We follow up on the last talk by going through a detailed proof of the construction of an almost disjoint family of finitely splitting trees (a.d.f.s. family) which stays maximal after forcing with countably supported product or iteration of Sacks forcing of any length. Remember that maximal a.d.f.s. families are in one-to-one correspondence with partitions of Baire space into compact sets. To this end we first prove the main fusion lemma which lets us construct a maximal a.d.f.s. family which is indestructible by countably supported product of Sacks forcing of size \(\aleph_0\). We then adapt the construction of a Sacks-indestructible maximal eventually different family by V. Fischer and D. Schrittesser to show that this family already satisfies the indestructibility properties of our theorem. If time permits we give an idea how to generalize the construction to other combinatorial families, for example maximal cofinitary groups.

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