On two problems of Erdös and Hechler: New methods in singular madness

For an infinite cardinal μ, MAD(μ) denotes the set of all cardinalities of nontrivial maximal almost disjoint families over μ. Erd̈os and Hechler proved in 1973 the consistency of μ, ∈ MAD(μ) for a singular cardinal μ and asked if it was ever possible for a singular μ that μ ∉ MAD(μ), and also whether 2 ^{cf μ} < μ ↔ μ ∈ MAD(μ) for every singular cardinal μ. We introduce a new method for controlling MAD(μ) for a singular μ and, among other new results about the structure of MAD(μ) for singular μ, settle both problems affirmatively.