A. Miller introduced in 1980 a forcing notion we refer to as a partition forcing. Although it is a variant of Sacks' perfect set forcing, it is closely related to Miller's rational perfect set forcing.
The talk is devoted to our application of partition forcing in a proof of consistency of \(\mathfrak{u}=\mathfrak{i}<\mathfrak{a}_T\). Here, \(\mathfrak{i}\) is the minimal cardinality of a maximal independent family, \(\mathfrak{u}\) a minimal size of an ultrafilter base, and \(\mathfrak{a}_T\) is the minimal cardinality of a maximal family of pairwise almost disjoint subtrees of \(2^{<\omega}\).
This is a joint work with Vera Fischer.