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The Ramsey Properties for Grassmannians Over ℝ, ℂ


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.

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

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Correspondence to Jordi Lopez-Abad.

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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.

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Bartošová, D., Lopez-Abad, J., Lupini, M. et al. The Ramsey Properties for Grassmannians Over ℝ, ℂ. Combinatorica 42, 9–69 (2022).

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Mathematics Subject Classification (2010)

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