Partitions of the Baire space into compact sets

We define a c.c.c. forcing which adds a maximal almost disjoint family of finitely splitting trees on $\omega $ (a.d.f.s. family) or equivalently a partition of the Baire space into compact sets of desired size. Further, we utilize this forcing to add arbitrarily many maximal a.d.f.s. families of arbitrary sizes at the same time, so that the spectrum of $\mathfrak a_{T} $ may be large.

Furthermore, under CH we construct a Sacks-indestructible maximal a.d.f.s. family (by countably supported iteration and product of any length), which answers a question of Newelski (1987). Also, we present an in-depth “isomorphism of names” argument to show that in generic extensions of models of CH by countably supported Sacks forcing there are no maximal a.d.f.s. families of size $\kappa $, where $\aleph _1 \lt \kappa \lt \mathfrak c $. Thus, we prove that in the generic extension the spectrum of $\mathfrak a_{T}$ is $\{\aleph _1, \mathfrak c\}$. Finally, we prove that Shelah’s (2004) ultrapower model for the consistency of $\mathfrak d \lt \mathfrak a $ also satisfies $\mathfrak a = \mathfrak a_{T} $. Thus, consistently $\mathfrak d \lt \mathfrak a= \mathfrak a_{T} $ may hold.