Countable Ordinals and Big Ramsey Degrees


In this paper we consider big Ramsey degrees of finite chains in countable ordinals. We prove that a countable ordinal has finite big Ramsey degrees if and only if it is smaller than ωω. Big Ramsey degrees of finite chains in all other countable ordinals are infinite.

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  1. [1]

    D. Devlin: Some partition theorems and ultrafilters on ω, Ph.D. Thesis, Dartmouth College, 1979.

  2. [2]

    P. Erdős, A. Hajnal: Solved and unsolved problems in set theory, in: Proceedings of the Tarski Symposium, (Proc. Sympos. Pure Math., UC Berkeley, CA, 1971), Vol. 25, 261–287. American Mathematical Society, Providence, 1974.

    Google Scholar 

  3. [3]

    R. Fraïssé: Theory of relations (Revised Ed), North-Holland Publishing Co., Amsterdam, 2000.

    MATH  Google Scholar 

  4. [4]

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

    Google Scholar 

  5. [5]

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

    Google Scholar 

  6. [6]

    F. Galvin: Private communication, July 2019.

  7. [7]

    R. L. Graham, B. L. Rothschild and J. H. Spencer: Ramsey Theory (2nd Ed), John Wiley & Sons, 1990.

  8. [8]

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

    Article  Google Scholar 

  9. [9]

    R. Laver: An Order Type Decomposition Theorem, Annals of Mathematics, Second Series, 98 (1973), 96–119.

    MathSciNet  Article  Google Scholar 

  10. [10]

    F. P. Ramsey: On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264–286.

    MathSciNet  Article  Google Scholar 

  11. [11]

    S. Todorčević: Oscillations of Sets of Integers. Advances in Applied Mathematics 20 (1998), 220–252.

    MathSciNet  Article  Google Scholar 

  12. [12]

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

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We would like to thank F. Galvin and S. Todorčević for their help and many insightful observations that led us to the solution of the problem for all countable ordinals.

The first author gratefully acknowledges the support of the Grant No. 174019 of the Ministry of Education, Science and Technological Development of the Republic of Serbia.

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Correspondence to Dragan Mašulović.

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Mašulović, D., Šobot, B. Countable Ordinals and Big Ramsey Degrees. Combinatorica 41, 425–446 (2021).

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Mathematics Subject Classification (2010)

  • 05D10
  • 03E10