Eprints of Rényi Institute

How to drive our families mad

Fuchino, Sakae and Geschke, Stefan and Soukup, Lajos (2011) How to drive our families mad. Arch. Math. Logic . (In Press)

[img]
Preview
PDF
221Kb

Abstract

Given a family \mathcal {F} of pairwise almost disjoint (ad) sets on a countable set S, we study families \tilde{\mathcal{F}} of maximal almost disjoint (mad) sets extending \mathcal{F}.
We define \mathfrak{a}^+(\mathcal{F}) to be the minimal possible cardinality of \tilde{\mathcal{F}}\setminus \cal F for such \tilde{\mathcal{F}} and \mathfrak{a}^+(\kappa)=\sup\{\mathfrak{a}^+(\mathcal{F}):|{\mathcal{F}|\le\kappa}\}. We show that all infinite cardinals less than or equal to the continuum 2^\omega can be represented as \mathfrak{a}^+(\mathcal{F}) for some ad \mathcal{F} and that the inequalities \aleph_1=\mathfrak{a}<\mathfrak{a}^+(\aleph_1)=2^\omega and \mathfrak{a}=\mathfrak{a}^+(\aleph_1)<2^\omega are both consistent.
We also give several constructions of mad families with some additional properties.

Item Type:Article
Keywords and phrases:cardinal invariants, almost disjoint number, Cohen model, destructible maximal almost disjoint family
Subjects:03E35 Consistency and independence results
Divisions:Research Divisions > Set theory and general topology
Last Modified:10 Jan 2011 16:07

Repository Staff Only: item control page