16 found
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  1.  40
    Forcing with Quotients.Michael Hrušák & Jindřich Zapletal - 2008 - Archive for Mathematical Logic 47 (7-8):719-739.
    We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.
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  2.  42
    Mathias–Prikry and Laver–Prikry Type Forcing.Michael Hrušák & Hiroaki Minami - 2014 - Annals of Pure and Applied Logic 165 (3):880-894.
    We study the Mathias–Prikry and Laver–Prikry forcings associated with filters on ω. We give a combinatorial characterization of Martinʼs number for these forcing notions and present a general scheme for analyzing preservation properties for them. In particular, we give a combinatorial characterization of those filters for which the Mathias–Prikry forcing does not add a dominating real.
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  3.  86
    Pair-Splitting, Pair-Reaping and Cardinal Invariants of F Σ -Ideals.Michael Hrušák, David Meza-Alcántara & Hiroaki Minami - 2010 - Journal of Symbolic Logic 75 (2):661-677.
    We investigate the pair-splitting number $\germ{s}_{pair}$ which is a variation of splitting number, pair-reaping number $\germ{r}_{pair}$ which is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants of F σ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.
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  4.  12
    Canjar Filters.Osvaldo Guzmán, Michael Hrušák & Arturo Martínez-Celis - 2017 - Notre Dame Journal of Formal Logic 58 (1):79-95.
    If $\mathcal{F}$ is a filter on $\omega$, we say that $\mathcal{F}$ is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is $F_{\sigma}$, solving a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters; in particular, we construct a $\mathsf{MAD}$ family such that the corresponding Mathias forcing adds a dominating real. This answers a question of Brendle. Then we prove that in all the “classical” models (...)
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  5.  6
    Katětov Order on Borel Ideals.Michael Hrušák - 2017 - Archive for Mathematical Logic 56 (7-8):831-847.
    We study the Katětov order on Borel ideals. We prove two structural theorems, one for Borel ideals, the other for analytic P-ideals. We isolate nine important Borel ideals and study the Katětov order among them. We also present a list of fundamental open problems concerning the Katětov order on Borel ideals.
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  6.  24
    Countable Fréchet Boolean Groups: An Independence Result.Jörg Brendle & Michael Hrušák - 2009 - Journal of Symbolic Logic 74 (3):1061-1068.
    It is relatively consistent with ZFC that every countable $FU_{fin} $ space of weight N₁ is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].
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  7.  3
    Preservation Theorems for Namba Forcing.Osvaldo Guzmán, Michael Hrušák & Jindřich Zapletal - 2021 - Annals of Pure and Applied Logic 172 (2):102869.
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  8.  22
    Ordering MAD Families a la Katětov.Michael Hrušák & Salvador García Ferreira - 2003 - Journal of Symbolic Logic 68 (4):1337-1353.
    An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size.
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  9.  7
    Cofinalities of Borel Ideals.Michael Hrušák, Diego Rojas-Rebolledo & Jindřich Zapletal - 2014 - Mathematical Logic Quarterly 60 (1-2):31-39.
  10.  9
    Construction with opposition: cardinal invariants and games.Jörg Brendle, Michael Hrušák & Víctor Torres-Pérez - 2019 - Archive for Mathematical Logic 58 (7):943-963.
    We consider several game versions of the cardinal invariants \, \ and \. We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that \ and \ are both relatively consistent with ZFC, where \ and \ are the principal game versions of \ and \, respectively. The corresponding question for \ remains open.
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  11.  4
    Restricted Mad Families.Osvaldo Guzmán, Michael Hrušák & Osvaldo Téllez - 2020 - Journal of Symbolic Logic 85 (1):149-165.
    Let ${\cal I}$ be an ideal on ω. By cov${}_{}^{\rm{*}}$ we denote the least size of a family ${\cal B} \subseteq {\cal I}$ such that for every infinite $X \in {\cal I}$ there is $B \in {\cal B}$ for which $B\mathop \cap \nolimits X$ is infinite. We say that an AD family ${\cal A} \subseteq {\cal I}$ is a MAD family restricted to${\cal I}$ if for every infinite $X \in {\cal I}$ there is $A \in {\cal A}$ such that $|X\mathop (...)
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  12.  55
    Cofinitary Groups, Almost Disjoint and Dominating Families.Michael Hrušák, Juris Steprans & Yi Zhang - 2001 - Journal of Symbolic Logic 66 (3):1259-1276.
    In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofinitary group of Sym(ω) is strictly greater than the cardinal numbers o and a.
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  13.  31
    Cardinal Invariants of Monotone and Porous Sets.Michael Hrušák & Ondřej Zindulka - 2012 - Journal of Symbolic Logic 77 (1):159-173.
    A metric space (X, d) is monotone if there is a linear order < on X and a constant c such that d(x, y) ≤ c d(x, z) for all x < y < z in X. We investigate cardinal invariants of the σ-ideal Mon generated by monotone subsets of the plane. Since there is a strong connection between monotone sets in the plane and porous subsets of the line, plane and the Cantor set, cardinal invariants of these ideals are (...)
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  14.  9
    Generic Existence of Mad Families.Osvaldo Guzmán-gonzález, Michael Hrušák, Carlos Azarel Martínez-Ranero & Ulises Ariet Ramos-garcía - 2017 - Journal of Symbolic Logic 82 (1):303-316.
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  15.  11
    Weak Partition Properties on Trees.Michael Hrušák, Petr Simon & Ondřej Zindulka - 2013 - Archive for Mathematical Logic 52 (5-6):543-567.
    We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ <ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa > \mathfrak{d}}$ (...)
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  16.  8
    Strong Measure Zero in Separable Metric Spaces and Polish Groups.Michael Hrušák, Wolfgang Wohofsky & Ondřej Zindulka - 2016 - Archive for Mathematical Logic 55 (1-2):105-131.
    The notion of strong measure zero is studied in the context of Polish groups and general separable metric spaces. An extension of a theorem of Galvin, Mycielski and Solovay is given, whereas the theorem is shown to fail for the Baer–Specker group Zω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}^{\omega}}}$$\end{document}. The uniformity number of the ideal of strong measure zero subsets of a separable metric space is examined, providing solutions to several problems of Miller and Steprāns :52–59, 2006).
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