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Maximal almost disjoint families of functions

100%

Raghavan D.

Fundamenta Mathematicae

2009

tom 204

nr 3

241-282

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.

Comparing the closed almost disjointness and dominating numbers

63%

Raghavan D., Shelah S.

Fundamenta Mathematicae

2012

tom 217

nr 1

73-81

We prove that if there is a dominating family of size ℵ₁, then there are ℵ₁ many compact subsets of $ω^{ω}$ whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Raghavan D.

1 Shelah S.

1 2012

1 2009

Wyniki wyszukiwania

Maximal almost disjoint families of functions

Comparing the closed almost disjointness and dominating numbers