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A TAIL CONE VERSION OF THE HALPERN–LÄUCHLI THEOREM AT A LARGE CARDINAL

Published online by Cambridge University Press:  08 April 2019

JING ZHANG
JING ZHANG*
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PENNSYLVANIA 15213, USA E-mail: jingzhang@cmu.edu
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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.

Keywords

Primary 03E0203E3503E55Halpern-Läuchli theoremRamsey theorytreesforcinglarge cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 473 - 496
Copyright
Copyright © The Association for Symbolic Logic 2019 

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