Maximal almost disjoint families of functions

Dilip Raghavan¹

¹ Department of Mathematics University of Toronto Toronto, ON M5S 2E4 Canada

Fundamenta Mathematicae, Tome 204 (2009) no. 3, pp. 241-282

Résumé

We study maximal almost disjoint (MAD) families of functions in $\omega^\omega$ 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 $\mathop{\rm cov}\nolimits({\cal M}) {\mathfrak a}_{\mathfrak e}$, then there no very MAD families. We answer a question of Kastermans by constructing a strongly MAD family from ${\mathfrak b} = {\mathfrak c}$. 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 ${\cal P}(\omega)$ implies that all strongly MAD families have size at most ${\aleph}_{1}$.

Détail

DOI : 10.4064/fm204-3-3

Keywords: study maximal almost disjoint mad families functions omega omega satisfy certain strong combinatorial properties particular study notions strongly mad families functions introduce study hierarchy combinatorial properties lying between strong madness madness proving conjecture brendle mathop cov nolimits cal mathfrak mathfrak there mad families answer question kastermans constructing strongly mad family mathfrak mathfrak study indestructibility properties strongly mad families prove strong madness strongly mad families preserved large class posets make ground model reals meager solve well known problem kellner shelah showing countable support iteration proper posets limit length does make ground model reals meager initial segment does finally prove weak freese nation property cal omega implies strongly mad families have size aleph