Analytic and coanalytic families of almost disjoint functions

If F ⊆ N N is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable H ⊆ N N, 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. §1. Introduction. It is a well known phenomenon of descriptive set theory that subsets of the reals requiring the axiom of choice in order to exist do not have nice descriptions. For example: • (Suslin [5]) No well ordering of an uncountable set of reals is analytic. • (Sierpinski) No ultrafilter is measurable or has the property of Baire. • (Talagrand [11]) The intersection of countably many nonmeasurable filters is nonmeasurable. • (Mathias [7]) There is no analytic maximal almost disjoint family. Since many variations on the theme of a maximal almost disjoint family have been explored, the last fact raises a series of questions about the definability properties of other such maximal families. It is the purpose of this paper to analyze one instance of this question for the case of almost disjoint families obtained from graphs. The following definition clarifies this. A family of functions F ⊆ N N will be said to be eventually different if for any two f and g in F there is some k such that f (n) = g(n) for n ≥ k. A maximal eventually different family is one which is maximal with respect to this property. The following question remains open: Question 1.1. Is there an analytic (or even closed) maximal, eventually different family?