In my talk, I would like to present joint work with Otmar Spinas, in which we show that it is consistent that \(\mathfrak{h}\) (the distributivity number of \(P(\omega)/fin\), in other words, the least number of maximal almost disjoint families without a common refinement) is strictly smaller than \(\mathrm{add}(s_0)\) (i.e., the least number of Marczewski null sets whose union is not Marczewski null, where a set is Marczewski null if each perfect set has a perfect subset disjoint from it). More explicitly, we show that this relation between \(\mathfrak{h}\) and \(\mathrm{add}(s_0)\) holds in the model obtained by a countable support iteration of length \(\omega_2\) of a specific kind of Sacks amoeba forcing which happens to have the pure decision and the Laver property, and therefore does not add Cohen reals. The model actually satisfies \(\mathfrak{h} = \mathrm{cov}(\mathcal{M}) < \mathfrak{b} = \mathfrak{s} = \mathrm{add}(s_0)\). (If time permits, I would also like to discuss why it seems easy to slightly modify the construction in such a way that the resulting model additionally satisfies the Borel Conjecture, but unclear how to modify it to make the Borel Conjecture fail.)

A video recording of this talk is available on Youtube.