Michael Hru\v s\' ak
MAD families and the rationals

Comment.Math.Univ.Carolinae 42,2 (2001) 345-352.

Abstract:Rational numbers are used to classify maximal almost disjoint (MAD) families of subsets of the integers. Combinatorial characterization of indestructibility of MAD families by the likes of Cohen, Miller and Sacks forcings are presented. Using these it is shown that Sacks indestructible MAD family exists in ZFC and that $\frak b =\frak c$ implies that there is a Cohen indestructible MAD family. It follows that a Cohen indestructible MAD family is in fact indestructible by Sacks and Miller forcings. A connection with Roitman's problem of whether $\frak d=\omega _1$ implies $\frak a=\omega _1$ is also discussed.

Keywords: maximal almost disjoint family; Cohen, Miller, Sacks forcing; cardinal invariants of the continuum
AMS Subject Classification: 03E17, 03E05, 03E20