Maximal almost disjoint families of functions

• Autorzy: Dilip Raghavan
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 03E50: Continuum hypothesis and Martin's axiom
• 03E17: Cardinal characteristics of the continuum
• 03E05: Other combinatorial set theory
• 03E40: Other aspects of forcing and Boolean-valued models
• 03E65: Other hypotheses and axioms
• 03E15: Descriptive set theory
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2009
• Tom: 204
• Numer: 3
• Strony: 241-282

Daty

wydano

2009

Twórcy

autor

Dilip Raghavan

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

Identyfikatory

DOI

10.4064/fm204-3-3