Almost disjoint families on large underlying sets

We show that, for any poset P, the existence of a P-indestructible mad family F ⊆ [ω]א0 is equivalent to the existence of such a family over אn for some/all n ∈ ω. Under the very weak square principle ¤∗∗∗ ω1,μ of Fuchino and Soukup [7] and cf([μ]א0 ,⊆) = μ+ for all limit cardinals μ of cofinality ω, the equivalence for any proper poset P transfers to all cardinals. That is, under these assumptions, if P is a proper poset, then there is a P-indestructible mad family on ω if and only if there is a P-indestructible mad family on some/all infinite cardinals κ.