# Strongly almost disjoint familes, revisited

A. Hajnal; Istvan Juhász; Saharon Shelah

Fundamenta Mathematicae (2000)

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

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topHajnal, 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

top- [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
- [EH] P. Erdős and A. Hajnal, On a property of families of sets, Acta Math. Acad. Sci. Hungar. 12 (1961), 87-124. Zbl0201.32801
- [Gr] J. Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41 (1976), 663-671. Zbl0347.02044
- [HJSh] A. Hajnal, I. Juhász and S. Shelah, Splitting strongly almost disjoint families, Trans. Amer. Math. Soc. 295 (1986), 369-387. Zbl0619.03033
- [Ka] A. Kanamori, The Higher Infinite, Springer, Berlin, 1994.
- [Ko] P. Komjáth, Families close to disjoint ones, Acta Math. Hungar. 43 (1984), 199-207. Zbl0541.03027
- [S] R. Solovay, Strongly compact cardinals and the GCH, in: Proc. Sympos. Pure Math. 25, Amer. Math. Soc., 1974, 365-372. Zbl0317.02083
- [W] N. H. Williams, Combinatorial Set Theory, Stud. Logic 91, North-Holland, Amsterdam, 1977.

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