Filters, Cohen sets and consistent extensions of the Erdős-Dushnik-Miller Theorem

Saharon Shelah; Lee J. Stanley

doi:10.2307/2586535

Published online by Cambridge University Press: 12 March 2014

Saharon Shelah and

Lee J. Stanley

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Saharon Shelah

Affiliation:

Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903, USA, E-mail: shelah@sunrise.huji.ac.il E-mail: shelah@math.rutgers.edu

Lee J. Stanley

Affiliation:

Lehigh University, Department of Mathematics, 14 E Packer Avenue, Bethlehem, PA 18015-3174, USA, E-mail: ljs4@lehigh.edu

Corresponding

E-mail address:

shelah@sunrise.huji.ac.il ;

shelah@math.rutgers.edu ;

ljs4@lehigh.edu

Abstract

We present two different types of models where, for certain singular cardinals λ of uncountable cofinality, λ → (λ, ω + 1)2, although λ is not a strong limit cardinal, We announce, here, and will present in a subsequent paper, [7], that, for example, consistently, and consistently, .

References

[1]Erdős, P., Hajnal, A., Mate, A., and Rado, R., Combinatorial set theory: Partition relations for cardinals, North-Holland, New York, 1984.Google Scholar

[2]Sheiah, S., Borel sets with large squares, Fundamenta Mathematica, submitted.Google Scholar

[3]Sheiah, S., Was Sierpinski right, I?, Israel Journal of Mathematics, vol. 62 (1988), pp. 335–380, paper #276.Google Scholar

[4]Sheiah, S., On for α < ω₂, Logic colloquium '90 (Oikkonen, J. and Väänänen, J., editors), Lecture Notes in Logic, no. 2, Springer-Verlag, Berlin, 1993, ASL summer meeting in Helsinki, pp. 281–289.Google Scholar

[5]Sheiah, S., Cardinal arithmetic, Oxford Logic Guides, no. 29, Oxford University Press, Oxford, 1994.Google Scholar

[6]Shelah, S. and Stanley, L., A theorem and some consistency results in partition calculus, Annals of Pure and Applied Logic, vol. 36 (1987), pp. 119–152.CrossRef Google Scholar

[7]Shelah, S. and Stanley, L., Consistent negative and positive partition relations for singular cardinals of uncountable cofinality, in preparation.Google Scholar

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Filters, Cohen sets and consistent extensions of the Erdős-Dushnik-Miller Theorem

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Filters, Cohen sets and consistent extensions of the Erdős-Dushnik-Miller Theorem

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