Strongly almost disjoint families, II

The relations M(kappa,lambda,mu)->B [resp. B(sigma)] meaning that if A subset [kappa]^lambda with |A|=kappa is mu-almost disjoint then A has property B [resp. has a sigma-transversal] had been introduced and studied under GCH by Erdos and Hajnal in 1961. Our two main results here say the following: Assume GCH and rho be any regular cardinal with a supercompact [resp. 2-huge] cardinal above rho. Then there is a rho-closed forcing P such that, in V^P, we have both GCH and M(rho^{(+rho+1)},rho^+,rho) not-> B [resp. M(rho^{(+rho+1)},lambda,rho) not-> B(rho^+) for all lambda =< rho^{(+rho+1)}].