Hajnal, András; Juhász, István; Shelah, Saharon Strongly almost disjoint families, revisited. (English) Zbl 0946.03057 Fundam. Math. 163, No. 1, 13-23 (2000). 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. Cited in 2 Documents MSC: 03E05 Other combinatorial set theory 03E35 Consistency and independence results 03E55 Large cardinals 03E50 Continuum hypothesis and Martin’s axiom Keywords:strongly almost disjoint family; property \(B\); \(\sigma\)-transversal; \(\rho\)-closed forcing; GCH; large cardinals Citations:Zbl 0201.32801 PDFBibTeX XMLCite \textit{A. Hajnal} et al., Fundam. Math. 163, No. 1, 13--23 (2000; Zbl 0946.03057) Full Text: EuDML