The Erdős–Rado Arrow for Singular Cardinals

Saharon Shelah

doi:10.4153/CMB-2009-015-8

Published online by Cambridge University Press: 20 November 2018

Saharon Shelah

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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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E-mail address:

shelah@math.huji.ac.il

Abstract

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 $\text{ }\!\!\lambda\!\!\text{ }\,\to \,{{\left( \text{ }\!\!\lambda\!\!\text{ ,}\,\omega \,\text{+}\,\text{1} \right)}^{2}}$ .

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, 56–66.CrossRef Google 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, 1624–1674.CrossRef Google 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, 259–271.CrossRef Google Scholar

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