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Cardinal characteristics of the continuum and partitions

  • William Chen
  • Shimon Garti
  • Thilo WeinertEmail author
Article
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Abstract

We prove that for regular cardinals κ, combinations of the stick principle at κ and certain cardinal characteristics at κ being κ+ cause partition relations such as κ+ → (κ+, (κ : 2))2 and (κ+)2 → (κ+κ, 4)2 to fail. Polarised partition relations are also considered, and the results are used to answer several problems posed by Garti, Larson and Shelah.

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References

  1. [1]
    J. E. Baumgartner, Almost-disjoint sets, the dense set problem and the partition calculus, Annals of Mathematical Logic 9 (1976), 401–439.MathSciNetCrossRefGoogle Scholar
  2. [2]
    J. E. Baumgartner, Remarks on partition ordinals, in Set Theory and its Applications (Toronto, ON, 1987), Lecture Notes in Mathematics, Vol. 1401, Springer, Berlin, 1989, pp. 5–17.Google Scholar
  3. [3]
    J. E. Baumgartner and A. Hajnal, A remark on partition relations for infinite ordinals with an application to finite combinatorics, in Logic and Combinatorics (Arcata, CA, 1985), Contemporary Mathematics, Vol. 65, AmericanMathematical Society, Providence, RI, 1987, pp. 157–167.MathSciNetCrossRefGoogle Scholar
  4. [4]
    A. R. Blass, Combinatorial cardinal characteristics of the continuum, in Handbook of Set Theory, Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 395–489.CrossRefGoogle Scholar
  5. [5]
    J. Brendle, Cardinal invariants of the continuum and combinatorics on uncountable cardinals, Annals of Pure and Applied Logic 144 (2006), 43–72.MathSciNetCrossRefGoogle Scholar
  6. [6]
    S. A. Broverman, J. N. Ginsburg, K. Kunen and F. D. Tall, Topologies determined by σ-ideals on ω 1, Canadian Journal of Mathematics 30 (1978), 1306–1312.MathSciNetCrossRefGoogle Scholar
  7. [7]
    P. E. Cohen, Partition generation of scales, Fundamenta Mathematicae 103 (1979), 77–82.MathSciNetCrossRefGoogle Scholar
  8. [8]
    B. Dushnik and E. W. Miller, Partially ordered sets, American Journal of Mathematics 63 (1941), 600–610.MathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Mathematica. Academiae Scientiarum Hungaricae 16 (1965), 93–196.MathSciNetCrossRefGoogle Scholar
  10. [10]
    P. Erdős and R. Rado, A partition calculus in set theory, Bulletin of the American Mathematical Society 62 (1956), 427–489.MathSciNetCrossRefGoogle Scholar
  11. [11]
    P. Erdős and A. Hajnal, Ordinary partition relations for ordinal numbers, Periodica Mathematica Hungarica 1 (1971), 171–185.MathSciNetCrossRefGoogle Scholar
  12. [12]
    P. Erdőos, A. Hajnal, A. Máté and R. Rado, Combinatorial Set Theory: Partition Relations for Cardinals, Studies in Logic and the Foundations of Mathematics, Vol. 106, North-Holland, Amsterdam, 1984.Google Scholar
  13. [13]
    V. Fischer and D. Soukup, More ZFC inequalities between cardinal invariants, preprint, https://arxiv.org/abs/1802.02791.
  14. [14]
    V. Fischer and J. Steprāns, The consistency of b = κ and s = κ +, Fundamenta Mathematicae 201 (2008), 283–293.MathSciNetCrossRefGoogle Scholar
  15. [15]
    S. Fuchino, S. Shelah and L. Soukup, Sticks and clubs, Annals of Pure and Applied Logic 90 (1997), 57–77.MathSciNetCrossRefGoogle Scholar
  16. [16]
    S. Garti and S. Shelah, Partition calculus and cardinal invariants, Journal of the Mathematical Society of Japan 66 (2014), 425–434.MathSciNetCrossRefGoogle Scholar
  17. [17]
    S. Garti and S. Shelah, Open and solved problems concerning polarized partition relations, Fundamenta Mathematicae 234 (2016), 1–14.MathSciNetCrossRefGoogle Scholar
  18. [18]
    S. Garti and S. Shelah, Random reals and polarized colorings, Studia Scientiarum Mathematicarum Hungarica 55 (2018), 203–212.MathSciNetCrossRefGoogle Scholar
  19. [19]
    A. Hajnal, Some results and problems on set theory, Acta Mathematica. Academiae Scientiarum Hungaricae 11 (1960), 277–298.MathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Hajnal, A negative partition relation, Proceedings of the National Academy of Sciences of the United States of America 68 (1971), 142–144.MathSciNetCrossRefGoogle Scholar
  21. [21]
    A. Hajnal and J. A. Larson, Partition relations, in Handbook of Set Theory. Vol. 1, Springer, Dordrecht, 2010, pp. 129–213.MathSciNetCrossRefGoogle Scholar
  22. [22]
    L. J. Halbeisen, Combinatorial Set Theory, Springer Monographs in Mathematics, Springer, Cham, 2017.CrossRefGoogle Scholar
  23. [23]
    H. Judah and S. Shelah, Killing Luzin and Sierpinski sets, Proceedings of the American Mathematical Society 120 (1994), 917–920.MathSciNetzbMATHGoogle Scholar
  24. [24]
    K. Kunen, Set Theory, Studies in Logic and the Foundations of Mathematics, Vol. 102, North-Holland, Amsterdam, 1980.Google Scholar
  25. [25]
    C. Lambie-Hanson and T. V. Weinert, Partitioning subsets of generalised scattered orders, Journal of the Mathematical Society of Japan 71 (2019), 425–434.MathSciNetCrossRefGoogle Scholar
  26. [26]
    J. A. Larson, An ordinal partition from a scale, in Set theory (Cura¸cao, 1995; Barcelona, 1996), Kluwer Academic, Dordrecht, 1998, pp. 109–125.Google Scholar
  27. [27]
    N. N. Luzin, Sur un probleme de M. Baire, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 158 (1914), 1258–1261.zbMATHGoogle Scholar
  28. [28]
    D. Raghavan and S. Shelah, Two inequalities between cardinal invariants, Fundmenta Mathematicae 237 (2017), 187–200.MathSciNetCrossRefGoogle Scholar
  29. [29]
    D. Raghavan and S. Todorcevic, Combinatorial dichotomies and cardinal invariants, Mathematical Research Letters 21 (2014), 379–401.MathSciNetCrossRefGoogle Scholar
  30. [30]
    D. Raghavan and S. Todorcevic, Suslin trees, the bounding number, and partition relations, Israel Journal of Mathematics 225 (2018), 771–796.MathSciNetCrossRefGoogle Scholar
  31. [31]
    S. Shelah, Colouring without triangles and partition relation, Israel Journal of Mathematics 20 (1975), 1–12.MathSciNetCrossRefGoogle Scholar
  32. [32]
    J. Takahashi, Two negative partition relations, Periodica Mathematica Hungarica 18 (1987), 1–6.MathSciNetCrossRefGoogle Scholar
  33. [33]
    S. Todorčević, Forcing positive partition relations, Transactions of the American Mathematical Society 280 (1983), 703–720.MathSciNetCrossRefGoogle Scholar
  34. [34]
    S. Todorčević, Partition Problems in Topology, Contemporary Mathematics, Vol. 84, American Mathematical Society, Providence, RI, 1989.Google Scholar
  35. [35]
    J. K. Truss, The noncommutativity of random and generic extensions, Journal fo Symbolic Logic 48 (1983), 1008–1012.MathSciNetCrossRefGoogle Scholar
  36. [36]
    N. H. Williams, Combinatorial Set Theory, Studies in Logic and the Foundations of Mathematics, Vol. 91, North-Holland, Amsterdam, 1977.Google Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsBen-Gurion-University of the NegevBeershebaIsrael
  2. 2.Institute of MathematicsThe Hebrew University of Jerusalem91904Israel
  3. 3.Kurt Gödel Research Centre for Mathematical LogicUniversity of ViennaVienneAutriche

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