Tight Eventually Different Families

TL;DR

This paper introduces the concept of tight eventually different families in Baire space, explores their properties, and uses them to analyze and construct models with specific cardinal characteristics of the continuum.

Contribution

It defines tight eventually different families, proves their existence under certain axioms, and computes related cardinal invariants in various models.

Findings

Existence of tight eventually different families under MA(σ-linked)

Explicit witnesses for cardinal characteristics in many models

Tight families are Cohen indestructible and non-analytic

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.