Cofinality of the laver ideal

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.