Forcing indestructibility of MAD families

Brendle, Jörg; Yatabe, Shunsuke

doi:10.1016/j.apal.2004.09.002

Cited by 32 publications

(36 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first step to construct a +-Ramsey MAD family is to prove that every Miller-indestructible MAD family has this property. If A is a MAD family and P is a partial order, then we say A is P-indestructible if A is still a MAD family after forcing with P. The destructibility of MAD families has become a very important area of research with many fundamental questions still open (the reader may consult [8], [9], or [4] to learn more about the indestructibility of MAD families and ideals). The following property of MAD families plays a fundamental role in the study of destructibility:…”

Section: Proof Let [ω]mentioning

confidence: 99%

See 1 more Smart Citation

There is a +-Ramsey MAD family

Guzmán

2019

Isr. J. Math.

View full text Add to dashboard Cite

We answer an old question of Michael Hrušák by constructing a +-Ramsey MAD family without the need of any additional axioms beyond ZFC. We also prove that every Miller-indestructible MAD family is +-Ramsey, this improves a result of Michael Hrušák. * Definition 1 1. By A ⊥ we denote the set of all X ⊆ ω such that A∪{X} is almost disjoint. IfIn [6] it is proved that there is a MAD family that is not +-Ramsey. On the other hand, +-Ramsey MAD families can be contructed under b = c, cov(M) = c, a < cov(M) or ♦ (b) (see [6] and [8]). Michael Hrušák asked the following, Problem 2 (Hrušák [6]) Is there a +-Ramsey MAD family in ZFC?In this note we will provide a positive answer to this question. In [20] (see also [7] and [15]) Saharon Shelah developed a novel and powerfull method to construct MAD families. He used it to prove that there is a completely separable MAD family if s ≤ a or a < s and a certain PCF-hypothesis holds. Our technique for constructing a +-Ramsey MAD is based on the technique of Shelah (however, in this case we were able to avoid the PCF-hypothesis). It is worth mentioning that the method of Shelah has been further developed in [19] and [15] where it is proved that weakly tight MAD families exist under s ≤ b. Our notation is mostly standard, the definition and basic properties of the cardinal invariants of the continuum used in this note can be found in [2]. PreliminariesA MAD family A is completely separable if for every X ∈ I (A) + there is A ∈ A such that A ⊆ X. This type of MAD families was introduced by Hechler in [5]. A year later, Shelah and Erdös asked the following question:Problem 3 (Erdös-Shelah) Is there a completely separable MAD family?It is easy to construct models where the previous question has a positive answer. It was shown by Balcar and Simon (see [1]) that such families exist assuming one of the following axioms: a = c, b = d, d ≤ a and s = ω 1 . In [20] (see also [7] and [15]) Shelah developed a novel and powerful method to construct completely separable MAD families. He used it to prove that there are such families if either s ≤ a or a < s and a certain (so called) PCF hypothesis

show abstract

Section: Proof Let [ω]mentioning

confidence: 99%

“…The converse of the previous result is not true in general, this will be shown in the corollary 27. It is known that every MAD family of size less than d is Miller indestructible (see [4]). We can then conclude the following unpublished result of Michael Hrušák, which he proved by completely different means.…”

Section: Proof Let [ω]mentioning

confidence: 99%

There is a +-Ramsey MAD family

Guzmán

2019

Isr. J. Math.

View full text Add to dashboard Cite

show abstract

“…Brendle and Yatabe [4] have studied P-indestructibility of MAD families of subsets of ω for various posets P. The focus of their work was to provide combinatorial characterizations of the property of being a P-indestructible MAD family of sets for some well known posets P. Here our focus is instead to find those posets P for which strongly MAD families of functions are strongly P-indestructible.…”

mentioning

confidence: 99%

“…In Section 3, we modify a construction of Hrušák [8], Kurilić [18] and Brendle and Yatabe [4] to show that strongly MAD families exist if b = c. In Section 4, we prove a conjecture of Brendle that if cov(M) < a e , there are no very MAD families. Together these results show that in the Laver model there is a strongly MAD family, but no very MAD families.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Maximal almost disjoint families of functions

Raghavan

2009

Fund. Math.

View full text Add to dashboard Cite

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) < ae, 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.

show abstract

Silver-Like Forcing Notions

Halbeisen

2012

Springer Monographs in Mathematics

View full text Add to dashboard Cite

Forcing indestructibility of MAD families

Cited by 32 publications

References 17 publications

There is a +-Ramsey MAD family

There is a +-Ramsey MAD family

Maximal almost disjoint families of functions

Silver-Like Forcing Notions

Contact Info

Product

Resources

About