Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T18:11:43.823Z Has data issue: false hasContentIssue false
  • English
  • Français

THE HALPERN–LÄUCHLI THEOREM AT A MEASURABLE CARDINAL

Published online by Cambridge University Press:  09 January 2018

NATASHA DOBRINEN  and
DAN HATHAWAY
NATASHA DOBRINEN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF DENVER 2280 S VINE ST DENVER, CO 80208, USAE-mail:[email protected]: http://cs.du.edu/∼ndobrine/
DAN HATHAWAY
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF DENVER 2280 S VINE ST DENVER, CO 80208, USAE-mail:[email protected]: http://mysite.du.edu/∼dhathaw2/
Get access Rights & Permissions [Opens in a new window]

Abstract

Several variants of the Halpern–Läuchli Theorem for trees of uncountable height are investigated. For κ weakly compact, we prove that the various statements are all equivalent, and hence, the strong tree version holds for one tree on any weakly compact cardinal. For any finite d ≥ 2, we prove the consistency of the Halpern–Läuchli Theorem on d many normal κ-trees at a measurable cardinal κ, given the consistency of a κ + d-strong cardinal. This follows from a more general consistency result at measurable κ, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.

Keywords

03E0203E0503E3503E4003E55Ramsey theorypartition relationsHalpern Läuchli Theoremmeasurable cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1560 - 1575
Copyright
Copyright © The Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Devlin, D., Some partition theorems and ultrafilters on ω, Ph.D. thesis, Dartmouth College, 1979.Google Scholar
DiPrisco, C. A., Llopis, J., and Todorcevic, S., Parametrized partitions of products of finite sets. Combinatorica, vol. 24 (2004), no. 2, pp. 209–232.CrossRefGoogle Scholar
Dobrinen, N., Forcing in Ramsey theory, RIMS Symposium on Infinite Combinatorics and Forcing Theory Conference Proceedings, vol. 17, 2017, submitted.Google Scholar
Dobrinen, N., Laflamme, C., and Sauer, N., Rainbow Ramsey simple structures. Discrete Mathematics, vol. 339 (2016), no. 11, pp. 2848–2855.Google Scholar
Dodos, P., Kanellopoulos, V., and Tyros, K., Dense subsets of products of finite trees. International Mathematics Research Notices, vol. 2013 (2013), no. 4, pp. 924–970.CrossRefGoogle Scholar
Džamonja, M., Larson, J., and Mitchell, W. J., A partition theorem for a large dense linear order. Israel Journal of Mathematics, vol. 171 (2009), pp. 237–284.CrossRefGoogle Scholar
Džamonja, M., Larson, J., and Mitchell, W. J., Partitions of large Rado graphs. Archive for Mathematical Logic, vol. 48 (2009), no. 6, pp. 579–606.Google Scholar
Erdős, P. and Rado, R., A partition calculus in set theory. Bulletin of the American Mathematical Society, vol. 62 (1956), pp. 427–489.Google Scholar
Friedman, S.-D. and Thompson, K., Perfect trees and elementary embeddings, this Journal, vol. 73 (2008), no. 3, pp. 906–918.Google Scholar
Hajnal, A. and Komjáth, P., A strongly non-Ramsey order type. Combinatorica, vol. 17 (1997), no. 2, pp. 363–367.CrossRefGoogle Scholar
Halpern, J. D. and Läuchli, H., A partition theorem. Transactions of the American Mathematical Society, vol. 124 (1966), pp. 360–367.CrossRefGoogle Scholar
Halpern, J. D. and Lévy, A., The Boolean prime ideal theorem does not imply the axiom of choice, Axiomatic Set Theory (Scott, Dana S., editor), Proceedings of the Symposium in Pure Mathematics, Vol. XIII, Part I, University of California, Los Angeles, CA; American Mathematical Society, 1971, pp. 83–134.Google Scholar
Kanamori, A., The Higher Infinite, second ed., Springer-Verlag, Berlin, 2003.Google Scholar
Laver, R., Products of infinitely many perfect trees. Journal of the London Mathematical Society (2), vol. 29 (1984), no. 4, pp. 385–396.Google Scholar
Milliken, K. R., A Ramsey theorem for trees. Journal of Combinatorial Theory, Series A, vol. 26 (1979), pp. 215–237.Google Scholar
Milliken, K. R., A partition theorem for the infinite subtrees of a tree. Transactions of the American Mathematical Society, vol. 263 (1981), no. 1, pp. 137–148.Google Scholar
Sauer, N., Coloring subgraphs of the Rado graph. Combinatorica, vol. 26 (2006), no. 2, pp. 231–253.Google Scholar
Shelah, S., Strong partition relations below the power set: Consistency – was Sierpinski right? II, Sets, Graphs and Numbers (Budapest, 1991), vol. 60, Colloquia Mathematica Societatis János Bolyai, North-Holland, 1991, pp. 637–688.Google Scholar
Todorcevic, S., Introduction to Ramsey Spaces, Princeton University Press, Princeton, NJ, 2010.Google Scholar
Todorchevich, S. and Farah, I., Some Applications of the Method of Forcing, Yenisei Series in Pure and Applied Mathematics, Yenisei, Moscow; Lycée, Troitsk, 1995.Google Scholar
Todorcevic, S. and Tyros, K., Subsets of products of finite sets of positive upper density. Journal of Combinatorial Theory, Series A, vol. 120 (2013), pp. 183–193.Google Scholar
Vlitas, D., A canonical partition theorem for uniform families of finite strong subtrees. Discrete Mathematics, vol. 335 (2014), pp. 45–65.Google Scholar
Vuksanovic, V., Canonical equivalence relations on ${\Bbb Q}^n $ . Order, vol. 20 (2003), no. 4, pp. 373–400.CrossRefGoogle Scholar