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2009 | 204 | 3 | 241-282
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Maximal almost disjoint families of functions

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We study maximal almost disjoint (MAD) families of functions in $ω^{ω}$ that satisfy certain strong combinatorial properties. In particular, we study the notions of strongly and very MAD families of functions. We introduce and study a hierarchy of combinatorial properties lying between strong MADness and very MADness. Proving a conjecture of Brendle, we show that if $cov(ℳ ) < 𝔞_{𝔢}$, then there no very MAD families. We answer a question of Kastermans by constructing a strongly MAD family from 𝔟 = 𝔠. Next, we study the indestructibility properties of strongly MAD families, and prove that the strong MADness of strongly MAD families is preserved by a large class of posets that do not make the ground model reals meager. We solve a well-known problem of Kellner and Shelah by showing that a countable support iteration of proper posets of limit length does not make the ground model reals meager if no initial segment does. Finally, we prove that the weak Freese-Nation property of 𝓟(ω) implies that all strongly MAD families have size at most ℵ₁.
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  • Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 Canada
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm204-3-3
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