Very Mad Families

MAD families and the rationals

Rational numbers are used to classify maximal almost disjoint (MAD) families of subsets of the integers. Combinatorial characterization of indestructibility of MAD families by the likes of Cohen, Miller and Sacks forcings are presented. Using these it is shown that Sacks indestructible MAD family exists in ZFC and that b = c implies that there is a Cohen indestructible MAD family. It follows th...

Mad spectra

The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on ω. We treat the problem to characterize those sets A which, in some forcing extension of the universe, can be the mad spectrum. We give a complete solution to this problem under the assumption θ = θ, where θ = min(A). §0. Introduction. Recall that A ⊆ [ω] is called almost disjoint (a. d. for short),...

Mad families constructed from perfect almost disjoint families

We prove the consistency of b > א1 together with the existence of a Π11definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Question 16]. For the proof we construct a mad family in L which is an א1-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Bor...

SEVERAL MAD FAMILIES AND THEIR NEIGHBORSTapani

Almost disjoint families on large underlying sets

We show that, for any poset P, the existence of a P-indestructible mad family F ⊆ [ω]א0 is equivalent to the existence of such a family over אn for some/all n ∈ ω. Under the very weak square principle ¤∗∗∗ ω1,μ of Fuchino and Soukup [7] and cf([μ]א0 ,⊆) = μ+ for all limit cardinals μ of cofinality ω, the equivalence for any proper poset P transfers to all cardinals. That is, under these assumpt...