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

Abstract

The classical Halpern-Läuchli theorem states that for any finite coloring of a finite product of finitely branching perfect trees of height ω, there exist strong subtrees sharing the same level set such that tuples in the product of the strong subtrees consisting of elements lying on the same level get the same color. Relative to large cardinals, we establish the consistency of a tail cone version of the Halpern-Läuchli theorem at a large cardinal (see Theorem 3.1), which, roughly speaking, deals with many colorings simultaneously and diagonally. Among other applications, we generalize a polarized partition relation on rational numbers due to Laver and Galvin to one on linear orders of larger saturation.

Forcing and the Halpern–läuchli Theorem

“…The Derived Tree Theorem is applied in Section 4 to show that small forcings preserve the Halpern-Läuchli Theorem and its tail-cone version. As the partition relation on finite products of κ-rationals holds in any model where the tail-cone version holds (by work of Zhang in [17]), our work shows that this partition relation is preserved by any further small forcing.…”

“…For inaccessible κ, we show they are preserved by forcings of size less than κ. Combining this with work of Zhang in [17] yields that the polarized partition relations associated with finite products of the κ-rationals are preserved by all forcings of size less than κ over models satisfying the Halpern-Läuchli Theorem at κ. We also show that the Halpern-Läuchli Theorem is preserved by <κ-closed forcings assuming κ is measurable, following some observed reflection properties.…”

“…Let κ be strongly inaccessible. There is a unique κsaturated linear order of size κ denoted Q κ , the κ-rationals [17]. Zhang proved in [17] that HL tc (d, <κ, κ) implies…”

“…In [2], we mapped out the implications between weaker and stronger forms of the Halpern-Läuchli Theorem on trees of uncountable height, and found a better upper bound for the consistency strength of the theorem holding for finitely many trees at a measurable cardinal. Building on work in [4] and [2], Zhang proved a stronger tail-cone version at measurable cardinals and applied it to obtain the analogue of Laver's partition relation for products of finitely many trees on a measurable cardinal (see [17]).…”

Monochromatic sumset without large cardinals

“…Remark 2.5. Different versions of Lemma 2.4 appeared in [10], Lemma 4.1 of [9], Claim 7.2.a of [1] and the appendix of [12]. We will use the fact that λ = ω to present a slightly simpler proof.…”