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Weak square bracket relations for Pκ(λ)

Published online by Cambridge University Press:  12 March 2014

Pierre Matet
Pierre Matet*
Affiliation:
Universite de Caen –CNRS, Laboratorie de Mathematiques, BP 5186, 14032 Caen Cedex, France, E-mail: matet@math.unicaen.fr
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Abstract

We study the partition relation that is a weakening of the usual partition relation . Our main result asserts that if κ is an uncountable strongly compact cardinal and , then does not hold.

Keywords

Pκ(λ)square bracket relations

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 3 , September 2008 , pp. 729 - 751
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[1]Landver, A., Singular Baire numbers and related topics, Ph.D. thesis, University of Wisconsin-Madison, 1990.Google Scholar
[2]Cummings, J., Foreman, M., and Magidor, M., Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 35–98.CrossRefGoogle Scholar
[3]Donder, H.D., Koepke, P., and Levinski, J.P., Some stationary subsets of P(λ), Proceedings of the American Mathematical Society, vol. 102 (1988), no. 4, pp. 1000–1004.Google Scholar
[4]Erdős, P., Hajnal, A., and Rado, R., Partition relations for cardinal numbers, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16 (1965), pp. 93–196.CrossRefGoogle Scholar
[5]Gitik, M. and Sheiah, S., On certain indestructibility of strong cardinals and a question of Hajnal, Archive for Mathematical Logic, vol. 28 (1989), pp. 35–42.CrossRefGoogle Scholar
[6]Jech, T., Set theory, 3rd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
[7]Jech, T. and Shelah, S., A partition theorem for pairs of finite sets, Journal of the American Mathematical Society, vol. 4 (1991), no. 4, pp. 647–656.CrossRefGoogle Scholar
[8]Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.CrossRefGoogle Scholar
[9]Johnson, C. A., Some partition relations for ideals on Pκλ, Acta Mathematica Hungarica, vol. 56 (1990), pp. 269–282.CrossRefGoogle Scholar
[10]Kanamori, A., On Silver's and related principles, Logic colloquium '80 (Dalen, D. Van, Lascar, D., and Smiley, J., editors), Studies in Logic and the Foundations of Mathematics, vol. 108, North-Holland, Amsterdam, 1982, pp. 153–172.Google Scholar
[11]Kanamori, A., The higher infinite, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994.Google Scholar
[12]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179–227.CrossRefGoogle Scholar
[13]Kunen, K., On the GCH at measurable cardinals, Logic colloquium '69 (Gandy, R.O. and Yates, C.E.M., editors), North-Holland, Amsterdam, 1971, pp. 107–110.CrossRefGoogle Scholar
[14]Matet, P., The covering number for category and partition relations on Pω(λ), Fundamenta Mathematicae, vol. 171 (2002), pp. 235–247.CrossRefGoogle Scholar
[15]Matet, P., A partition property of a mixed type for P κ(λ), Mathematical Logic Quarterly, vol. 49 (2003), pp. 1–14.CrossRefGoogle Scholar
[16]Matet, P., Covering for category and combinatorics on P κ(λ), Journal of the Mathematical Society of Japan, vol. 58 (2006), pp. 153–181.CrossRefGoogle Scholar
[17]Matet, P., Part(κ, λ) and Part*(κ, λ), Set theory, Trends in Mathematics, Birkhäuser, Basel, 2006, pp. 319–342.Google Scholar
[18]Matet, P., Strong compactness and a partition properly, Proceedings of the American Mathematical Society, vol. 134 (2006), pp. 2147–2152.CrossRefGoogle Scholar
[19]Matet, P., Game ideals, Annals of Pure and Applied Logic, to appear.Google Scholar
[20]Matet, P., Large cardinals and covering numbers, preprint.Google Scholar
[21]Matet, P. and Péan, C., Distributivity properties on Pω(λ), Discrete Mathematics, vol. 291 (2005), pp. 143–154.CrossRefGoogle Scholar
[22]Matet, P., Péan, C., and Shelah, S., Cofinality of normal ideals on P κ(λ) I, Preprint.Google Scholar
[23]Matet, P., Péan, C., and Shelah, S., Cofinality of normal ideals on P κ(λ). II, Israel Journal of Mathematics, vol. 150 (2005), pp. 253–283.CrossRefGoogle Scholar
[24]Matet, P., Péan, C., and Todorcevic, S., Prime ideals on P ω(λ) with the partition property, Archive for Mathematical Logic, vol. 41 (2002), pp. 743–764.CrossRefGoogle Scholar
[25]Matet, P., Rosłanowski, A., and Shelah, S., Cofinality of the nonstationary ideal, Transactions of the American Mathematical Society, vol. 357 (2005), pp. 4813–4837.CrossRefGoogle Scholar
[26]Matet, P. and Shelah, S., Cardinal invariants for κ and partition relations for Pκ(λ), Preprint.Google Scholar
[27]Menas, T. K., A combinatorial property of p κλ, this Journal, vol, 41 (1976), pp. 225–234.Google Scholar
[28]Shelah, S., Cardinal arithmetic for skeptics, Bulletin of the American Mathematical Society, vol. 26 (1992), pp. 197–210.CrossRefGoogle Scholar
[29]Shelah, S., Cardinal arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, Oxford, 1994.CrossRefGoogle Scholar
[30]Shelah, S., Further cardinal arithmetic, Israel Journal of Mathematics, vol. 95 (1996), pp. 61–114.CrossRefGoogle Scholar
[31]Shelah, S., On the existence of large subsets of [λ]<κ which contain no unbounded non-stationary subsets, Archive for Mathematical Logic, vol. 41 (2002), pp. 207–213.CrossRefGoogle Scholar
[32]Shore, R. A., Square bracket partition relations in L, Fundamenta Mathematicae, vol. 84 (1974), pp. 101–106.CrossRefGoogle Scholar
[33]Solovay, R. M., Real-valued measurable cardinals, Axiomatic set theory (Scott, D.S., editor), Proceedings of Symposia in Pure Mathematics, Vol. 13, American Mathematical Society, Providence, R.I., 1971, pp. 397–428.CrossRefGoogle Scholar
[34]Todorčević, S., Coherent sequences, Handbook of set theory (Foreman, M., Kanamori, A., and Magidor, M., editors), Kluwer, Dordrecht, to appear.Google Scholar
[35]Todorčević, S., Partitioning pairs of countable ordinals, Acta Mathematica, vol. 159 (1987), pp. 261–294.CrossRefGoogle Scholar
[36]Todorčević, S., Partitioning pairs of countable sets, Proceedings of the American Mathematical Society, vol. 111 (1991), pp. 841–844.CrossRefGoogle Scholar
[37]Velleman, D., Partitioning pairs of countable sets of ordinals, this Journal, vol. 55 (1990), pp. 1019–1021.Google Scholar