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. (shrink)

We extend the work of Fischer et al. [6] by presenting a method for controlling cardinal characteristics in the presence of a projective wellorder and 2ℵ0>ℵ2. This also answers a question of Harrington [9] by showing that the existence of a Δ31 wellorder of the reals is consistent with Martinʼs axiom and 2ℵ0=ℵ3.

We show that is consistent with the existence of a -definable wellorder of the reals and a -definable ω-mad subfamily of [ω]ω.