Strongly almost disjoint familes, revisited

A. Hajnal; Istvan Juhász; Saharon Shelah

Fundamenta Mathematicae (2000)

  • Volume: 163, Issue: 1, page 13-23
  • ISSN: 0016-2736

Abstract

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The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if A ⊂ [ κ ] λ with |A|=κ is μ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let ϱ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ϱ. Then there is a ϱ-closed forcing P such that, in V P , we have both GCH and M ( ϱ ( + ϱ + 1 ) , ϱ + , ϱ ) ↛ B [resp. M ( ϱ ( + ϱ + 1 ) , λ , ϱ ) ↛ B ( ϱ + ) for all λ ≤ ϱ ( + ϱ + 1 ) ] . These show that, consistently, the results of [EH] are sharp. The necessity of using large cardinals follows from the results of [Ko], [HJSh] and [BDJShSz].

How to cite

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Hajnal, A., Juhász, Istvan, and Shelah, Saharon. "Strongly almost disjoint familes, revisited." Fundamenta Mathematicae 163.1 (2000): 13-23. <http://eudml.org/doc/212425>.

@article{Hajnal2000,
abstract = {The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if $A⊂[κ]^λ$ with |A|=κ is μ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let ϱ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ϱ. Then there is a ϱ-closed forcing P such that, in $V^P$, we have both GCH and $M(ϱ^\{(+ϱ+1)\},ϱ^+,ϱ) ↛ B$ [resp. $M(ϱ^\{(+ϱ+1)\},λ,ϱ) ↛ B(ϱ^+)$ for all $λ ≤ ϱ^\{(+ϱ+1)\}]$. These show that, consistently, the results of [EH] are sharp. The necessity of using large cardinals follows from the results of [Ko], [HJSh] and [BDJShSz].},
author = {Hajnal, A., Juhász, Istvan, Shelah, Saharon},
journal = {Fundamenta Mathematicae},
keywords = {strongly almost disjoint family; property B; σ-transversal; property ; -transversal; -closed forcing; GCH; large cardinals},
language = {eng},
number = {1},
pages = {13-23},
title = {Strongly almost disjoint familes, revisited},
url = {http://eudml.org/doc/212425},
volume = {163},
year = {2000},
}

TY - JOUR
AU - Hajnal, A.
AU - Juhász, Istvan
AU - Shelah, Saharon
TI - Strongly almost disjoint familes, revisited
JO - Fundamenta Mathematicae
PY - 2000
VL - 163
IS - 1
SP - 13
EP - 23
AB - The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if $A⊂[κ]^λ$ with |A|=κ is μ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let ϱ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ϱ. Then there is a ϱ-closed forcing P such that, in $V^P$, we have both GCH and $M(ϱ^{(+ϱ+1)},ϱ^+,ϱ) ↛ B$ [resp. $M(ϱ^{(+ϱ+1)},λ,ϱ) ↛ B(ϱ^+)$ for all $λ ≤ ϱ^{(+ϱ+1)}]$. These show that, consistently, the results of [EH] are sharp. The necessity of using large cardinals follows from the results of [Ko], [HJSh] and [BDJShSz].
LA - eng
KW - strongly almost disjoint family; property B; σ-transversal; property ; -transversal; -closed forcing; GCH; large cardinals
UR - http://eudml.org/doc/212425
ER -

References

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  1. [BDJShSz] Z. T. Balogh, S. W. Davis, W. Just, S. Shelah and J. Szeptycki, Strongly almost disjoint sets and weakly uniform bases, Preprint no. 12 (1997/98), Hebrew Univ. Jerusalem, Inst. of Math. Zbl0960.03039
  2. [EH] P. Erdős and A. Hajnal, On a property of families of sets, Acta Math. Acad. Sci. Hungar. 12 (1961), 87-124. Zbl0201.32801
  3. [Gr] J. Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41 (1976), 663-671. Zbl0347.02044
  4. [HJSh] A. Hajnal, I. Juhász and S. Shelah, Splitting strongly almost disjoint families, Trans. Amer. Math. Soc. 295 (1986), 369-387. Zbl0619.03033
  5. [Ka] A. Kanamori, The Higher Infinite, Springer, Berlin, 1994. 
  6. [Ko] P. Komjáth, Families close to disjoint ones, Acta Math. Hungar. 43 (1984), 199-207. Zbl0541.03027
  7. [S] R. Solovay, Strongly compact cardinals and the GCH, in: Proc. Sympos. Pure Math. 25, Amer. Math. Soc., 1974, 365-372. Zbl0317.02083
  8. [W] N. H. Williams, Combinatorial Set Theory, Stud. Logic 91, North-Holland, Amsterdam, 1977. 

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