Skip to main content
SpringerLink
Log in

A short note on the characterization of countable chains with finite big Ramsey spectra

  • Published:
Order Aims and scope Submit manuscript

Abstract

In this short note we confirm the deep structural correspondence between the complexity of a countable scattered chain (\(=\) strict linear order) and its big Ramsey combinatorics: we show that a countable scattered chain has finite big Ramsey degrees if and only if it is of finite Hausdorff rank. This also provides a complete characterization of countable chains whose big Ramsey spectra are finite. We expand the notion of big Ramsey spectrum to monomorphic structures and give a sufficient condition for a monomorphic countable structure to have finite big Ramsey spectrum.

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

Availability of data and materials

No datasets were used or generated in the course of the preparation of the paper.

References

  1. Boyland, J., Gasarch, W., Hurtig, N., Rust, R.: Big Ramsey Degrees of Countable Ordinals. arXiv:2305.07192

  2. Devlin, D.: Some partition theorems and ultrafilters on \(\omega \). Ph.D. Thesis, Dartmouth College, (1979)

  3. Erdős, P., Hajnal, A.: On a classification of denumerable order types and an application to the partition calculus. Fundam. Math. 51, 117–129 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fraïssé, R.: Theory of relations, Revised edition, With an appendix by Norbert Sauer. Studies in Logic and the Foundations of Mathematics, 145. North-Holland, Amsterdam, (2000)

  5. Galvin, F.: Partition theorems for the real line. Notices Amer. Math. Soc. 15, 660 (1968)

    Google Scholar 

  6. Galvin, F.: Errata to “Partition theorems for the real line." Notices Amer. Math. Soc. 16, 1095 (1969)

  7. Gheysens, M.: Dynamics and structure of groups of homeomorphisms of scattered spaces. Enseign. Math. 67(1/2), 161–186 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hausdorff, F.: Grundzüge einer Theorie der Geordnete Mengen. Math. Ann. 65, 435–505 (1908)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kechris, A.S., Pestov, V.G., Todorčević, S.: Fraïssé limits, Ramsey theory and topological dynamics of automorphism groups. GAFA Geometric Funct. Anal. 15, 106–189 (2005)

    Article  MATH  Google Scholar 

  10. Laver, R.: An Order Type Decomposition Theorem. Ann. Math. Second Series 98, 96–119 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mašulović D.: Ramsey degrees: big v. small. Eur. J. Comb. 95, 103323 (2021)

  12. Mašulović, D.: Big Ramsey spectra of countable chains. Order 40, 237–256 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mašulović, D., Šobot, B.: Countable ordinals and big Ramsey degrees. Combinatorica 41, 425–446 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pouzet, M.: Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d’une relation. Z. Math. Logik Grundlag. Math. 27, 289–332 (1981)

    Article  MATH  Google Scholar 

Download references

Funding

The first author was supported by the http://www.fields.utoronto.ca/ Fields Institute for Research in Mathematical Sciences.The second author was supported by the Science Fund of the Republic of Serbia, Grant No. 7750027: Set-theoretic, model-theoretic and Ramsey-theoretic phenomena in mathematical structures: similarity and diversity – SMART.

Author information

Authors and Affiliations

  1. Fields Institute for Research in Mathematical Science, 222 College St, Toronto, ON M5T 3J1, Canada

    Keegan Dasilva Barbosa

  2. Department of Mathematics and informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 3, Novi Sad, 21000, Serbia

    Dragan Mašulović

  3. School of Computer Science, University of Auckland, Auckland, New Zealand

    Rajko Nenadov

Authors
  1. Keegan Dasilva Barbosa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dragan Mašulović
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rajko Nenadov
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors have contributed to the main result. All authors have reviewed the paper.

Corresponding author

Correspondence to Dragan Mašulović.

Ethics declarations

Competing interests

No competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Keegan Dasilva Barbosa and Rajko Nenadov have contributed equally to this work.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barbosa, K.D., Mašulović, D. & Nenadov, R. A short note on the characterization of countable chains with finite big Ramsey spectra. Order (2023). https://doi.org/10.1007/s11083-023-09647-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11083-023-09647-5

Keyword

Search

Navigation