Cohen-stable families of subsets of integers

Abstract

A maximal almost disjoint (mad) family ⊆ [ω]ω is Cohen-stable if and only if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-unstable. It is shown that a mad family. .is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the sets G[A]. A ∈ are nowhere dense. An ℵ0-mad family, . is a mad family with the property that given any countable family ℬ ⊂ [ω]ω such that each element of ℬ meets infinitely many elements of in an infinite set there is an element of meeting each element of ℬ in an infinite set. It is shown that Cohen-stable mad families exist if and only if there exist ℵ0-mad families. Either of the conditions b = c or a < cov() implies that there exist Cohen-stable mad families. Similar results are obtained for splitting families. For example, a splitting family. . is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the boundaries of the sets G[S], S ∈ are nowhere dense. Also. Cohen-stable splitting families of cardinality ≤ κ exist if and only if ℵ0-splitting families of cardinality ≤ κ exist.

Maximal almost disjoint families of functions

“…The notion of a strongly MAD family of subsets of ω was introduced in Malykhin [19]. It has been further studied by Kurilić [18] and Hrušák and García Ferreira [9]. The definition of this concept is identical to our Definition 2, but with ω ω replaced everywhere by [ω] ω , and with the additional requirement that the family be infinite.…”

“…Strongly MAD families were introduced by Steprāns [15], who showed that they cannot be analytic, though the same notion had been considered earlier by Malykhin [19] in the context of MAD families of subsets of ω and further studied by Kurilić [18] and Hrušák and García Ferreira [9]. Soon after, Zhang and Kastermans [14] introduced a strengthening of this notion, which they called very MAD family.…”

“…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.…”

“…Hrušák [8], Kurilić [18] and Brendle and Yatabe [4] construct a Cohenindestructible MAD family of sets from b = c. Our construction was inspired by theirs, although our presentation is different. We will inductively construct a strongly MAD family in c steps.…”

“…As a warm up, we first show that the Cohen poset strongly preserves all strongly MAD families. Kurilić [18] and Hrušák and García Ferreira [9] showed that a strongly MAD family in [ω] ω (see Definition 70) stays maximal after Cohen forcing. While it is possible to modify their proof to get a proof of Theorem 36, we give a different proof which foreshadows the proof of the more general Theorem 52.…”