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Universal graphs at the successor of a singular cardinal

Published online by Cambridge University Press:  12 March 2014

Mirna Džamonja  and
Saharon Shelah
Mirna Džamonja
Affiliation:
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK, E-mail: M.Dzamonja@uea.ac.uk, URL: http://www.mth.uea.ac.uk/people/md.html
Saharon Shelah
Affiliation:
Institute of Mathematics, Hebrew University of Jerusalem, 91904 Givat Ram, Israel Rutgers University, New Brunswick, NJ, USA, E-mail: shelah@sunset.huji.ac.il, URL: http://www.math.rutgers.edu/~shelarch
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Abstract

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are μ++ graphs on μ+ that taken jointly are universal for the graphs on μ+, while .

The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.

Keywords

Successor of singulariterated forcinguniversal graph
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 366 - 388
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[ChKe]Chang, C. C. and Keisler, H. J., Model theory, 1 ed., North-Holland, 1973, latest edition 1990.Google Scholar
[FuKo]Füredi, Z. and Komjath, P., Nonexistence of universal graphs without some trees, Combinatorics vol. 17 (1997), pp. 163–171.CrossRefGoogle Scholar
[GrSh 174]Grossberg, R. and Shelah, S., On universal locally finite groups, Israel Journal of Mathematics, vol. 44 (1983), pp. 289–302.CrossRefGoogle Scholar
[DjSh 614]Džamonja, M. and Shelah, S., On the existence of universals, submitted.Google Scholar
[DjSh 710]Džamonja, M. and Shelah, S., On properties of theories which preclude the existence of universal models, submitted.Google Scholar
[GiSh 597]Gitik, M. and Shelah, S., On densities of box products, Topology and its Applications, vol. 88 (1998), no. 3, pp. 219–237.CrossRefGoogle Scholar
[KaReSo]Kanamori, A., Reindhart, W., and Solovay, R., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73–116.Google Scholar
[Kj]Kojman, M., Representing embeddability as set inclusion, Journal of LMS (2nd series), (1998), no. 158, (58) (2), pp. 257–270.Google Scholar
[KoSh 492]Komjath, P. and Shelah, S., Universal graphs without large cliques, Journal of Combinatorial Theory (Series B), (1995), pp. 125–135.CrossRefGoogle Scholar
[KjSh 409]Kojman, M. and Shelah, S., Non-existence of universal orders in many cardinals, this Journal, vol. 57 (1992), pp. 875–891.Google Scholar
[KjSh 447]Kojman, M. and Shelah, S., The universality spectrum of stable unsuperstable theories, Annals of Pure and Applied Logic, vol. 58 (1992), pp. 57–92.CrossRefGoogle Scholar
[La]Laver, R., Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), no. 4, pp. 385–388.CrossRefGoogle Scholar
[Ma 1]Magidor, M., On the singular cardinals problem I, Israel Journal of Mathematics, vol. 28 (1977), pp. 1–31.CrossRefGoogle Scholar
[Ma 2]Magidor, M., On the singular cardinals problem II, Annals of Mathematic, vol. 106 (1977), pp. 517–549.Google Scholar
[Ma3]Magidor, M., Changing cofinality of cardinals, Fundamenta Mathematicae, vol. XCIX (1978), pp. 61–71.CrossRefGoogle Scholar
[MkSh 274]Mekler, A. and Shelah, S., Uniformization principles, this Journal, vol. 54 (1989), no. 2, pp. 441–459.Google Scholar
[Ra]Radin, L., Adding closed cofinal sequences to large cardinals, Annals of Mathematical Logic, vol. 22 (1982), pp. 243–261.CrossRefGoogle Scholar
[Rd]Rado, R., Universal graphs and universal functions, Acta Arithmetica, vol. 9 (1964), pp. 331–340.CrossRefGoogle Scholar
[Sh -f]Shelah, S., Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer, 1998.CrossRefGoogle Scholar
[Sh 80]Shelah, S., A weak generalization of MA to higher cardinals, Israel Journal of Mathematics, vol. 30 (1978), pp. 297–308.CrossRefGoogle Scholar
[Sh 93]Shelah, S., Simple unstable theories, Annals of Mathematical Logic, vol. 19 (1980), pp. 177–204.CrossRefGoogle Scholar
[Sh 175a]Shelah, S., Universal graphs without instances of CH: revisited, Israel Journal of Mathematics, vol. 70 (1990), pp. 69–81.CrossRefGoogle Scholar
[Sh 457]Shelah, S., The universality spectrum: Consistency for more classes, Combinatorics, Paul Erdös is eighty, vol. 1, pp. 403–420, Bolyai Society Mathematical Studies, 1993, Proceedings of the Meeting in honour of Paul Erdös, Ketzhely, Hungary 7. 1993. An elaborated version available from http://www.math.rutgers.edu/~shelarch.Google Scholar
[Sh 500]Shelah, S., Toward classifying unstable theories, Annals of Pure and Applied Logic, vol. 80 (1996), pp. 229–255.CrossRefGoogle Scholar
[Sh 546]Shelah, S., Was Sierpinski right IV, this Journal, vol. 65 (2000), no. 3, pp. 1031–1054.Google Scholar