# Sh:697

- Hajnal, A., Juhász, I., & Shelah, S. (2000).
*Strongly almost disjoint families, revisited*. Fund. Math.,**163**(1), 13–23. arXiv: math/9812114 MR: 1750332 -
Abstract:

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 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\le\varrho^{(+\varrho+1)}).

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Bib entry

@article{Sh:697, author = {Hajnal, Andr{\'a}s and Juh{\'a}sz, Istv{\'a}n and Shelah, Saharon}, title = {{Strongly almost disjoint families, revisited}}, journal = {Fund. Math.}, fjournal = {Fundamenta Mathematicae}, volume = {163}, number = {1}, year = {2000}, pages = {13--23}, issn = {0016-2736}, mrnumber = {1750332}, mrclass = {03E05 (03E02 03E35)}, note = {\href{https://arxiv.org/abs/math/9812114}{arXiv: math/9812114}}, arxiv_number = {math/9812114}, keyword = {Strongly almost disjoint family, property B, $\sigma$-transversal.} }