Partition properties of subsets of Pκλ

Masahiro Shioya

Fundamenta Mathematicae (1999)

  • Volume: 161, Issue: 3, page 325-329
  • ISSN: 0016-2736

Abstract

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Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any f : ∪ n < ω [ X ] ⊂ n → γ with X ⊂ P κ λ unbounded and 1 < γ < κ there is an unbounded Y ∪ X with | f ' ' [ Y ] ⊂ n | = 1 for any n < ω.

How to cite

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Shioya, Masahiro. "Partition properties of subsets of Pκλ." Fundamenta Mathematicae 161.3 (1999): 325-329. <http://eudml.org/doc/212409>.

@article{Shioya1999,
abstract = {Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any $f : ∪_\{n < ω\}[X]^\{n\}_⊂ → γ$ with $X⊂P_κλ$ unbounded and 1 < γ < κ there is an unbounded Y ∪ X with $|f^\{\prime \prime \}[Y]^n_⊂| = 1$ for any n < ω.},
author = {Shioya, Masahiro},
journal = {Fundamenta Mathematicae},
keywords = {forcing; partition property; supercompact cardinal},
language = {eng},
number = {3},
pages = {325-329},
title = {Partition properties of subsets of Pκλ},
url = {http://eudml.org/doc/212409},
volume = {161},
year = {1999},
}

TY - JOUR
AU - Shioya, Masahiro
TI - Partition properties of subsets of Pκλ
JO - Fundamenta Mathematicae
PY - 1999
VL - 161
IS - 3
SP - 325
EP - 329
AB - Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any $f : ∪_{n < ω}[X]^{n}_⊂ → γ$ with $X⊂P_κλ$ unbounded and 1 < γ < κ there is an unbounded Y ∪ X with $|f^{\prime \prime }[Y]^n_⊂| = 1$ for any n < ω.
LA - eng
KW - forcing; partition property; supercompact cardinal
UR - http://eudml.org/doc/212409
ER -

References

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  1. [1] Y. Abe, Combinatorics for small ideals on P κ λ , Math. Logic Quart. 43 (1997), 541-549. Zbl0897.03049
  2. [2] Y. Abe, private communication. 
  3. [3] J. Baumgartner, Iterated forcing, in: Surveys in Set Theory, A. Mathias (ed.), London Math. Soc. Lecture Note Ser. 87, Cambridge Univ. Press, Cambridge, 1983, 1-59. 
  4. [4] C. Di Prisco and W. Zwicker, Flipping properties and supercompact cardinals, Fund. Math. 109 (1980), 31-36. Zbl0464.03047
  5. [5] R. Engelking and M. Karłowicz, Some theorems of set theory and their topological consequences, ibid. 57 (1965), 275-285. Zbl0137.41904
  6. [6] T. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1973), 165-198. Zbl0262.02062
  7. [7] T. Jech and S. Shelah, A partition theorem for pairs of finite sets, J. Amer. Math. Soc. 4 (1991), 647-656. Zbl0744.03048
  8. [8] C. Johnson, Some partition relations for ideals on P κ λ , Acta Math. Hungar. 56 (1990), 269-282. Zbl0733.03039
  9. [9] S. Kamo, Ineffability and partition property on P κ λ , J. Math. Soc. Japan 49 (1997), 125-143. 
  10. [10] A. Kanamori, The Higher Infinite, Springer, Berlin, 1994. Zbl0813.03034
  11. [11] R. Laver, Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel J. Math. 29 (1978), 385-388. Zbl0381.03039
  12. [12] M. Magidor, Combinatorial characterization of supercompact cardinals, Proc. Amer. Math. Soc. 42 (1974), 279-285. Zbl0279.02050
  13. [13] P. Matet, handwritten notes. 
  14. [14] T. Menas, A combinatorial property of P κ λ , J. Symbolic Logic 41 (1976), 225-234. Zbl0331.02045

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