December 2006 A polarized partition relation for weakly compact cardinals using elementary substructures
Albin L. Jones
J. Symbolic Logic 71(4): 1342-1352 (December 2006). DOI: 10.2178/jsl/1164060459

Abstract

We show that if κ is a weakly compact cardinal, then

(\vector{κ⁺}{κ}) → ((\vector{α}{κ})m (\vector{κⁿ}{κ})μ))1,1

for any ordinals α < κ⁺ and μ < κ, and any finite ordinals m and n. This polarized partition relation represents the statement that for any partition

κ × κ⁺ = ⋃i < m Ki ∪ ⋃j < μ Lj

of κ × κ⁺ into m + μ pieces either there are A ∈ [κ]κ, B ∈ [κ⁺]α, and i < m with A × B ⊆ Ki or there are C ∈ [κ]κ, D ∈ [κ⁺]κⁿ, and j < μ with C × D ⊆ Lj. Related results for measurable and almost measurable κ are also investigated. Our proofs of these relations involve the use of elementary substructures of set models of large fragments of ZFC.

Citation

Download Citation

Albin L. Jones. "A polarized partition relation for weakly compact cardinals using elementary substructures." J. Symbolic Logic 71 (4) 1342 - 1352, December 2006. https://doi.org/10.2178/jsl/1164060459

Information

Published: December 2006
First available in Project Euclid: 20 November 2006

zbMATH: 1109.03043
MathSciNet: MR2275863
Digital Object Identifier: 10.2178/jsl/1164060459

Subjects:
Primary: Primary 03E02, 05D10; Secondary 04A20

Keywords: elementary substructures , measurable cardinals , normal ultrafilters , polarized partition relations , Ramsey theory , transfinite numbers , weakly compact cardinals

Rights: Copyright © 2006 Association for Symbolic Logic

JOURNAL ARTICLE
11 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

Vol.71 • No. 4 • December 2006
Back to Top