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Idiosynchromatic Poetry

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Abstract

We prove equivalences of asymmetric partition relations involving natural numbers and products of weakly compact cardinals к, infinite cardinals λ<к and natural numbers to certain classes of finitary problems in the theory of edge-coloured digraphs. We are able to determine three classes of Ramsey numbers exactly, typical examples in these classes are r(ω 22,3), r(кλ2,3) and r(кλ3,3). We moreover provide general upper bounds for r(ω 2 m,3) and r(кλm,n).

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References

  1. J. E. Baumgartner: Improvement of a partition theorem of Erdös and Rado, J. Combinatorial Theory Ser. A 17 (1974), 134–137.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. E. Baumgartner: Remarks on partition ordinals, in: Set theory and its applications (Toronto, ON, 1987), volume 1401 of Lecture Notes in Math., 5–17. Springer Berlin, 1989.

    Chapter  Google Scholar 

  3. J.-C. Bermond: Some Ramsey numbers for directed graphs, Discrete Math. 9 (1974), 313–321.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Bang-Jensen and G. Gutin: Digraphs, Springer Monographs in Mathematics. Springer-Verlag London Ltd. London, second edition, 2009. Theory, algorithms and applications

    Book  MATH  Google Scholar 

  5. F. Chung and R. Graham: Erdős on graphs, A K Peters Ltd. Wellesley, MA, 1998. His legacy of unsolved problems.

    Google Scholar 

  6. C. Darby and J. A. Larson: A positive partition relation for a critical countable ordinal, in preparation.

  7. P. Erdős and L. Moser: On the representation of directed graphs as unions of orderings, Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964), 125–132.

    Google Scholar 

  8. P. Erdős and R. Rado: A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427–489.

    Article  MathSciNet  Google Scholar 

  9. R. E. Greenwood, Jr. and A. M. Gleason: Combinatorial relations and chromatic graphs, Canad. J. Math. 7 (1955), 1–7.

    Article  MATH  MathSciNet  Google Scholar 

  10. C. M. Grinstead and Sam M. Roberts: On the Ramsey numbers R(3; 8) and R(3; 9), J. Combin. Theory Ser. B 33 (1982), 27–51.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Georgakopoulos and P. Sprüssel: On 3-coloured tournaments, preprint, April 2009.

    Google Scholar 

  12. J. E. Graver and J. Yackel: Some graph theoretic results associated with Ramsey’s theorem, J. Combinatorial Theory 4 (1968), 125–175.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Hajnal and J. A. Larson: Handbook of set theory. Vol. 1, chapter Partition relations, pages Vol. 1: xiv+736, Springer Dordrecht, 2010.

  14. L. Haddad and G. Sabbagh: Calcul de certains nombres de Ramsey généralisés, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A1233–A1234.

    MathSciNet  Google Scholar 

  15. L. Haddad and G. Sabbagh: Nouveaux résultats sur les nombres de Ramsey généralisés, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A1516–A1518.

    MathSciNet  Google Scholar 

  16. L. Haddad and G. Sabbagh: Sur une extension des nombres de Ramsey aux ordinaux, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A1165–A1167.

    MathSciNet  Google Scholar 

  17. J. G. Kalbfleisch: Construction of special edge-chromatic graphs, Canad. Math. Bull. 8 (1965), 575–584.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. G. Kalbfleisch: Chromatic Graphs and Ramsey’s theorem, PhD thesis, University of Waterloo, January 1966.

    Google Scholar 

  19. A. Kanamori: The higher infinite, Springer Monographs in Mathematics. Springer-Verlag Berlin, second edition, 2003. Large cardinals in set theory from their beginnings.

    MATH  Google Scholar 

  20. G. Kéry: On a theorem of Ramsey, Mat. Lapok 15 (1964), 204–224.

    MATH  MathSciNet  Google Scholar 

  21. J. A. Larson: A short proof of a partition theorem for the ordinal ω ω, Ann. Math. Logic 6 (1973/1974), 129–145.

    Article  MATH  MathSciNet  Google Scholar 

  22. J. A. Larson and W. J. Mitchell: On a problem of Erdős and Rado, Ann. Comb. 1 (1997), 245–252.

    Article  MATH  MathSciNet  Google Scholar 

  23. E. C. Milner: Partition relations for ordinal numbers, Canad. J. Math. 21 (1969), 317–334.

    Article  MATH  MathSciNet  Google Scholar 

  24. B. D. McKay and Z. K. Min: The value of the Ramsey number R(3; 8), J. Graph Theory 16 (1992), 99–105.

    Article  MATH  MathSciNet  Google Scholar 

  25. B. D. McKay and S. P. Radziszowski: R(4;5)=25, J. Graph Theory 19 (1995), 309–322.

    Article  MATH  MathSciNet  Google Scholar 

  26. B. D. McKay and S. P. Radziszowski: Subgraph counting identities and Ramsey numbers, J. Combin. Theory Ser. B 69 (1997), 193–209.

    Article  MATH  MathSciNet  Google Scholar 

  27. E. Nosal: On a partition relation for ordinal numbers, J. London Math. Soc. (2) 8 (1974), 306–310.

    Article  MATH  MathSciNet  Google Scholar 

  28. E. Nosal: On arrow relations w(k,w) 2(2) m,n,k less than w): A study in the partition calculus, ProQuest LLC Ann Arbor, MI, 1975. Thesis (Ph.D.) University of Calgary (Canada).

    Google Scholar 

  29. E. Nosal: Partition relations for denumerable ordinals, J. Combin. Theory Ser. B 27 (1979), 190–197.

    Article  MATH  MathSciNet  Google Scholar 

  30. K. Piwakowski and Stanis law P. Radziszowski: 30≤R(3,3,4)≤31, J. Combin. Math. Combin. Comput. 27 (1998), 135–141.

    MathSciNet  Google Scholar 

  31. S. P. Radziszowski: Small Ramsey numbers, Electron. J. Combin. 1 (1994), Dynamic Survey 1, 30 (electronic).

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

    Article  MathSciNet  Google Scholar 

  33. K. B. Reid and E. T. Parker: Disproof of a conjecture of Erdős and Moser on tournaments, J. Combinatorial Theory 9 (1970), 225–238.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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  1. Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 62, 53115, Bonn, Germany

    Thilo V. Weinert

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Weinert, T.V. Idiosynchromatic Poetry. Combinatorica 34, 707–742 (2014). https://doi.org/10.1007/s00493-011-2980-1

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  • DOI: https://doi.org/10.1007/s00493-011-2980-1

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