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Forcing many positive polarized partition relations between a cardinal and its powerset

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel and Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway NJ 08854-8019, USA, E-mail: shelah@math.huji.ac.il., E-mail: shelah@math.rutgers.edu
Lee J. Stanley
Affiliation:
Department of Mathematics, Lehigh University, 14 E. Packer Avenue, Bethlehem, PA, 18015-3174, USA, E-mail: ljs4@lehigh.edu

Abstract

A fairly quotable special, but still representative, case of our main result is that for 2 ≤ n < ω, there is a natural number m(n) such that, the following holds. Assume GCH: If λ < μ are regular, there is a cofinality preserving forcing extension in which 2λ = μ and, for all σ < λκ < η such that η(+m(n)−+)μ,

This generalizes results of [3], Section 1. and the forcing is a “many cardinals” version of the forcing there.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2001

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References

[1]Komjath, P. and Shelah, S., On Taylor's problem, Acta Mathematica Hungarica, vol. 70 (1996), pp. 217225.CrossRefGoogle Scholar
[2]Rabus, M. and Shelah, S., Consistency of partition relations, submitted to Annals of Pure and Applied Logic.Google Scholar
[3]Shelah, S., Was Sierpiński right, I?, Israel Journal of Mathematics, vol. 62 (1988), pp. 355380.CrossRefGoogle Scholar
[4]Shelah, S., Strong partition relations below the power set: Consistency, was Sierpinski right, II?, Sets, graphs, and numbers: Proceedings of the conference on set theory and its applications in honor of A. Hajnal and V. T. Sos (Budapest), Colloquia Mathematica Societatis, vol. 60, Janos Bolyai, 01 1991, pp. 637638.Google Scholar
[5]Shelah, S., Consistency of positive partition theorems for graphs and models, Set theory and its applications (Toronto, ON, 1987) (Steprans, J. and Watson, S., editors), Lecture Notes in Mathematics, vol. 1401, Springer, Berlin-New York, 1989, pp. 167193.Google Scholar
[6]Shelah, S., Was Sierpinski right, III? Can continuum–c.c. times c.c.c. be continuum–c.c.?, Annals of Pure and Applied Logic, vol. 78 (1996), pp. 259269.CrossRefGoogle Scholar
[7]Shelah, S., Was Sierpiński right, IV?, submitted to this Journal.Google Scholar
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