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THE HALPERN–LÄUCHLI THEOREM AT A MEASURABLE CARDINAL

Published online by Cambridge University Press:  09 January 2018

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/

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.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2017 

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