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Combinatorics related to the continuum

Research Project

Project/Area Number 17540116
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKobe University

Principal Investigator

BRENDLE Jorg  Kobe U, Graduate School of Science and Technology, Associate Professor, 大学院自然科学研究科, 助教授 (70301851)

Co-Investigator(Kenkyū-buntansha) FUCHINO Sakae  Chubu University, Faculty of Engineering, Professor, 工学部, 教授 (30292098)
Project Period (FY) 2005 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2006: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsFoundations of Mathematics / Topology / Set Theory / Forcing Theory / Infinitary Combinatorics / Descriptive Set Theory / Cardinal Invariants of the Continuum / プール代数 / 強制法 / ブール代数 / 国際研究者交流 / メキシコ
Research Abstract

We investigate the combinatorial structure of the reals and its interplay with forcing theory, as well as other areas of set theory. Particular focus is put on structures like P(ω)/fin.
(1) Distributivity numbers. Using finite support iteration of Laver forcing L_F with respect to a filter F, we prove the consistency of η(P(ω)/fin x P(ω)/fin) < η(C^ω/fin) where η(A) is the distributivity number of a Boolean algebra A, and C is the Cohen algebra. This answers a question of Dow.
(2) Groupwise density numbers. Let g be the groupwise density number, and let g_f be the groupwise density number for ideals. We show the consistency of g < g_f, thus answering a question of Mildenberger.
(3) Topological groups. Let G = ([ω]^<<ω>, Δ) be the group of finite subsets of the natural numbers ω equipped with symmetric difference Δ as group multiplication. In joint work with Michael Hrusak, we obtain the consistency of the statement for all ω_1-generated filters F on ω, the group topology on G corresponding to F is not Frechet.
(4) Forcing theory. Using a novel iteration technique for ccc forcing, we obtain a new proof of Shelah's result saying that u < a is consistent where u is the ultrafilter number and a, the almost disjointness number.
(5) Mad families with strong combinatorial properties. In joint work with Greg Piper, we construct, under the continuum hypothesis CH, a maximal almost disjoint (mad) family which is simultaneously a ο-set, as well as a mad family which is concentrated on a countable subset. This confirms two conjectures of A. Miller.
(6) Homogeneity properties of product-like models. In joint work with Sakae Fuchino, we investigate several combinatorial principles which hold in generic extensions by partial-orders which admit many automorphisms. In particular, we show the homogeneity principle HP(κ) implies the combinatorial principle C^s(κ) of Juhasz, Soukup and Szentmiklossy, and we prove HP(N_2) holds in product models like the Cohen model.

Report

(3 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • Research Products

    (41 results)

All 2007 2006 2005 Other

All Journal Article (41 results)

  • [Journal Article] HAD FAMILIES WITH NRONG COMBINATORIAC PROPERTIES2007

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Funda menta Hathematicae 193

      Pages: 7-21

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] INDEPENDENCE FOR DISTRIBUTIVITY NUMBERS2007

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Algebra, Logic, Set Theory Studies in Logic (B. Lowe, ed.)(College Publications) 4

      Pages: 63-84

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] ALMOST DISJOINT FAMILIESON LARGE UNDERLYING SETS2007

    • Author(s)
      SAKAE FUCHINO
    • Journal Title

      京都大学数理解析研究所講究録 1530

      Pages: 5-16

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] MAD families with strong combinatorial properties2007

    • Author(s)
      Jorg Brendle
    • Journal Title

      Fundamenta Mathematicae 193

      Pages: 7-21

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Independence for distributivity numbers2007

    • Author(s)
      Jorg Brendle
    • Journal Title

      Algebra, Logic, Set Theory. Festschrift fur Ulrich Feigner zum 65. Geburtstag (B. Lowe, ed.), Studies in Logic (College Publications) 4

      Pages: 63-84

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Distinguishing groupwise density numbers2007

    • Author(s)
      Jorg Brendle
    • Journal Title

      Monatshefte fur Mathematik

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Almost disjoint families on large underlying sets2007

    • Author(s)
      Sakae Fuchino
    • Journal Title

      Kyoto Daigaku Surikaiseki Kenkyusho Kokyuroku 1530

      Pages: 5-16

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] MAD families with strong combinatorial properties2007

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Fundamenta Mathematicae 193

      Pages: 7-21

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Independence for Distributivity Numbers2007

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Algebra, Logic, Set Theory, Studies in Logic, College Publications London (Benedikt Lowe, ed.) 4

      Pages: 63-84

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Distinguishing groupwise density numbers2007

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Monatshefte fur Mathematik (未定)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Almost disjoint families on large underlying sets2007

    • Author(s)
      SAKAE FUCHINO
    • Journal Title

      数理解析研究所講究録 1530

      Pages: 5-16

    • Related Report
      2006 Annual Research Report
  • [Journal Article] CONVERSE DUAL CARDINALS2006

    • Author(s)
      JORG BRENDLE
    • Journal Title

      THE Journal of Symbolic Cogic 71-1

      Pages: 22-34

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] DISTRIBUTIVITY NUMBERS OF P(w)/fim AND ITS FRIENDS2006

    • Author(s)
      JORG BRENDLE
    • Journal Title

      京都大学数理解析研究所講究録 1471

      Pages: 9-18

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] VAN DouWEN'S DSAGRAM FOR DENSE SETS of RATIONAls2006

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Annals of Pure and Applied Logic 143

      Pages: 54-69

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] CARDINAL SNVARIANTS OF THE CONTINUUM AND COMBINATORICS ON UNCouNTABLE CARDINALS2006

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Annals of Pure and Applied Logic 144

      Pages: 43-72

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] κ^κ IN LIGHT OF THE TUKEY ORDERING2006

    • Author(s)
      SAKAE FUCHINO
    • Journal Title

      京都大学数理解析研究所講究録 1471

      Pages: 19-34

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] MODELS OF REAL-VALUED MEASURABILITY2006

    • Author(s)
      SAKAE FUCHINO
    • Journal Title

      Annals of Pure and Applied Logic 142

      Pages: 380-397

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Converse dual cardinals2006

    • Author(s)
      Jorg Brendle
    • Journal Title

      The Journal of Symbolic Logic 71-1

      Pages: 22-34

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Distributivity numbers of P(ω)/fin and its friends2006

    • Author(s)
      Jiirg Brendle
    • Journal Title

      Kyoto Daigaku Surikaiseki Kenkyusho Kokyuroku 1471

      Pages: 9-18

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Van Douwen's diagram for dense sets of rationales2006

    • Author(s)
      Jorg Brendle
    • Journal Title

      Annals of Pure and Applied Logic 143

      Pages: 54-69

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Cardinal invariants of the continuum and combinatorics on uncountable cardinals2006

    • Author(s)
      Jorg Brendle
    • Journal Title

      Annals of Pure and Applied Logic 144

      Pages: 43-72

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] k^k in light of the Tukey ordering2006

    • Author(s)
      Sakae Fuchino
    • Journal Title

      Kyoto Daigaku Surikaiseki Kenkyusho Kokyuroku 1471

      Pages: 19-34

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Models of real-valued measurability2006

    • Author(s)
      Sakae Fuchino
    • Journal Title

      Annals of Pure and Applied Logic 142

      Pages: 380-397

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Van Douwen's diagram for dense sets of rationals2006

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Annals of Pure and Applied Logic 143

      Pages: 54-69

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Cardinal invariants of the continuum and combinatorics on uncountable cardinals2006

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Annals of Pure and Applied Logic 144

      Pages: 43-72

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Models of real-valued measurability2006

    • Author(s)
      SAKAE FUCHINO
    • Journal Title

      Annals of Pure and Applied Logic 142

      Pages: 380-397

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Converse dual cardinals2006

    • Author(s)
      JORG BRENDLE
    • Journal Title

      The Journal of Symbolic Logic 71・1

      Pages: 22-34

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Distributivity numbers of P(omega)/fin and its friends2006

    • Author(s)
      JORG BRENDLE
    • Journal Title

      数理解析研究所講究録 (未定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Kappa^kappa in light of the Tukey ordering2006

    • Author(s)
      SAKAE FUCHINO
    • Journal Title

      数理解析研究所講究録 (未定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] TEMPLATES AND ITERATIONS2005

    • Author(s)
      JORG BRENDLE
    • Journal Title

      京都大学数理解析研究所講究録 1423

      Pages: 1-12

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] PRINC(κ,λ), C^s(κ), HP(κ) etc.2005

    • Author(s)
      SAKAE FUCHINO
    • Journal Title

      京都大学数理解析研究所講究録 1423

      Pages: 13-27

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] DESTRUCTIBIUTY OF STATIONARY SUBSETS OF P_κ(λ)2005

    • Author(s)
      SAKAE FUCHINO
    • Journal Title

      Mathematical Logic Quanterly 51-6

      Pages: 560-569

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Templates and iterations. Luminy 2002 lecture notes2005

    • Author(s)
      Jorg Brendle
    • Journal Title

      Kyoto Daigaku Surikaiseki Kenkyusho Kokyuroku 1423

      Pages: 1-12

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] PRINC (γ,λ), C^s ( K ), HP (k) etc. and variants of the bounding number2005

    • Author(s)
      Sakae Fuchino
    • Journal Title

      Kyoto Daigaku Surikaiseki Kenkyusho Kokyuroku 1423

      Pages: 13-27

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Destructibility of stationary subsets of P_K (λ)2005

    • Author(s)
      Salute Fuchino
    • Journal Title

      Mathematical Logic Quarterly 51-6

      Pages: 560-569

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Templates and iterations. Luminy 2002 lecture notes2005

    • Author(s)
      JORG BRENDLE
    • Journal Title

      数理解析研究所講究録 1423

      Pages: 1-12

    • Related Report
      2005 Annual Research Report
  • [Journal Article] PRINC(kappa, lambda), C^s(kappa), HP(kappa) etc. and variants of the bounding number2005

    • Author(s)
      SAKAE FUCHINO
    • Journal Title

      数理解析研究所講究録 1423

      Pages: 13-27

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Destructibility of stationary subsets of P_kappa lambda2005

    • Author(s)
      SAKAE FUCHINO
    • Journal Title

      Mathematical Logic Quarterly 51・6

      Pages: 560-569

    • Related Report
      2005 Annual Research Report
  • [Journal Article] DISTINGUISHING GRouPWISE DENSITY NUHBERS

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Mouats heffe fur Hathemafik (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Cardinals invariants of the continuum and combinatorics on uncountable cardinals

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Annals of Pure and Applied Logic (未定)(掲載確定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Van Douwen's diagram for dense sets of rationals

    • Author(s)
      JORG BRENDLE
    • Journal Title

      Annals of Pure and Applied Logic (未定)(掲載確定)

    • Related Report
      2005 Annual Research Report

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Published: 2005-04-01   Modified: 2021-04-07  

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