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.
"Analytic and coanalytic families of almost disjoint functions." J. Symbolic Logic 73 (4) 1158 - 1172, December 2008. https://doi.org/10.2178/jsl/1230396911