Tight eventually different families

Generalizing the notion of a tight almost disjoint family, we introduce the notions of a tight eventually different family of functions in Baire space and a tight eventually different set of permutations of ω. Such sets strengthen maximality, exist under MA(σ−centered) and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals ae and ap in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including ae=ap=d<aT, ae=ap<d=aT, ae=ap=i<u, and ae=ap=a<non(N)=cof(N). We also show that there are Π11 tight eventually different families and tight eventually different sets of permutations in L thus obtaining the above inequalities alongside Π11 witnesses for ae=ap=ℵ1

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Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic.