Abstract
We show that, consistently, there can be maximal subtrees of \(\mathcal{P}(\omega )\) and \(\mathcal{P}(\omega ) / {\mathrm {fin}}\) of arbitrary regular uncountable size below the size of the continuum \({\mathfrak c}\). We also show that there are no maximal subtrees of \(\mathcal{P}(\omega ) / {\mathrm {fin}}\) with countable levels. Our results answer several questions of Campero-Arena et al. (Fund Math 234:73–89, 2016).
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Partially supported by Grant-in-Aid for Scientific Research (C) 15K04977, Japan Society for the Promotion of Science, and by Michael Hrušák’s Grants, CONACyT Grant No. 177758 and PAPIIT Grant IN-108014.
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Brendle, J. Maximal trees. Arch. Math. Logic 57, 421–428 (2018). https://doi.org/10.1007/s00153-017-0575-2
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DOI: https://doi.org/10.1007/s00153-017-0575-2