Geodesic
MAD families and the rationals
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 2, pp. 345-352 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 03E05, 03E17, 03E20
Keywords: maximal almost disjoint family; Cohen; Miller; Sacks forcing; cardinal invariants of the continuum 
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     author = {Hru\v{s}\'ak, Michael},
     title = {MAD families and the rationals},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {345--352},
     year = {2001},
     volume = {42},
     number = {2},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2001_42_2_a10/}
}
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Hrušák, Michael. MAD families and the rationals. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 2, pp. 345-352. http://geodesic.mathdoc.fr/item/CMUC_2001_42_2_a10/