Ramsey theory for highly connected monochromatic subgraphs

Abstract

An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey’s Theorem asserting that in any coloring of the edges of a complete graph there exist large highly connected subgraphs all of whose edges are colored by the same color.

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References

  1. 1.

    Elekes, M., Soukup, D., Soukup, L., Szentmiklóssy, Z.: Decompositions of edge-colored infinite complete graphs into monochromatic paths. Discrete Math. 340, 2053–2069 (2017)

    MathSciNet  Article  Google Scholar 

  2. 2.

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

  3. 3.

    P. Erdős and S. Kakutani, On non-denumerable graphs, Bull. Amer. Math. Soc., 49 (1943).

  4. 4.

    P. Erdős and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc.,56 (1956).

  5. 5.

    R. L. Graham, B. L. Rothschild and J. H. Spencer, Ramsey Theory, John Wiley & Sons (New York, 1980)

  6. 6.

    A. Hajnal, P. Komjáth, L. Soukup and I. Szalkai, Decompositions of edge colored infinite complete graphs, in: Combinatorics (Eger, 1987), Colloq. Math. Soc. János Bolyai, 52 (1988)

  7. 7.

    L. Harrington and S. Shelah, Some exact equiconsistency results in set theory, Notre Dame J. Formal Logic, 26 (1985).

  8. 8.

    R. B. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic, 4 (1972).

  9. 9.

    A. Kanamori, The Higher Infinite: Large Cardinals in Set Theory from their Beginnings, Springer-Verlag (Berlin, 1994)

  10. 10.

    P. Komjáth, On the connectivity of infinite graphs, Acta Math. Hungar., 154 (2018).

  11. 11.

    C. Lambie-Hanson, Squares and covering matrices, Ann. Pure Appl. Logic, 165 (2014).

  12. 12.

    W. Mader, Connectivity and edge-connectivity in finite graphs, Surveys in Combinatorics, B. Bollobás (ed.), Cambridge University Press (Cambridge, 1979)

  13. 13.

    D. W. Matula, Ramsey theory for graph connectivity, J. Graph Theory, 7 (1983).

  14. 14.

    K. Menger, Zur allgemeinen Kurventheorie, Fund. Math., 10 (1927).

  15. 15.

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

  16. 16.

    S. Shelah, Was Sierpiński right? Israel J. Math., 62 (1988).

  17. 17.

    W. Sierpiński, Sur un problème de la thèorie des relations, Ann. Scuola Norm. Sup. Pisa, 2 (1933).

  18. 18.

    D. Soukup, Trees, ladders, and graphs, J. Combin. Theory, Series B, 115 (2015)

  19. 19.

    S.Todorcevic, Partitioning pairs of countable ordinals, Acta Math., 159 (1987).

  20. 20.

    S.Todorcevic, Walks on Ordinals and their Characteristics, Birkhäuser (Basel, 2007)

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Acknowledgements

The first author would like to thank Chris Lambie-Hanson for valuable discussion of the consistency strengths of failures of \(\lambda\)-stationary \(\square(\mu)\) sequences to exist. The authors would like to thank the referee for a prompt, alert, and intelligent reading. This paper is number 1157 in the third authors publication list.

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Correspondence to M. Hrušák.

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The research of the second author was supported by a PAPIIT grant No. IN100317 and CONACyT grant No. A1-S-16164.

The research of the third author was partially supported by NSF grant 136974 and by ERC grant 338821.

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Bergfalk, J., Hrušák, M. & Shelah, S. Ramsey theory for highly connected monochromatic subgraphs. Acta Math. Hungar. 163, 309–322 (2021). https://doi.org/10.1007/s10474-020-01058-x

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Key words and phrases

  • Ramsey theory
  • k-connected graph
  • highly connected graph
  • Mahlo cardinal
  • weakly compact cardinal

Mathematics Subject Classification

  • 03E02
  • 03E10