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Higher-dimensional Delta-systems

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Abstract

We investigate higher-dimensional \(\Delta\)-systems indexed by finite sets of ordinals, isolating a particular definition thereof and proving a higher-dimensional version of the classical \(\Delta\)-system lemma. We focus in particular on systems that consist of sets of ordinals, in which case useful order-theoretic uniformities can be ensured. We then present three applications of these higher-dimensional \(\Delta\)-systems to problems involving the interplay between forcing and partition relations on the reals.

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References

  1. Baumgartner, J.E.: Canonical partition relations. J. Symbolic Logic 40(4), 541–554 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bergfalk, J., Hrušák, M., Shelah, S.: Ramsey theory for highly connected monochromatic subgraphs. Acta Math. Hungar. 163(1), 309–322 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bergfalk, J., Lambie-Hanson, C.: Simultaneously vanishing higher derived limits. Forum Math. Pi, 9:Paper No. e4, 31 (2021)

  4. Chang, C.C., Keisler, H.J.: Model theory, volume 73 of Studies in Logic and the Foundations of Mathematics., 3rd edn. North-Holland Publishing Co., Amsterdam (1990)

    Google Scholar 

  5. Dobrinen, N., Hathaway, D.: The Halpern-Läuchli theorem at a measurable cardinal. J. Symb. Log. 82(4), 1560–1575 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Džamonja, M., Larson, J.A., Mitchell, W.J.: A partition theorem for a large dense linear order. Israel J. Math. 171, 237–284 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Erdős, P., Hajnal, A., Rado, R.: Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. 16, 93–196 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  8. Erdős, P., Hajnal, A., Máté, A., Rado, R.: Combinatorial set theory: partition relations for cardinals, volume 106 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam (1984)

    MATH  Google Scholar 

  9. Erdös, P., Rado, R.: A combinatorial theorem. J. London Math. Soc. 25, 249–255 (1950)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Hindman, N., Leader, I., Strauss, D.: Pairwise sums in colourings of the reals. Abh. Math. Semin. Univ. Hambg. 87(2), 275–287 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kumar, A., Raghavan, D.: Separating families and order dimension of Turing degrees. Ann. Pure Appl. Logic 172(5), 102911 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kunen, K.: Set theory, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam-New York (1980) An introduction to independence proofs

  14. Lambie-Hanson, C.: A note on a result of Zhang about monochromatic sumsets of reals. 2020. Unpublished note, available at https://users.math.cas.cz/~lambiehanson/sumset_note.pdf

  15. Shanin, N.A.: A theorem from the general theory of sets. C. R. (Doklady) Acad. Sci. URSS (N.S.) 53, 399–400 (1946)

    MathSciNet  MATH  Google Scholar 

  16. Shelah, S.: Strong partition relations below the power set: consistency; was Sierpiński right? II. In: Sets, graphs and numbers (Budapest, 1991), volume 60 of Colloq. Math. Soc. János Bolyai, pp. 637–668. North-Holland, Amsterdam (1992)

  17. Shelah, S.: Consistency of positive partition theorems for graphs and models. In: Set theory and its applications (Toronto, ON, 1987), volume 1401 of Lecture Notes in Math., pp. 167–193. Springer, Berlin (1989)

  18. Simpson, S.G.: Model-theoretic proof of a partition theorem. Not. Am. Math. Soc. 17(6), 964 (1970)

    Google Scholar 

  19. Todorčević, S.: Reals and positive partition relations. In: Logic, methodology and philosophy of science, VII (Salzburg, 1983), volume 114 of Stud. Logic Found. Math., pp. 159–169. North-Holland, Amsterdam (1986)

  20. Todorčević, S.: Walks on ordinals and their characteristics, volume 263 of Progress in Mathematics. Birkhäuser Verlag, Basel (2007)

    Book  Google Scholar 

  21. Zhang, J.: A tail cone version of the Halpern-Läuchli theorem at a large cardinal. J. Symb. Log. 84(2), 473–496 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, J.: Monochromatic sumset without large cardinals. Fund. Math. 250(3), 243–252 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the anonymous referee for an exceptionally thorough and careful reading and a great number of corrections and suggestions that have significantly improved the exposition of the paper.

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  1. Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, Praha 1, Czechia

    Chris Lambie-Hanson

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  1. Chris Lambie-Hanson
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Correspondence to Chris Lambie-Hanson.

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Lambie-Hanson, C. Higher-dimensional Delta-systems. Order 40, 173–197 (2023). https://doi.org/10.1007/s11083-022-09602-w

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