Abstract
Yurii Khomskii observed that \({{\mathrm{cof}}}(l^0)>\mathfrak {c}\) assuming \(\mathfrak {b}=\mathfrak {c}\) and he asked whether the inequality \({{\mathrm{cof}}}(l^0)>\mathfrak {c}\) is provable in ZFC. We find several conditions that imply some variants of this inequality for tree ideals. Applying a recent result of Brendle, Khomskii, and Wohofsky we show that \(l^0\) satisfies some of these conditions and consequently, \({{\mathrm{cof}}}(l^0)=\mathfrak {d}(^\mathfrak {c}l^0)\ge \mathfrak {d}(^\mathfrak {c}\mathfrak {c})>\mathfrak {c}\). We also prove that if the cellularity of a Boolean algebra B is hereditarily \(\ge \kappa \), then every \(\kappa \)-sequence in \(B^+\) has a \(\kappa \)-subsequence with a disjoint refinement.
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Miroslav Repický was supported by Grants APVV-0269-11 and VEGA 1/0097/16.
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Repický, M. Cofinality of the laver ideal. Arch. Math. Logic 55, 1025–1036 (2016). https://doi.org/10.1007/s00153-016-0510-y
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DOI: https://doi.org/10.1007/s00153-016-0510-y