Journal of Symbolic Logic 66 (1):257-270 (2001)

Abstract
A maximal almost disjoint (mad) family $\mathscr{A} \subseteq [\omega]^\omega$ is Cohen-stable if and only if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-unstable. It is shown that a mad family, A, is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the sets G[A], A ∈A are nowhere dense. An ℵ 0 -mad family, A, is a mad family with the property that given any countable family $\mathscr{B} \subset [\omega]^\omega$ such that each element of B meets infinitely many elements of A in an infinite set there is an element of A meeting each element of B in an infinite set. It is shown that Cohen-stable mad families exist if and only if there exist ℵ 0 -mad families. Either of the conditions b = c or $\mathfrak{a} ) implies that there exist Cohen-stable mad families. Similar results are obtained for splitting families. For example, a splitting family, S, is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the boundaries of the sets G[S], S ∈S are nowhere dense. Also, Cohen-stable splitting families of cardinality ≤ κ exist if and only if ℵ 0 -splitting families of cardinality ≤ κ exist
Keywords Cohen Forcing   Mad Families   Splitting Families
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DOI 10.2307/2694920
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Splittings.A. Kamburelis & B. W’Glorz - 1996 - Archive for Mathematical Logic 35 (4):263-277.

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Citations of this work BETA

Projective Mad Families.Sy-David Friedman & Lyubomyr Zdomskyy - 2010 - Annals of Pure and Applied Logic 161 (12):1581-1587.
Forcing Indestructibility of MAD Families.Jörg Brendle & Shunsuke Yatabe - 2005 - Annals of Pure and Applied Logic 132 (2):271-312.
Covering properties of $$omega $$ω -mad families.Leandro Aurichi & Lyubomyr Zdomskyy - 2020 - Archive for Mathematical Logic 59 (3):445-452.
Mad Families, Forcing and the Suslin Hypothesis.Miloš S. Kurilić - 2004 - Archive for Mathematical Logic 44 (4):499-512.
Splitting Families and Forcing.Miloš S. Kurilić - 2007 - Annals of Pure and Applied Logic 145 (3):240-251.

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