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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, USA E-mail: Natasha.Dobrinen@du.edu URL: http://cs.du.edu/∼ndobrine/
DAN HATHAWAY
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF DENVER 2280 S VINE ST DENVER, CO 80208, USA E-mail: Daniel.Hathaway@du.edu URL: http://mysite.du.edu/∼dhathaw2/
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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

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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 

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