The Halpern-Läuchli Theorem at a Measurable Cardinal

Abstract

Several variants of the Halpern-Läuchli 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-Läuchli 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.

15 pages

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• Forcing in Ramsey theory

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• Forcing in Ramsey theory