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Strong negative partition above the continuum

  • Saharon Shelah (a1) (a2) (a3)
Abstract

For e.g. λ = μ+, μ regular, λ larger than the continuum, we prove a strong nonpartition result (stronger than λ → [λ; λ]2). As a consequence, the product of two topological spaces of cellularity <λ may have cellularity λ, or, in equivalent formulation, the product of two λ-c.c. Boolean algebras may lack the λ-c.c. Also λ-S-spaces and λ-L-spaces exist. In fact we deal not with successors of regular λ but with regular λ above the continuum which has a nonreflecting stationary subset of ordinals with uncountable cofinalities; sometimes we require λ to be not strong limit.

The paper is self-contained. On the nonpartition results see the closely related papers of Todorčević [T1], Shelah [Sh276] and [Sh261], and Shelah and Steprans [ShSt1].

On the cellularity of products see Todorčević [T2] and [T3], where such results were obtained for (e.g.) cf and ; the class of cardinals he gets is quite disjoint from ours. In [Sh282] such results were obtained for more successors of singulars (mainly λ+, λ > 2cf λ). Also, concerning S and L spaces, Todorčević gets existence.

Todorčević's work on cardinals like relies on [Sh68] (see more in [ShA2, Chapter XIII]) (the scales appearing in the proof of ). The problem was stressed in a preliminary version of the surveys of Juhasz and Monk. We give a detailed proof for one strong nonpartition theorem (1.1) and then give various strengthenings. We then use 1.10 to get the consequences (in 1.11 and 1.12).

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References
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[R]Roitman Judy, Basic S and L, Handbook of set-theoretic topology (Kunen K. and Vaughan J. E., editors), North-Holland, Amsterdam, 1984, pp. 295326.
[ShA2]Shelah S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982. (See pp. 298300.)
[Sh68]Shelah S., Jónsson algebras in successor cardinals, Israel Journal of Mathematics, vol. 30 (1978), pp. 5764.
[Sh261]Shelah S., A graph which embeds all small graphs on any large set of vertices, Annals of Pure and Applied Logic, vol. 38 (1988), pp. 171183.
[Sh276]Shelah S., Was Sierpiński right? I, Israel Journal of Mathematics, vol. 62 (1988), pp. 355380.
[Sh282]Shelah S., Successor of singulars, productivity of chain conditions and cofinalities of reduced products of cardinals, Israel Journal of Mathematics, vol. 62 (1988), pp. 213256.
[Sh327]Shelah S., Strong negative partition relation below the continuum, Acta Mathematica Hungarica (to appear).
[Sh355]Shelah S., ℵω+1has a Jónsson algebra (to appear).
[Sh365]Shelah S., There are Jónsson algebras in many inaccessibles (to appear).
[ShSt1]Shelah S. and Steprans Y., Extra-special p-groups, Annals of Pure and Applied Logic, vol. 34 (1987), pp. 8797.
[T1]Todorčević S., Partitioning pairs of uncountable ordinals, Acta Mathematica, vol. 159 (1987), pp. 261294.
[T2]Todorčević S., Remark on chain conditions in products, Compositio Mathematica, vol. 55 (1985), pp. 295302.
[T3]Todorčević S., Remarks on cellularity in products, Compositio Mathematica, vol. 57 (1986), pp. 357372.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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