How to drive our families mad

2006/11/24 by Fuchino, Sakaé, Geschke, Stefan, Guzman, Osvaldo +1
#03E35 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.math/0611744

Abstract

Given a family F of pairwise almost disjoint sets on a countable set S, we study maximal almost disjoint (mad) families F⁺ extending F. We define a⁺(F) to be the minimal possible cardinality of F⁺∖ F for such F⁺, and a⁺(κ)=sup\a⁺(F): |F| ≤ κ\. We show that all infinite cardinal less than or equal to the continuum continuum can be represented as a⁺(F) for some almost disjoint F and that the inequalities ℵ₁=a

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