Strongly almost disjoint families, {I}{I}

by Hajnal and Juhasz and Shelah. [HJSh:697]
Fundamenta Math, 2000
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 varrho be any regular cardinal with a supercompact [resp. 2-huge] cardinal above varrho . Then there is a varrho-closed forcing P such that, in V^P, we have both GCH and M(varrho^{(+ varrho +1)}, varrho^+, varrho) nrightarrow B (resp. M(varrho^{(+ varrho +1)}, lambda, varrho) nrightarrow B(varrho^+) for all lambda <= varrho^{(+ varrho +1)}).

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