Strongly almost disjoint familes, revisited

• Autorzy: A. Hajnal , Istvan Juhász , Saharon Shelah
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

strongly almost disjoint family; property B; σ-transversal

Kategorie tematyczne

• 03E50: Continuum hypothesis and Martin's axiom
• 03E55: Large cardinals
• 03E05: Other combinatorial set theory
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2000
• Tom: 163
• Numer: 1
• Strony: 13-23

Daty

wydano

2000

otrzymano

1998-12-07

poprawiono

1999-09-05

Twórcy

autor

A. Hajnal

• Department of Mathematics, Rutgers University New Brunswick, NJ 08903, U.S.A.

autor

Istvan Juhász

• Mathematical Institute of the Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary

autor

Saharon Shelah

• Department of Mathematics, Rutgers University New Brunswick, NJ 08903, U.S.A.
• 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.