Maximal almost disjoint families of functions

Dilip Raghavan. "Maximal almost disjoint families of functions." Fundamenta Mathematicae 204.3 (2009): 241-282. <http://eudml.org/doc/282755>.

@article{DilipRaghavan2009,
abstract = {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 $cov(ℳ ) < _\{\}$, then there no very MAD families. We answer a question of Kastermans by constructing a strongly MAD family from = . 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 (ω) implies that all strongly MAD families have size at most ℵ₁.},
author = {Dilip Raghavan},
journal = {Fundamenta Mathematicae},
keywords = {maximal almost disjoint family; cardinal invariants; iterated forcing; preservation theorem; Freese-Nation property},
language = {eng},
number = {3},
pages = {241-282},
title = {Maximal almost disjoint families of functions},
url = {http://eudml.org/doc/282755},
volume = {204},
year = {2009},
}

TY - JOUR
AU - Dilip Raghavan
TI - Maximal almost disjoint families of functions
JO - Fundamenta Mathematicae
PY - 2009
VL - 204
IS - 3
SP - 241
EP - 282
AB - 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 $cov(ℳ ) < _{}$, then there no very MAD families. We answer a question of Kastermans by constructing a strongly MAD family from = . 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 (ω) implies that all strongly MAD families have size at most ℵ₁.
LA - eng
KW - maximal almost disjoint family; cardinal invariants; iterated forcing; preservation theorem; Freese-Nation property
UR - http://eudml.org/doc/282755
ER -