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), if a ∩ b is finite for all a, b ∈ A, a ̸= b. Such A is called maximal almost disjoint (mad for short), if it is maximal with respect to ⊆ among a. d. families. An easy diagonalization shows that every infinite mad family is uncountable. The well-known cardinal invariant a is defined as the minimal cardinality of an infinite mad family. Over the past decades, much work has been done to understand this cardinal. We only mention [Sh700] and [Br]. In the first one, the consistency of d < a with ZFC was proved, in the second one, which further develops the ideas of [Sh700], it was shown that consistently a = אω. It is natural to study mad families in more general ways, e. g. investigate the mad spectrum, i. e. the set A of all infinite cardinals that are the cardinality of some mad family. This problem has been attacked already in the early period of forcing by Hechler [H]. There are two obvious restrictions A must satisfy. Firstly, A has 2א0 as its maximum, and, secondly, A is closed under singular limits (see [H, Theorem 3.1]). For the first one, notice that there are always a. d. and hence mad families of size 2א0 . For the second one, if μ is a singular 2010 Mathematics Subject Classification. Primary: 03E35, 03E17;