Skip to main content

The Ramsey Properties for Grassmannians Over ℝ, ℂ

Abstract

In this note we study and obtain factorization theorems for colorings of matrices and Grassmannians over ℝ and ℂ, which can be considered metric versions of the Dual Ramsey Theorem for Boolean matrices and of the Graham-Leeb-Rothschild Theorem for Grassmannians over a finite field.

This is a preview of subscription content, access via your institution.

References

  1. F. Albiac and N. J. Kalton: Topics in Banach space theory, second ed., Graduate Texts in Mathematics, vol. 233, Springer, [Cham], 2016.

    Book  Google Scholar 

  2. D. Bartošová, J. Lopez-Abad, M. Lupini and B. Mbombo: The Ramsey property for Banach spaces and Choquet simplices, and applications, C. R. Math. Acad. Sci. Paris 355 (2017), 1242–1246.

    MathSciNet  Article  Google Scholar 

  3. D. Bartošová, J. Lopez-Abad, M. Lupini and B. Mbombo: The Ramsey property for Banach spaces, Choquet simplices, and their noncommutative analogs, preprint, 2017.

  4. D. Bartošová, J. Lopez-Abad, M. Lupini and B. Mbombo: The Ramsey property for Banach spaces and Choquet simplices, Journal of the European Mathematical Society (2019), in press; arXiv:1708.01317.

  5. M. Fabian, P. Habala, P. Hájek, V. M. Santalucía, J. Pelant and V. Zizler: Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001.

    Book  Google Scholar 

  6. V. Ferenczi, J. Lopez-Abad, B. Mbombo and S. Todorcevic: Amalgamation and Ramsey properties of Lp spaces, Adv. Math. 369 (2020), 107–190.

    Article  Google Scholar 

  7. T. Giordano and V. Pestov: Some extremely amenable groups related to operator algebras and ergodic theory, Journal of the Institute of Mathematics of Jussieu 6 (2007), 279–315.

    MathSciNet  Article  Google Scholar 

  8. R. L. Graham, K. Leeb and B. L. Rothschild: Ramsey’s theorem for a class of categories, Proceedings of the National Academy of Sciences of the United States of America 69 (1972), 119–120.

    MathSciNet  Article  Google Scholar 

  9. R. L. Graham and B. L. Rothschild: Ramsey’s theorem for n-parameter sets, Transactions of the American Mathematical Society 159 (1971).

  10. R. L. Graham, B. L. Rothschild and J. H. Spencer: Ramsey theory, second ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1990, A Wiley-Interscience Publication.

    MATH  Google Scholar 

  11. M. Gromov and V. D. Milman: A topological application of the isoperimetric inequality, American Journal of Mathematics 105 (1983), 843–854.

    MathSciNet  Article  Google Scholar 

  12. V. I. Gurariĭ: Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Siberian Mathematical Journal 7 (1966), 1002–1013.

    MathSciNet  Article  Google Scholar 

  13. N. J. Kalton and M. I. Ostrovskii: Distances between Banach spaces, Forum Mathematicum 11 (2008), 17–48.

    MathSciNet  Google Scholar 

  14. A. S. Kechris, V. Pestov and S. Todorcevic: Fraïsse limits, Ramsey theory, and topological dynamics of automorphism groups, Geometric and Functional Analysis 15 (2005), 106–189.

    MathSciNet  Article  Google Scholar 

  15. J. Melleray and T. Tsankov: Extremely amenable groups via continuous logic, arXiv:1404.4590 (2014).

  16. V. D. Milman and G. Schechtman: Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986, with an appendix by M. Gromov.

    MATH  Google Scholar 

  17. J. Nešetřil: Ramsey theory, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, 1331–1403.

    MATH  Google Scholar 

  18. M. I. Ostrovskii: Topologies on the set of all subspaces of a Banach space and related questions of Banach space geometry, Quaestiones Math. 17 (1994), 259–319.

    MathSciNet  Article  Google Scholar 

  19. V. Pestov: Dynamics of infinite-dimensional groups, University Lecture Series, vol. 40, American Mathematical Society, Providence, RI, 2006.

    Book  Google Scholar 

  20. W. Rudin: Functional analysis, second ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.

    MATH  Google Scholar 

  21. G. Schechtman: Almost isometric Lp subspaces of Lp(0, 1), The Journal of the London Mathematical Society 20 (1979), 516–528.

    MathSciNet  Article  Google Scholar 

  22. J. H. Spencer: Ramsey’s theorem for spaces, Trans. Amer. Math. Soc. 249 (1979), 363–371.

    MathSciNet  MATH  Google Scholar 

  23. S. Todorcevic: Introduction to Ramsey spaces, Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, NJ, 2010.

    Book  Google Scholar 

Download references

Acknowledgments

We would like to express our gratitude to the anonymous referees for their careful review and valuable comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jordi Lopez-Abad.

Additional information

The first author was supported by the grant FAPESP 2013/14458-9.

The second author was partially supported by the grant MTM2016-76808-P (Spain) and the Fapesp Grant 2013/24827-1 (Brazil).

The third author was partially supported by the NSF Grant DMS-1600186.

The fourth author was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) postdoctoral grant, processo 12/20084-1.

This work was initiated during a visit of J. Lopez-Abad to the Universidade de Sao Pãulo in 2014, and continued during visits of D. Bartošová and J. Lopez-Abad to the Fields Institute in the Fall 2014, a visit of M. Lupini to the Instituto de Ciencias Matemáticas in the Spring 2015, and a visit of all the authors at the Banff International Research Station in occasion of the Workshop on Homogeneous Structures in the Fall 2015. The hospitality of all these institutions is gratefully acknowledged.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bartošová, D., Lopez-Abad, J., Lupini, M. et al. The Ramsey Properties for Grassmannians Over ℝ, ℂ. Combinatorica 42, 9–69 (2022). https://doi.org/10.1007/s00493-020-4264-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-020-4264-0

Mathematics Subject Classification (2010)

  • Primary 05D10
  • 37B05
  • Secondary 05A05
  • 46B20