The Halpern-läUchli Theorem at a Measurable cardinal

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.