Martin's axiom and separated mad families
type: Item
instance of: scholarly article
title: Martin's axiom and separated mad families
zbMATH Open document ID: 1272.03150
author: S. Shelah
author: Alan Dow
publication date: 2012-12-07T00:00:00Z
review text: Two families of subsets of \(\omega\) are said to be separated if there is a set which almost contains every element of one of the families, while it is almost disjoint from every element of the other family. An almost disjoint family is said to be \(\omega_1\)-separated if any disjoint pair of \(\leq\omega_1\)-sized subsets are separated. A maximal almost disjoint family \(\mathcal{A}\subseteq [\omega]^\omega\) is \(\omega_1\)-special if for every subfamily \(\mathcal{A^\prime}\in [\mathcal{A}]^{\leq \omega_1}\) there is a function \(c:[\mathcal{A}^\prime]^{<\omega}\to\omega\) and a linear ordering \(<\) of \(\mathcal{A}^\prime\) such that for each \(n\in\omega\) and \(n\)-tuples \(B_1<\cdots<B_n\), \(C_1,\cdots<C_n\) from \(\mathcal{A}^\prime\) if \(c({B_1,\cdots,B_n})= c({C_1,\cdots,C_n})=k\) then for all \(i\neq j\) the intersection \(B_i\cap C_j\) is contained in \(k\). Two families of subsets of \(\omega\) are said to be orthogonal if each element of one of the families is almost disjoint from each element of the other family. In [Izv. Akad. Nauk SSSR, Ser. Mat. 11, 403--410 (1947; Zbl 0029.34703)], \textit{N. N. Luzin} constructed an orthogonal pair of \(\aleph_1\)-sized almost disjoint subsets of \(\omega\) which can not be separated. In [\textit{A. Dow}, Can. Math. Bull. 54, No. 2, 270--276 (2011; Zbl 1244.54064)], it is shown that PFA implies that every maximal almost disjoint family contains a Luzin sequence, and so an \(\omega_1\)-sized subfamily, which has the property that each of its uncountable pairs cannot be separated. In the present paper the authors obtain a model of MA in which \(\mathfrak{c}=\omega_2\) and there is an \(\omega_1\)-separated maximal almost disjoint family. The proof uses a result of Silver according to which \(\text{MA}_{\omega_1}\) implies that every \(\omega_1\)-special family is \(\omega_1\)-separated. In order to obtain their main result, using a finite support iteration of ccc posets of length \(\omega_2\), the authors forc…
reviewed by: Vera Fischer
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 6113990
zbMATH Keywords: maximal almost disjoint family
zbMATH Keywords: Martin's axiom
Wikidata QID: Q114017189
MaRDI profile type: Publication
full work available at URL: s12215-011-0078-7
OpenAlex ID: W2121334121
cites work: Lusin sequences under CH and under Martin's axiom
cites work: Sequential order under PFA
cites work: Q5794332
cites work: Internal cohen extensions
cites work: Strongs-Sequences and Variations on Martin's Axiom
cites work: Q4882397
DOI: 10.1007/S12215-011-0078-7
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 6113990