Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-15T05:20:21.907Z Has data issue: false hasContentIssue false

Weak square bracket relations for Pκ(λ)

Published online by Cambridge University Press:  12 March 2014

Pierre Matet*
Affiliation:
Universite de Caen –CNRS, Laboratorie de Mathematiques, BP 5186, 14032 Caen Cedex, France, E-mail: [email protected]

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 3598.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. 10001004.Google Scholar
[4]Erdős, P., Hajnal, A., and Rado, R., Partition relations for cardinal numbers, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16 (1965), pp. 93196.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. 3542.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. 647656.CrossRefGoogle Scholar
[8]Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229308.CrossRefGoogle Scholar
[9]Johnson, C. A., Some partition relations for ideals on Pκλ, Acta Mathematica Hungarica, vol. 56 (1990), pp. 269282.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. 153172.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. 179227.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. 107110.CrossRefGoogle Scholar
[14]Matet, P., The covering number for category and partition relations on Pω(λ), Fundamenta Mathematicae, vol. 171 (2002), pp. 235247.CrossRefGoogle Scholar
[15]Matet, P., A partition property of a mixed type for P κ(λ), Mathematical Logic Quarterly, vol. 49 (2003), pp. 114.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. 153181.CrossRefGoogle Scholar
[17]Matet, P., Part(κ, λ) and Part*(κ, λ), Set theory, Trends in Mathematics, Birkhäuser, Basel, 2006, pp. 319342.Google Scholar
[18]Matet, P., Strong compactness and a partition properly, Proceedings of the American Mathematical Society, vol. 134 (2006), pp. 21472152.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. 143154.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. 253283.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. 743764.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. 48134837.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. 225234.Google Scholar
[28]Shelah, S., Cardinal arithmetic for skeptics, Bulletin of the American Mathematical Society, vol. 26 (1992), pp. 197210.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. 61114.CrossRefGoogle Scholar
[31]Shelah, S., On the existence of large subsets of [λ] which contain no unbounded non-stationary subsets, Archive for Mathematical Logic, vol. 41 (2002), pp. 207213.CrossRefGoogle Scholar
[32]Shore, R. A., Square bracket partition relations in L, Fundamenta Mathematicae, vol. 84 (1974), pp. 101106.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. 397428.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. 261294.CrossRefGoogle Scholar
[36]Todorčević, S., Partitioning pairs of countable sets, Proceedings of the American Mathematical Society, vol. 111 (1991), pp. 841844.CrossRefGoogle Scholar
[37]Velleman, D., Partitioning pairs of countable sets of ordinals, this Journal, vol. 55 (1990), pp. 10191021.Google Scholar