Strongly almost disjoint familes, revisited

The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if $A⊂[κ]^λ$ with |A|=κ 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].