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The Erdős–Rado Arrow for Singular Cardinals

Published online by Cambridge University Press:  20 November 2018

Saharon Shelah*
Affiliation:
Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel, and Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA e-mail: shelah@math.huji.ac.il
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Abstract

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We prove in $\text{ZFC}$ that if $\text{cf}\left( \text{ }\lambda \text{ } \right)>{{\aleph }_{0}}$ and ${{2}^{\text{cf}\left( \text{ }\!\!\lambda\!\!\text{ } \right)}}\,<\,\text{ }\!\!\lambda\!\!\text{ }$, then $\text{ }\!\!\lambda\!\!\text{ }\,\to \,{{\left( \text{ }\!\!\lambda\!\!\text{ ,}\,\omega \,\text{+}\,\text{1} \right)}^{2}}$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Erdős, P., Hajnal, A., Maté, A., and Rado, R., Combinatorial set theory: partition relations for cardinals. Studies in Logic and the Foundations of Mathematics 106, North Holland Publishing Co, Amsterdam, 1984.Google Scholar
[2] Shelah, S., A note on cardinal exponentiation. J. Symbolic Logic 45(1980), no. 1, 5666.CrossRefGoogle Scholar
[3] Shelah, S., Cardinal Arithmetic. Oxford Logic Guides 29, Oxford University Press, New York, 1994.Google Scholar
[4] Shelah, S., Applications of PCF theory. J. Symbolic Logic 65(2000), no. 4, 16241674.CrossRefGoogle Scholar
[5] Shelah, S. and Stanley, L., Filters, Cohen sets and consistent extensions of the Erdős–Dushnik–Miller theorem. J. Symbolic Logic 65(2000), no. 1, 259271.CrossRefGoogle Scholar
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