Strongly almost disjoint familes, revisited

The relations M(K, λ, μ) → B [resp. B(σ)] meaning that if A ⊂ [K]^λ with \A\ = K is μ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let ρ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ρ. Then there is a ρ-closed forcing P such that, in V^P, we have both GCH and M(ρ^(+ρ+1), ρ⁺, ρ) →B [resp. M(ρ^(+ρ+1), λ, ρ) → B(ρ⁺) for all λ ≤ ρ^(+ρ+1)]. These show that, consistently, the results of [EH] are sharp. The necessity of using large cardinals follows from the results of [Ko], [HJSh] and [BDJShSz].