Tight Eventually Different Families

Abstract: Generalizing the notion of a tight almost disjoint family, we introduce the notions of a {\em tight eventually different} family of functions in Baire space and a {\em tight eventually different set of permutations} of $\omega$. Such sets strengthen maximality, exist under $\mathsf{MA} (\sigma {\rm -linked})$ 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 $\mathfrak{a}_e$ and $\mathfrak{a}_p$ in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including $\mathfrak{a}_e = \mathfrak{a}_p = \mathfrak{d} < \mathfrak{a}_T$, $\mathfrak{a}_e = \mathfrak{a}_p < \mathfrak{d} = \mathfrak{a}_T$, $\mathfrak{a}_e = \mathfrak{a}_p = \mathfrak{u} < non(\mathcal N) = cof(\mathcal N)$ and $\mathfrak{a}_e = \mathfrak{a}_p =\mathfrak{i} < \mathfrak{u}$. We also show that there are $\Pi^1_1$ tight eventually different families and tight eventually different sets of permutations in $L$ thus obtaining the above inequalities alongside $\Pi^1_1$ witnesses for $\mathfrak{a}_e = \mathfrak{a}_p = \aleph_1$. Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic.