Skip to main content
SpringerLink
Log in

Remarks on partition ordinals

  • Conference paper
  • First Online:
Set Theory and its Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1401))

Abstract

After a brief survey of the theory of partition ordinals, i.e., ordinals α such that α → (α,n)2 for all n < ω, it is shown that MA(ℵ1) implies that ω1ω and ω1ω2 are partition ordinals. This contrasts with an old result of Erdős and Hajnal that α ↛ (α,3)2 holds for both these ordinals under the Continuum Hypothesis.

Preparation of this paper was partially supported by National Science Foundation grant number DMS-8704586.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Baumgartner, Partition relations for uncountable cardinals, Israel J. Math. 21 (1975), 296–307.

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Devlin and H. Johnsbråten, The Souslin Problem, Lecture Notes in Mathematics 405, Springer-Verlag, 1974.

    Google Scholar 

  3. P. Erdős and A. Hajnal, Ordinary partition relations for ordinal numbers, Periodica Math. Hung. 1 (1971), 171–185.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Erdős, A. Hajnal, A. Máté and R. Rado, Combinatorial Set Theory: Partition Relations for Cardinals, North-Holland, 1984.

    Google Scholar 

  5. F. Galvin and J. Larson, Pinning countable ordinals, Fund. Math. 82 (1975), 357–361.

    MathSciNet  MATH  Google Scholar 

  6. J. Larson, Partition theorems for certain ordinal products, in Infinite and Finite sets, Coll. Math. Soc. J. Bolyai 10, Keszthely, Hungary, 1973, 1017–1024.

    Google Scholar 

  7. J. Larson, A counter-example in the partition calculus for an uncountable ordinal, Israel J. Math. 36 (1980), 287–299.

    Article  MathSciNet  MATH  Google Scholar 

  8. T. Miyamoto, Ph.D. dissertation, Dartmouth College, 1987.

    Google Scholar 

  9. S. Shelah and L. Stanley, A theorem and some consistency results in partition calculus, to appear.

    Google Scholar 

  10. E. Specker, Teilmengen von Mengen mit Relationen, Comment. Math. Helv. 31 (1957), 302–314.

    Article  MathSciNet  MATH  Google Scholar 

  11. N. H. Williams, Combinatorial Set Theory, North-Holland, 1977.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dartmouth College, 03755, Hanover, NH, USA

    James E. Baumgartner

Authors
  1. James E. Baumgartner
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Juris Steprāns Stephen Watson

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

Baumgartner, J.E. (1989). Remarks on partition ordinals. In: Steprāns, J., Watson, S. (eds) Set Theory and its Applications. Lecture Notes in Mathematics, vol 1401. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097328

Download citation

  • DOI: https://doi.org/10.1007/BFb0097328

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51730-6

  • Online ISBN: 978-3-540-46795-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation