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A polarized partition relation for weakly compact cardinals using elementary substructures

Published online by Cambridge University Press:  12 March 2014

Albin L. Jones
Albin L. Jones*
Affiliation:
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KANSAS, LAWRENCE, KS 66045-2142, USA
*
2153 Oakdale Rd., Pasadena, MD 21122-5715, USA, E-mail: alj@mojumi.net, URL: http://www.mojumi.net/~alj
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Abstract

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

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

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, , 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.

Keywords

elementary substructures, measurable cardinals, normal ultrafilters, polarized partition relations, Ramsey theory, transfinite numbers, weakly compact cardinals
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1342 - 1352
Copyright
Copyright © Association for Symbolic Logic 2006

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References

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