### 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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