The Halpern-L\"auchli Theorem at a Measurable Cardinal

Natasha Dobrinen; Dan Hathaway

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arXiv · 1608.00592

The Halpern-L\"auchli Theorem at a Measurable Cardinal

Natasha Dobrinen·Dan Hathaway

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Abstract

Several variants of the Halpern-L\"auchli Theorem for trees of uncountable height are investigated. For $\kappa$ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one tree on any infinite cardinal. For any finite $d \ge 2$, we prove the consistency of the Halpern-L\"auchli Theorem on $d$ many $\kappa$-trees at a measurable cardinal $\kappa$, given the consistency of a $\kappa+d$-strong cardinal. This follows from a more general consistency result at measurable $\kappa$, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.

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BibTeX RIS

Natasha Dobrinen, Dan Hathaway. 2016-08-01. The Halpern-L\"auchli Theorem at a Measurable Cardinal. https://arxiv.org/abs/1608.00592

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