Strongly almost disjoint functions
type: Item
instance of: scholarly article
title: Strongly almost disjoint functions
zbMATH Open document ID: 0873.03046
DOI: 10.1007/BF02774029
author: Jindřich Zapletal
published in: Israel Journal of Mathematics
publication date: 1997-09-28T00:00:00Z
review text: A family of functions from \( \omega_1 \) to \( \omega \) is \textsl{strongly almost disjoint} if for any distinct \( f \) and \( g \) in the family, there are only finitely many \( \alpha \in \omega_1 \) such that \( f(\alpha) = g(\alpha) \). It is known that: (1) There is a strongly almost disjoint family of cardinality \( \aleph_1 \); and (2) The negation of Chang's Conjecture is equivalent to the existence of a c.c.c.\ poset that forces the existence of a strongly disjoint family of cardinality \( \aleph_2 \) [\textit{S. Todorcevic}, Can. J. Math. 43, 832-851 (1991; Zbl 0776.03024)]. The author considers the possible existence of even larger strongly almost disjoint families, and shows that for any cardinal \( \kappa \), there is a cardinal-preserving forcing notion that forces the existence of a strongly almost disjoint family of cardinality \( \kappa \). He assumes the Continuum Hypothesis in the ground model, and uses the side condition method due to \textit{S. Todorcevic} [Partition problems in topology (Contemp. Math. 84) (1989; Zbl 0659.54001)]. He remarks that it is also possible to obtain the same result by iterated forcing, and that this approach can be generalized to give a consistency proof of the existence of an arbitrarily large strongly almost disjoint family of functions from \( \omega_{n+1} \) to \( \omega_n \) (\( n \in \omega \)).
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 1011391
zbMATH Keywords: strongly almost disjoint family
zbMATH Keywords: side condition method
zbMATH Keywords: cardinal-preserving forcing notion
zbMATH Keywords: continuum hypothesis
reviewed by: J. Takahashi
MaRDI profile type: Publication
cites work: Almost-disjoint sets the dense set problem and the partition calculus
cites work: Q3852172
cites work: A note on canonical functions
cites work: Proper forcing
cites work: Partition Problems in Topology
cites work: Remarks on Martin's Axiom and the Continuum Hypothesis
full work available at URL: https://doi.org/10.1007/bf02774029
OpenAlex ID: W1963777171
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description: scientific article; zbMATH DE number 1011391