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Switch to: Citations
References in:
Van Douwen’s diagram for dense sets of rationals
Annals of Pure and Applied Logic 143 (1-3):54-69 (2006)
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We make a more systematic study of the van Douwen diagram for cardinal coefficients related to combinatorial properties of partitions of natural numbers. |
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Let J be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal J * ∣B as follows: X ∈ J * ∣B if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any x ∈ R we have $\{y \in R: \langle x,y\rangle \in B\} \in \mathscr{J}$ . We show that there exists a family $\mathscr{A} \subset \mathscr{J}^\ast\mid\mathscr{B}$ of power ω (...) |
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If dominating functions in ω ω are adjoined repeatedly over a model of GCH via a finite-support c.c.c. iteration, then in the resulting generic extension there are no long towers, every well-ordered unbounded family of increasing functions is a scale, and the splitting number s (and hence the distributivity number h) remains at ω 1. |
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There is an optimal way of increasing certain cardinal invariants of the continuum. |
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We investigate some notions of splitting families and estimate sizes of the corresponding cardinal coefficients. In particular we solve a problem of P. Simon. |
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We investigate the set (ω) of partitions of the natural numbers ordered by ≤* where A ≤* B if by gluing finitely many blocks of A we can get a partition coarser than B. In particular, we determine the values of a number of cardinals which are naturally associated with the structure ((ω),≥*), in terms of classical cardinal invariants of the continuum. |
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We study the I-ultrafilters on ω, where I is a collection of subsets of a set X, usually R or ω 1 . The I-ultrafilters usually contain the P-points, often as a small proper subset. We study relations between I-ultrafilters for various I, and closure of I-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether I-ultrafilters always exist. |
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We study the $I$-ultrafilters on $\omega$, where $I$ is a collection of subsets of a set $X$, usually $\mathbb{R}$ or $\omega_1$. The $I$-ultrafilters usually contain the $P$-points, often as a small proper subset. We study relations between $I$-ultrafilters for various $I$, and closure of $I$-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether $I$-ultrafilters always exist. |
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