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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A polarized partition relation for cardinals of countable cofinality


Author: Albin L. Jones
Journal: Proc. Amer. Math. Soc. 136 (2008), 1445-1449
MSC (2000): Primary 03E05, 05D10; Secondary 05A18
Published electronically: November 30, 2007
MathSciNet review: 2367118
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if $ \operatorname{cf}{\kappa} = \omega$ and $ \lambda = 2^{<\kappa}$, then

$\displaystyle \left( \begin{matrix} \lambda^+ \\ \lambda \end{matrix}\right) \t... ...egin{matrix} \lambda^+ & \alpha \\ \lambda & \kappa \end{matrix}\right)^{1,1} $

for all $ \alpha < \omega_1$. This polarized partition relation holds if for every partition $ \lambda \times \lambda^+ = K_0 \cup K_1$ either there are $ B_0 \in [\lambda]^{\lambda}$ and $ A_0 \in [\lambda^+]^{\lambda^+}$ with $ B_0 \times A_0 \subseteq K_0$ or there are $ B_1 \in [\kappa]^{\lambda}$ and $ A_1 \in [\alpha]^{\lambda^+}$ with $ B_1 \times A_1 \subseteq K_1$.


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Additional Information

Albin L. Jones
Affiliation: 2153 Oakdale Rd., Pasadena, Maryland 21122
Email: alj@mojumi.net

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09143-5
PII: S 0002-9939(07)09143-5
Keywords: Transfinite cardinal, countable cofinality, elementary substructure, transfinite ordinal, polarized partition relation, Ramsey theory, regular cardinal, singular cardinal
Received by editor(s): October 13, 2006
Received by editor(s) in revised form: February 15, 2007
Published electronically: November 30, 2007
Communicated by: Julia Knight
Article copyright: © Copyright 2007 Albin L. Jones