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Tytuł artykułu

Strongly almost disjoint familes, revisited

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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].
Twórcy
autor
  • Mathematical Institute of the Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary, juhasz@math-inst.hu
  • Department of Mathematics, Rutgers University New Brunswick, NJ 08903, U.S.A., shelah@math.huji.ac.il
  • Institute of Mathematics, The Hebrew University, 91904 Jerusalem
Bibliografia
  • [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.
  • [EH] P. Erdős and A. Hajnal, On a property of families of sets, Acta Math. Acad. Sci. Hungar. 12 (1961), 87-124.
  • [Gr] J. Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41 (1976), 663-671.
  • [HJSh] A. Hajnal, I. Juhász and S. Shelah, Splitting strongly almost disjoint families, Trans. Amer. Math. Soc. 295 (1986), 369-387.
  • [Ka] A. Kanamori, The Higher Infinite, Springer, Berlin, 1994.
  • [Ko] P. Komjáth, Families close to disjoint ones, Acta Math. Hungar. 43 (1984), 199-207.
  • [S] R. Solovay, Strongly compact cardinals and the GCH, in: Proc. Sympos. Pure Math. 25, Amer. Math. Soc., 1974, 365-372.
  • [W] N. H. Williams, Combinatorial Set Theory, Stud. Logic 91, North-Holland, Amsterdam, 1977.
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Bibliografia
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bwmeta1.element.bwnjournal-article-fmv163i1p13bwm
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