2003)

Revista Matemática Complutense

This is an expository paper about constructions of locally compact, Hausdorff, scattered spaces whose Cantor-Bendixson height has cardinality greater than their Cantor-Bendixson width.

Convexity ranks in higher dimensions

Menachem Kojman (2000)

Fundamenta Mathematicae

A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a Polish linear...

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