Analytic and coanalytic families of almost disjoint functions

Abstract

If ℱ ⊆ ^ℕℕ is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable ℌ⊆ ^ℕℕ, no member of which is covered by finitely many functions from ℱ, there is f ∈ ℱ such that for all h∈ ℌ there are infinitely many integers k such that f(k) = h(k). However if V=L then there exists a coanalytic family of pairwise eventually different functions satisfying this strong maximality condition.