Skip to main content
SpringerLink
Log in

Cardinal characteristics of the continuum and partitions

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. E. Baumgartner, Almost-disjoint sets, the dense set problem and the partition calculus, Annals of Mathematical Logic 9 (1976), 401–439.

    Article  MathSciNet  Google Scholar 

  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. 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.

    Article  MathSciNet  Google Scholar 

  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.

    Chapter  Google Scholar 

  5. J. Brendle, Cardinal invariants of the continuum and combinatorics on uncountable cardinals, Annals of Pure and Applied Logic 144 (2006), 43–72.

    Article  MathSciNet  Google Scholar 

  6. S. A. Broverman, J. N. Ginsburg, K. Kunen and F. D. Tall, Topologies determined by σ-ideals on ω1, Canadian Journal of Mathematics 30 (1978), 1306–1312.

    Article  MathSciNet  Google Scholar 

  7. P. E. Cohen, Partition generation of scales, Fundamenta Mathematicae 103 (1979), 77–82.

    Article  MathSciNet  Google Scholar 

  8. B. Dushnik and E. W. Miller, Partially ordered sets, American Journal of Mathematics 63 (1941), 600–610.

    Article  MathSciNet  Google Scholar 

  9. P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Mathematica. Academiae Scientiarum Hungaricae 16 (1965), 93–196.

    Article  MathSciNet  Google Scholar 

  10. P. Erdős and R. Rado, A partition calculus in set theory, Bulletin of the American Mathematical Society 62 (1956), 427–489.

    Article  MathSciNet  Google Scholar 

  11. P. Erdős and A. Hajnal, Ordinary partition relations for ordinal numbers, Periodica Mathematica Hungarica 1 (1971), 171–185.

    Article  MathSciNet  Google Scholar 

  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.

  13. V. Fischer and D. Soukup, More ZFC inequalities between cardinal invariants, preprint, https://arxiv.org/abs/1802.02791.

  14. V. Fischer and J. Steprāns, The consistency of b = κ and s = κ+, Fundamenta Mathematicae 201 (2008), 283–293.

    Article  MathSciNet  Google Scholar 

  15. S. Fuchino, S. Shelah and L. Soukup, Sticks and clubs, Annals of Pure and Applied Logic 90 (1997), 57–77.

    Article  MathSciNet  Google Scholar 

  16. S. Garti and S. Shelah, Partition calculus and cardinal invariants, Journal of the Mathematical Society of Japan 66 (2014), 425–434.

    Article  MathSciNet  Google Scholar 

  17. S. Garti and S. Shelah, Open and solved problems concerning polarized partition relations, Fundamenta Mathematicae 234 (2016), 1–14.

    Article  MathSciNet  Google Scholar 

  18. S. Garti and S. Shelah, Random reals and polarized colorings, Studia Scientiarum Mathematicarum Hungarica 55 (2018), 203–212.

    Article  MathSciNet  Google Scholar 

  19. A. Hajnal, Some results and problems on set theory, Acta Mathematica. Academiae Scientiarum Hungaricae 11 (1960), 277–298.

    Article  MathSciNet  Google Scholar 

  20. A. Hajnal, A negative partition relation, Proceedings of the National Academy of Sciences of the United States of America 68 (1971), 142–144.

    Article  MathSciNet  Google Scholar 

  21. A. Hajnal and J. A. Larson, Partition relations, in Handbook of Set Theory. Vol. 1, Springer, Dordrecht, 2010, pp. 129–213.

    Article  MathSciNet  Google Scholar 

  22. L. J. Halbeisen, Combinatorial Set Theory, Springer Monographs in Mathematics, Springer, Cham, 2017.

    Book  Google Scholar 

  23. H. Judah and S. Shelah, Killing Luzin and Sierpinski sets, Proceedings of the American Mathematical Society 120 (1994), 917–920.

    MathSciNet  MATH  Google Scholar 

  24. K. Kunen, Set Theory, Studies in Logic and the Foundations of Mathematics, Vol. 102, North-Holland, Amsterdam, 1980.

  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.

    Article  MathSciNet  Google Scholar 

  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. N. N. Luzin, Sur un probleme de M. Baire, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 158 (1914), 1258–1261.

    MATH  Google Scholar 

  28. D. Raghavan and S. Shelah, Two inequalities between cardinal invariants, Fundmenta Mathematicae 237 (2017), 187–200.

    Article  MathSciNet  Google Scholar 

  29. D. Raghavan and S. Todorcevic, Combinatorial dichotomies and cardinal invariants, Mathematical Research Letters 21 (2014), 379–401.

    Article  MathSciNet  Google Scholar 

  30. D. Raghavan and S. Todorcevic, Suslin trees, the bounding number, and partition relations, Israel Journal of Mathematics 225 (2018), 771–796.

    Article  MathSciNet  Google Scholar 

  31. S. Shelah, Colouring without triangles and partition relation, Israel Journal of Mathematics 20 (1975), 1–12.

    Article  MathSciNet  Google Scholar 

  32. J. Takahashi, Two negative partition relations, Periodica Mathematica Hungarica 18 (1987), 1–6.

    Article  MathSciNet  Google Scholar 

  33. S. Todorčević, Forcing positive partition relations, Transactions of the American Mathematical Society 280 (1983), 703–720.

    Article  MathSciNet  Google Scholar 

  34. S. Todorčević, Partition Problems in Topology, Contemporary Mathematics, Vol. 84, American Mathematical Society, Providence, RI, 1989.

  35. J. K. Truss, The noncommutativity of random and generic extensions, Journal fo Symbolic Logic 48 (1983), 1008–1012.

    Article  MathSciNet  Google Scholar 

  36. N. H. Williams, Combinatorial Set Theory, Studies in Logic and the Foundations of Mathematics, Vol. 91, North-Holland, Amsterdam, 1977.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion-University of the Negev, Beersheba, 8410501, Israel

    William Chen

  2. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Shimon Garti

  3. Kurt Gödel Research Centre for Mathematical Logic, University of Vienna, Währinger Straße 25, 1090, Vienne, Autriche

    Thilo Weinert

Authors
  1. William Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shimon Garti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thilo Weinert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thilo Weinert.

Additional information

A portion of this research was undertaken while both the first and the last author were postdoctoral fellows at Ben-Gurion University of the Negev. They would like to thank Ben-Gurion University of the Negev and the Israel Science Foundation which supported this research (grant #1365/14). The second author was supported by the European Research Council, grant 338821. The last author moreover thanks the FWF for supporting him during the revision of this paper through the grants numbered I3081 and Y1012-N35.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, W., Garti, S. & Weinert, T. Cardinal characteristics of the continuum and partitions. Isr. J. Math. 235, 13–38 (2020). https://doi.org/10.1007/s11856-019-1942-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-019-1942-y

Search

Navigation