Tight Eventually Different Families

Abstract

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 $\omega $ . Such sets strengthen maximality, exist under $\mathsf {MA} (\sigma \mathrm {-centered})$ 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 $\mathfrak {a}_e$ 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 $\mathfrak {a}_e = \mathfrak {a}_p = \mathfrak {d} < \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p < \mathfrak {d} = \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p =\mathfrak {i} < \mathfrak {u}$ , and $\mathfrak {a}_e=\mathfrak {a}_p = \mathfrak {a} < non(\mathcal N) = cof(\mathcal N)$ . 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.