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

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, and hence, the strong tree version holds for one tree on any weakly compact 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.