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Strongly almost disjoint families, revisited. (English) Zbl 0946.03057

Summary: The relations \(M(\kappa,\lambda,\mu)\to B\) [resp. \(B (\sigma)]\) meaning that if \({\mathcal A}\subset [\kappa]^\lambda\) with \(|{\mathcal A} |=\kappa\) is \(\mu\)-almost disjoint then \({\mathcal A}\) has property \(B\) [resp. has a \(\sigma\)-transversal] had been introduced and studied under GCH by P. Erdős and A. Hajnal [Acta Math. Acad. Sci. Hung. 12, 87-123 (1961; Zbl 0201.32801)]. Our two main results here say the following:
Assume GCH and let \(\rho\) 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 \(M(\rho^{(+ \rho+1)}, \rho^+, \rho) \nrightarrow B\) [resp. \(M(\rho^{(+\rho+1)},\lambda,\rho)\nrightarrow B(\rho^+)\) for all \(\lambda\leq \rho^{(+\rho +1)}]\).
These show that, consistently, the results of Erdős and Hajnal [loc. cit.] are sharp. The necessity of using large cardinals follows from results of P. Komjáth and of the authors.

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
03E55 Large cardinals
03E50 Continuum hypothesis and Martin’s axiom

Citations:

Zbl 0201.32801
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