The Halpern–läuchli Theorem at a Measurable Cardinal

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. We show that the strong tree version holds for one tree on any infinite 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.

A Tail Cone Version of the Halpern–läuchli Theorem at a Large Cardinal

“…We will mainly be concerned with the versions of the Halpern-Läuchli theorem considered in [4]. In Section 2, we show at the level of a weakly compact cardinal, the 1-dimensional Halpern-Läuchli theorem holds for < κ many colors, extending a previous result by Dobrinen and Hathaway [4] where the number of colors is finite. In Section 3, we establish the consistency of the tail cone version of the Halpern-Läuchli theorem.…”

“…Hence whenever a sequence of strong subtrees is produced, we will always make it clear whether they share the same level set. Theorem 1.15 (Dobrinen and Hathaway [4]). For κ a weakly compact cardinal, d ∈ ω, δ < κ, the following are equivalent.…”

“…The dense set version enables the construction of desired strong subtrees in a rather straightforward way. Definition 1.14 (Dobrinen and Hathaway [4]). Let d ∈ ω, κ inaccessible and δ < κ be given.…”

Forcing and the Halpern–läuchli Theorem

“…While we were writing [2] we discovered the derived tree theorem and the proof in this section, which answers the question in the negative. In the meantime, Zhang discovered a different proof of the consistency of (∀σ < κ) HL(2, σ, κ) for a κ that is not weakly compact.…”

A Ramsey theorem for pairs in trees

“…The Halpern-Läuchli Theorem has produced many variations and generalizations, such as Milliken's theorem [7] on finite partitions of strong trees, countable colorings of perfect trees [6], and a dual version [11]. For a fuller discussion of the theorem and its variants, see [3] and the references therein.…”