# Tight Eventually Different Families

Vera Fischer  Institut für Mathematik, Kurt Gödel Research Center, Universität
Wien, Kolingasse 14-16, 1090 Wien, AUSTRIA<EMAIL_ADDRESS>and
Corey Bacal Switzer  Institut für Mathematik, Kurt Gödel Research Center,
Universität Wien, Kolingasse 14-16, 1090 Wien, AUSTRIA
<EMAIL_ADDRESS>
###### 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 $\mathsf{MA}(\sigma{\rm-centered})$ 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.

###### Key words and phrases:

Cardinal characteristics; MAD Families; Eventually Different Families

###### 2000 Mathematics Subject Classification:

03E17, 03E35, 03E50

_Acknowledgements:_ The authors would like to thank the Austrian Science Fund
(FWF) for the generous support through grant number Y1012-N35.

##  1\. Introduction

Two infinite subsets of $\omega$, $A,B\in[\omega]^{\omega}$ are almost
disjoint if $A\cap B$ is finite. A family
$\mathcal{A}\subseteq[\omega]^{\omega}$ is almost disjoint if its elements are
pairwise almost disjoint. Such a family is maximal or MAD if it is almost
disjoint but not properly included in any other almost disjoint family. The
least size of an infinite MAD family is denoted by $\mathfrak{a}$ and is one
of the most studied cardinal invariants of the continuum. See [2, Section 8]
for more on $\mathfrak{a}$.

The cardinal $\mathfrak{a}$ has many relatives that have also been studied in
the literature. These essentially come in two forms: either we can replace
“finite” in the definition of almost disjoint with belonging to another ideal
or we can insist that the MAD family come equipped with additional structure.
Some examples of this are as follows:

  1. (1)

$\mathfrak{a}_{e}$, the least size of an eventually different family of
functions in $\omega^{\omega}$.

  2. (2)

$\mathfrak{a}_{p}$, the least size of an eventually different set of
permutations of $\omega$.

  3. (3)

$\mathfrak{a}_{T}$, the least size of an almost disjoint family of finitely
branching trees $T\subseteq\omega^{<\omega}$. This cardinal is equivalent to
the least size of an uncountable partition of any perfect Polish space into
compact sets.

  4. (4)

$\mathfrak{a}_{g}$, the least size of a maximal cofinitary group111Recall that
a subgroup $G\leq S(\omega)$ is cofinitary if every non-identity element has
only finitely many fixed points. It’s not hard to see that any cofinitary
group consists of eventually different permutations..

Here, recall that a set $\mathcal{E}\subseteq\omega^{\omega}$ is eventually
different if for all distinct $f,g\in\mathcal{E}$ there is a $k<\omega$ so
that for all $n>k$ $f(n)\neq g(n)$. Denote this situation by $f\neq^{*}g$.
Dually if $f\neq^{*}g$ fails then we say that $f$ and $g$ are infinitely often
equal, denoted $f=^{\infty}g$. An eventually different family is maximal if it
is not contained in any strictly larger such family. The same can be said in
the case of $\mathfrak{a}_{p}$ with the additional stipulation that every
element is a bijection of $\omega$ with itself. For more examples of relatives
of $\mathfrak{a}$ see [11, 13].

In general the relationship between these cardinals remains murky. It’s known
that $\mathfrak{a}$ can be consistently less than all of them, but it’s not
known if the reverse inequality is consistent for any of them. It remains open
if $\mathfrak{a}_{e}=\mathfrak{a}_{p}$ is provable in $\mathsf{ZFC}$. In this
paper we contribute to the project of separating and understanding these
invariants. We provide several models of
$\mathfrak{a}_{e}=\mathfrak{a}_{p}=\aleph_{1}$ while allowing other related
cardinals to be $\aleph_{2}$ in a controlled way. These results are obtained
using new strengthenings of maximality for eventually different families of
functions and permutations respectively.

A fruitful strengthening of maximality for almost disjoint families in
$[\omega]^{\omega}$ if given by the notion of a tight MAD family. An almost
disjoint family $\mathcal{A}$ is tight if given any countable set
$\\{B_{n}\;|\;n<\omega\\}$ of $\mathcal{I}(\mathcal{A})^{+}$ sets there is a
single $C\in\mathcal{I}(\mathcal{A})$ so that $B_{n}\cap C$ is infinite for
all $n<\omega$. Tight MAD families exist under $\mathfrak{b}=2^{\aleph_{0}}$
(see [10]), are Cohen indestructible ([10, Corollary 3.2]) and, under certain
conditions are preserved by countable support iterations of proper forcing
notions (see [11]).

On the other hand, the preservation of the maximality of maximal eventually
different families under forcing iterations are not that well understood.
Using the parametrized $\diamondsuit$ principles of [17], Kastermans and Zhang
show that $\mathfrak{a}_{g}=\mathfrak{a}_{e}$ in the Miller model (see [14]).
Their construction can be easily modified to establish the analogous result
for maximal families of eventually different functions. A more recent study in
the area is the work of the first author with D. Schrittesser, see [7], where
they establish the existence (in the Constructible Universe $L$) of a co-
analytic Sacks-indestructible maximal eventually different family. Note that
the techniques of [14] and [7] are completely different, and moreover
completely different than the techniques developed in the current article.

The results of the current paper improve and generalize the above, as we
produce a uniform framework for the preservation of maximal families of
eventually different functions, a framework, which applies to a long list of
partial orders, including Sacks forcing, Miller rational perfect tree forcing,
partition forcing, infinitely often equal forcing, Shelah’s poset for
destroying the maximality of a given maximal ideal, as well as their countable
support iterations.

For an eventually different family $\mathcal{E}$, we denote by
$\mathcal{I}_{T}(\mathcal{E})$ the family of all trees
$T\subseteq{\omega^{<\omega}}$ so that there is $t\in T$ and a finite
$X\in[\mathcal{E}]^{<\omega}$ such that $\bigcup T_{t}\subseteq\bigcup X$,
where $T_{t}=\\{s\in T:s\subseteq t\hbox{ or }t\subseteq s\\}$. We refer to
this family, as the tree ideal generated by $\mathcal{E}$ (see Definition
2.1). Then $\mathcal{I}_{T}(\mathcal{E})^{+}$ is defined as the collection of
all trees $T\subseteq\omega^{<\omega}$ with the property that for each $t\in
T$, $\bigcup T_{t}$ is not almost covered by finitely many functions from
$\mathcal{E}$. Finally, we say that an eventually different family
$\mathcal{E}$ is tight, if for every
$\\{T_{n}\\}_{n\in\omega}\subseteq\mathcal{I}_{T}^{+}(\mathcal{E})$ there is a
single $g\in\mathcal{E}$ with the property that $\forall n\in\omega\forall
t\in T_{n}$ there is a branch $g_{n,t}$ of $T_{n}$ such that $t\subseteq
g_{n,t}$ and for infinitely many $m$, $g_{n,t}(m)=g(m)$ (see Definition 2.2).

Note that the notion of tightness for an eventually different family is a
natural strengthening of maximality for such families (see Proposition 2.3) as
well as a strengthening of $\omega$-maximality (see [9]). As shown in
Proposition 5.4, tight eventually different families exist under
$\hbox{{MA}}(\sigma\hbox{-centered})$ and so in particular under CH. In
Section 4, we introduce strong preservation of a tight mad families and show
that the countable support iteration of proper forcing notions, each iterand
of which strongly preserves a tight mad family, also preserve the tightness of
an eventually different family. Equipped with the above techniques, we
establish:

######  Theorem 1.1.

The following inequalities are all consistent and in each case
$\mathfrak{a}_{e}=\mathfrak{a}_{p}=\aleph_{1}$ is witnessed by a tight
eventually different family and a tight eventually different set of
permutations respectively.

  1. (1)

$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}<\mathfrak{d}=\mathfrak{a}_{T}=2^{\aleph_{0}}$

  2. (2)

$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{d}<\mathfrak{a}_{T}=2^{\aleph_{0}}$

  3. (3)

$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{d}=\mathfrak{u}<non(\mathcal{N})=cof(\mathcal{N})=2^{\aleph_{0}}$.

  4. (4)

$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{i}=cof(\mathcal{N})<\mathfrak{u}$.

Moreover, if we work over the constructible universe, we can provide co-
analytic witnesses of cardinality $\aleph_{1}$ to each of
$\mathfrak{a},\mathfrak{a}_{e},\mathfrak{a}_{p},\mathfrak{i},\mathfrak{u}$ in
the above inequalities.

The rest of this paper is organized as follows. In the remaining part of the
introduction we record some preliminaries for later use. In Section 2 we
introduce tight eventually different families, which is of central focus for
the entire paper. In Section 3 we prove that tight eventually different
families are Cohen indestructible and never analytic. Section 4 contain our
preservation results for such families. In Section 5, we observe that the
discussion of tight eventually different families applies mutatis mutandis to
its generalization for eventually different sets of permutations. Next we turn
to applications. In Sections 6 through 9 we prove that several well known
forcing notions strongly preserve the tightness of eventually different
families of functions and permutations, and conclude the consistency of the
inequalities from the above Theorem. In Section 10 we study the definability
properties of tight families of functions and provide co-analytic witnesses
for the results described in 1.1. We conclude the paper with open questions
for future research.

###  1.1. Preliminaries

Our first preliminary involves recalling the ideal associated to an almost
disjoint family. Recall that given an almost disjoint family $\mathcal{A}$ the
ideal associated to $\mathcal{A}$ is the set $\mathcal{I}(\mathcal{A})$
consisting of all $B\in[\omega]^{\omega}$ for which there is a finite set
$\\{A_{0},...,A_{n-1}\\}\subseteq\mathcal{A}$ so that
$B\subseteq^{*}\bigcup_{k<n}A_{k}$. This ideal is important when considering
how MAD families persist (or not) to forcing extensions. Often this takes the
following form. If ${\mathord{\mathbb{P}}}$ is some forcing notion and
$\dot{X}$ is a ${\mathord{\mathbb{P}}}$-name for an element of
$\mathcal{I}(\mathcal{A})^{+}$ then its outer hull, the set of all $m$ for
which some condition forces $\check{m}\in\dot{X}$ is an element of
$\mathcal{I}(\mathcal{A})^{+}$ in $V$. Many arguments involving MAD families
in forcing extensions consider this set at key points. One of the main
technical observations of this paper involves the introduction of a similar
ideal associated to an eventually different family. However, the naive
generalization, thinking of an eventually different family as collection of
subsets of $P(\omega^{2})$ does not work and instead we consider an ideal
generated by the eventually different family consisting of trees
$T\subseteq\omega^{<\omega}$. This is described in detail in Section 2, see
also Remark 1.

We conclude this introduction by recording some of what was known already
concerning $\mathfrak{a}$, $\mathfrak{a}_{e}$, $\mathfrak{a}_{p}$,
$\mathfrak{a}_{T}$, $\mathfrak{b}$ and $\mathfrak{d}$. To start we recall the
known provable inequalities that will be used throughout the paper.

######  Fact 1.2.

The following inequalities are provable in $\mathsf{ZFC}$.

  1. (1)

$non(\mathcal{M})\leq\mathfrak{a}_{e}$

  2. (2)

([5, Theorem 2.2] $non(\mathcal{M})\leq\mathfrak{a}_{p}$

  3. (3)

([2, Proposition 8.4]) $\mathfrak{b}\leq\mathfrak{a}$

  4. (4)

([21, Theorem 2.5]) $\mathfrak{d}\leq\mathfrak{a}_{T}$

###### Proof.

All of these have been proved in the literature (see the citations) with the
exception of $non(\mathcal{M})\leq\mathfrak{a}_{e}$. This inequality however
follows from the well known fact that $non(\mathcal{M})$ is equal to the
bounding number for $\neq^{*}$, i.e. $non(\mathcal{M})$ is the least size of a
set of reals $A\subseteq\omega^{\omega}$ for which there is no $f\in A$ which
is eventually different from all $g\in A$, see [1, Theorem 2.4.7]. Indeed,
given this, suppose $A\subseteq\omega^{\omega}$ is eventually different and
$|A|<non(\mathcal{M})$. Then, there must be a real $f\in\omega^{\omega}$
eventually different from every element of $A$, so $A$ is not maximal. ∎

Note that since $\mathfrak{b}\leq non(\mathcal{M}),\mathfrak{d}$ the bounding
number is actually a lower bound on all the relatives of $\mathfrak{a}$ (see
[3] for a discussion of this). It’s also known that both $non(\mathcal{M})$
and $\mathfrak{d}$ are independent of $\mathfrak{a}$, and that
$non(\mathcal{M})$ and $\mathfrak{a}_{T}$ are independent (see below).
Moreover $\mathfrak{d}$ and $\mathfrak{a}_{e}/\mathfrak{a}_{p}$ are
independent. In the case of $\mathfrak{a}_{p}$ this was known, though perhaps
never written down. Namely, $non(\mathcal{M})=2^{\aleph_{0}}$ in the random
model, and therefore $\mathfrak{a}_{e}=\mathfrak{a}_{p}=2^{\aleph_{0}}$ while
$\mathfrak{d}=\aleph_{1}$. On the other hand Kastermans and Zhang proved in
[14] that $\mathfrak{a}_{p}=\aleph_{1}$ in the Miller model, where it’s well
known that $\mathfrak{d}=\aleph_{2}$. Their proof uses the parametrized
diamonds of [17], and is completely different than the proof we give of this
result in Section 5 below. Though it’s not stated in their paper, Kastermans
and Zhang’s proof that $\mathfrak{a}_{p}=\aleph_{1}$ in the Miller model can
also be easily augmented to show that $\mathfrak{a}_{e}=\aleph_{1}$ in the
Miller model as well. In Section 5 we give a different proof that
$\mathfrak{a}_{e}=\aleph_{1}$ in the Miller model as well hence giving a new
proof of the independence of $\mathfrak{d}$ and $\mathfrak{a}_{e}$.

We finish this section by noting what is known about consistent inequalities
between $\mathfrak{a}_{e}$, $\mathfrak{a}_{p}$, $\mathfrak{a}_{T}$ and
$\mathfrak{d}$.

######  Fact 1.3.

The inequality
$\mathfrak{a}_{T}=\mathfrak{d}<\mathfrak{a}_{e}=\mathfrak{a}_{p}$ holds in the
random model, in particular it is consistent.

###### Proof.

That $\mathfrak{d}=\aleph_{1}$ holds in the random model is well known, see,
for example [1, Model 7.6.8]. Kunen and, independently, Stern showed that
$\mathfrak{a}_{T}=\aleph_{1}$ in the random model, see [15, Theorem 5].
Meanwhile since $non(\mathcal{M})\leq\mathfrak{a}_{e},\mathfrak{a}_{p}$ and
$non(\mathcal{M})=2^{\aleph_{0}}$ in the random model, we have that
$\mathfrak{a}_{e}=\mathfrak{a}_{p}=2^{\aleph_{0}}$. ∎

It is also not hard to obtain the consistency of
$\mathfrak{d}<\mathfrak{a}_{T}=\mathfrak{a}_{e}=\mathfrak{a}_{p}$.

######  Proposition 1.4.

It is consistent that
$\mathfrak{d}=\aleph_{1}<\mathfrak{a}_{T}=\mathfrak{a}_{e}=\mathfrak{a}_{p}=\aleph_{2}$.

To describe this model we recall the partition forcing introduced in [15] (see
also [6] and [11, Section 1.6]). This forcing will be discussed in more length
in Section 7. Given an uncountable partition of size $\aleph_{1}$ of
$2^{\omega}$ into closed sets
$\mathcal{K}=\\{C_{\alpha}\;|\;\alpha<\omega_{1}\\}$ let
$\mathbb{P}(\mathcal{K})$ be the set of all perfect trees $p$ so that for all
$\alpha$ $[p]\cap C_{\alpha}$ is nowhere dense in $[p]$. The order is
inclusion. This forcing was first investigated by Miller who showed that it is
proper, has the Laver property and adds a real not in the evaluation of any
$C_{\alpha}$ in the extension, thus killing the fact that $\mathcal{K}$ is a
partition of $2^{\omega}$. It follows that, via an appropriate bookkeeping
device, an $\omega_{2}$-length countable support iteration of forcing notions
of the form ${\mathord{\mathbb{P}}}(\mathcal{K})$ will force
$\mathfrak{a}_{T}=\aleph_{2}$. Later Spinas, [21, Lemma 2.7] proved that this
forcing notion is $\omega^{\omega}$-bounding. Black-boxing these facts the
proof of Proposition 1.4 is as follows.

###### Proof.

Assume $\mathsf{CH}$ in the ground model. Define a countable support iteration
$\langle{\mathord{\mathbb{P}}}_{\alpha},\dot{{\mathord{\mathbb{Q}}}}_{\alpha}\;|\;\alpha<\omega_{2}\rangle$
so that for even $\alpha$
$\Vdash_{\alpha}$“$\dot{{\mathord{\mathbb{Q}}}}_{\alpha}$ is random forcing”
and, using some appropriate bookkeeping device, for odd $\alpha$
$\Vdash_{\alpha}$ “$\dot{{\mathord{\mathbb{Q}}}}_{\alpha}$ is partition
forcing for some uncountable partition of $2^{\omega}$”. In the resulting
model $non(\mathcal{M})=2^{\aleph_{0}}$ because of the random reals added and
hence $\mathfrak{a}_{e}=\mathfrak{a}_{p}=2^{\aleph_{0}}$. Similarly
$\mathfrak{a}_{T}=2^{\aleph_{0}}$ because of the partition forcing iterands.
Finally note that since all iterands are $\omega^{\omega}$-bounding the entire
iteration is $\omega^{\omega}$-bounding and so $\mathfrak{d}=\aleph_{1}$. ∎

Collectively, Fact 1.3, Proposition 1.4 and Theorems 6.3 and 7.5 show no
provable inequalities exist between $\mathfrak{a}_{e}/\mathfrak{a}_{p}$,
$\mathfrak{d}$ and $\mathfrak{a}_{T}$ aside from
$\mathfrak{d}\leq\mathfrak{a}_{T}$. This is stated more precisely as Corollary
7.6 below.

##  2\. Tight Eventually Different Families

In this section we introduce a strong version of maximality for eventually
different families which can be preserved by countable support iterations of
proper forcing notions. Towards this we introduce some new terminology.
Throughout this section fix an eventually different family $\mathcal{E}$.
Recall that a tree on $\omega$ is a set $T\subseteq\omega^{<\omega}$ which is
closed downwards in the sense that if $s$ is an initial segment of $t\in T$
then $s\in T$. Throughout we will only restrict our attention to pruned trees
i.e. trees $T$ in which every finite sequence $t\in T$ has a proper extension.
Unless stated otherwise every tree in this article is assumed to be pruned.
Note that if $X$ is a set of functions then $\bigcup X\subseteq\omega^{2}$ and
similarly if $T\subseteq\omega^{<\omega}$ is a tree then $\bigcup
T\subseteq\omega^{2}$. Therefore it makes sense to talk about a set of
functions covering a tree, namely we say that a set of functions $X$ covers a
tree $T$ if $\bigcup T\subseteq\bigcup X$ and $X$ almost covers $T$ if
$\bigcup T\subseteq^{*}\bigcup X$. Given a tree $T$ and a node $t\in T$ let
$T_{t}=\\{s\in T\;|\;s\subseteq t\;{\rm or}\;t\subseteq s\\}$.

######  Definition 2.1.

[The Tree Ideal generated by $\mathcal{E}$] The tree ideal generated by
$\mathcal{E}$, denoted $\mathcal{I}_{T}(\mathcal{E})$, is the set of all trees
$T\subseteq\omega^{<\omega}$ so that there is a $t\in T$ and a finite set
$X\in[\mathcal{E}]^{<\omega}$ so that $\bigcup T_{t}\subseteq^{*}\bigcup X$.

Dually a tree $T\subseteq\omega^{<\omega}$ is in
$\mathcal{I}_{T}(\mathcal{E})^{+}$ if for each $t\in T$ it’s not the case that
$\bigcup T_{t}$ can be almost covered by finitely many functions from
$\mathcal{E}$.

######  Remark 1.

The use of the word “ideal” here is somewhat misleading as
$\mathcal{I}_{T}(\mathcal{E})$ is not an idea, nor does it generate one. In
fact it is not closed under unions. The terminology is meant to draw the
analogy with the ideal generated by a MAD family, for which
$\mathcal{I}_{T}(\mathcal{E})$ serves a similar purpose.

If we have a tree $T\subseteq\omega^{<\omega}$ and a real
$g\in\omega^{\omega}$ we say that $g$ densely diagonalizes $T$ if for each
$t\in T$ there is an $s\supsetneq t$ so that $s\in T$ and there is a $k\in{\rm
dom}(s)\setminus{\rm dom}(t)$ so that $s(k)=g(k)$. In other words, each node
of $T$ sits on a branch which is infinitely often equal to $g$.

The main definition in this paper is the following.

######  Definition 2.2.

An eventually different family $\mathcal{E}$ is tight if given any sequence of
countably many trees $\\{T_{n}\;|\;n<\omega\\}$ so that
$T_{n}\in\mathcal{I}_{T}(\mathcal{E})^{+}$ for all $n<\omega$ there is a
single $g\in\mathcal{E}$ which densely diagonalizes all the $T_{n}$’s.

The point is that this strengthens maximality.

######  Proposition 2.3.

If $\mathcal{E}$ is tight then it is maximal.

###### Proof.

Suppose $\mathcal{E}$ is tight but not maximal and let $h\notin\mathcal{E}$ be
eventually different from every element of $\mathcal{E}$. Consider now the
tree $T_{h}=\\{h\upharpoonright n\;|\;n<\omega\\}$. This tree is in
$\mathcal{I}_{T}(\mathcal{E})^{+}$ since if we could cover it by finitely many
functions from $\mathcal{E}$ then there would be an $f\in\mathcal{E}$ so that
for infinitely many $n$ $f(n)=(h\upharpoonright n+1)(n)$ i.e. $f=^{\infty}h$.
But then by tightness there is a $g\in\mathcal{E}$ which densely diagonalizes
$T_{h}$. But this just means that $g=^{\infty}h$, contradiction. ∎

Later we will see that in $\mathsf{ZFC}$, there are maximal eventually
different families that are not tight, see Theorem 3.3 below.

######  Remark 2.

The intuition behind using trees, as opposed to simply functions is as
follows. Often in preservation arguments involving maximal sets in
$[\omega]^{\omega}$ (MAD families, maximal independent families etc) one is
given a forcing notion ${\mathord{\mathbb{P}}}$, a condition
$p\in{\mathord{\mathbb{P}}}$ and a ${\mathord{\mathbb{P}}}$-name $\dot{X}$ so
that $p\Vdash\dot{X}\in[\omega]^{\omega}\check{}$ and needs to “reflect”
$\dot{X}$ to the ground model. This is usually done using the outer hull of
$\dot{X}$ with respect to $p$, namely the set
$\dot{X}_{p}:=\\{m\;|\;p\nVdash\check{m}\notin\dot{X}\\}$. The issue in trying
to import this idea to a space of functions is that the outer hull of a
function (viewed as a subset of $\omega^{2}$) is no longer necessarily a
function. However, if we have a name $\dot{f}$ for an element of
$\omega^{\omega}$ and consider instead the set of
$\\{t\in\omega^{<\omega}\;|\;p\nVdash\check{t}\nsubseteq\dot{f}\\}$ then this
set forms a subtree of $\omega^{<\omega}$. It is for this reason that we need
to work with trees. This idea is central to the argument used to prove Theorem
4.2 below as well as in the applications given in Sections 3 and 6 - 9.

We need to show that this definition is consistent, i.e. it is consistent that
there are tight eventually different families. In fact, much more may be true.

######  Theorem 2.4.

Assume $\mathsf{MA}(\sigma{\rm-centered})$. Every eventually different family
$\mathcal{E}_{0}$ of size ${<}2^{\aleph_{0}}$ is contained in a tight
eventually different family. In particular $\mathsf{CH}$ implies that tight
eventually different families exist.

The theorem makes use of a forcing notion we introduce now. Let $\mathcal{E}$
be an eventually different family (not necessarily maximal). Define the
forcing notion ${\mathord{\mathbb{P}}}_{\mathcal{E}}$ to be the set of all
pairs $(s,E)$ so that the following hold.

  1. (1)

$s\in\omega^{<\omega}$

  2. (2)

$E\in[\mathcal{E}]^{<\omega}$

The order on ${\mathord{\mathbb{P}}}_{\mathcal{E}}$ is defined as follows. We
let $(s_{1},E_{1})\leq(s_{0},E_{0})$ if and only if:

  1. (1)

$s_{1}\supseteq s_{0}$ and $E_{1}\supseteq E_{0}$.

  2. (2)

If $k\in{\rm dom}(s_{1})\setminus{\rm dom}(s_{0})$ then $s_{1}(k)\neq f(k)$
for each $f\in E_{0}$.

This forcing notion is the same as eventually different forcing $\mathbb{E}$
from [1], but with the second coordinate restricted to $\mathcal{E}$. It is
clear that this forcing notion is $\sigma$-centered since any two conditions
$(s,E)$ and $(s,F)$ with the same first coordinate are compatible and in fact
strengthened by $(s,E\cup F)$.

We will show that ${\mathord{\mathbb{P}}}_{\mathcal{E}}$ adds a real
eventually different from every element of $\mathcal{E}$ and which densely
diagonalizes each $T\in\mathcal{I}_{T}(\mathcal{E})^{+}$ in the ground model.
This follows from the following lemma.

######  Lemma 2.5.

The following sets are dense in
${\mathord{\mathbb{P}}}_{\mathcal{E},\vec{T}}$.

  1. (1)

For each $k<\omega$ the set of conditions $(s,E)$ so that $k\in{\rm dom}(s)$.

  2. (2)

For each $T\in\mathcal{I}_{T}(\mathcal{E})^{+}$, and $t\in T$ the set of
conditions $(s,E)$ so that there is a $t^{\prime}\in T$ which extends $t$ and
a $k\in{\rm dom}(t^{\prime})\setminus{\rm dom}(t)$ so that
$s(k)=t^{\prime}(k)$.

  3. (3)

For each $f\in\mathcal{E}$ the set of conditions $(s,E)$ so that $f\in E$.

Note that condition 1 ensures the generic is a total function on $\omega$,
condition 2 ensures the function densely diagonalizes all the trees from
$\mathcal{I}_{T}(\mathcal{E})^{+}$ in the ground model and condition 3
guarantees that the generic function is eventually different from every
element of $\mathcal{E}$.

###### Proof.

Let $(s,E)$ be a condition. We will kill three birds with one stone and show
that there is a stronger condition in all three dense sets at once. First, by
the explanation of $\sigma$-centeredness given above,
$(s,E\cup\\{f\\})\leq(s,E)$ and is in the last dense set. Now, Fix
$T\in\mathcal{I}_{T}(\mathcal{E})^{+}$ and $t\in T$. By assumption we know
that $T_{t}$ cannot be almost covered by $\bigcup E\cup\\{f\\}$. It follows
that there is a node $t^{\prime}\supsetneq t$ in $T$ and an $l_{t}\in{\rm
dom}(t^{\prime})\setminus{\rm dom}(t)$ so that $t^{\prime}(l_{t})\neq
f^{\prime}(l_{t})$ for all $f^{\prime}\in E\cup\\{f\\}$. Let
$s^{\prime}=s\cup\\{(l_{t},t^{\prime}(l_{t}))\\}$. Now, fix $k\in\omega$. If
$k\notin{\rm dom}(s^{\prime})$ then let
$s^{\prime\prime}=s^{\prime}\cup\\{(k,l_{k})\\}$ where $l_{k}$ is the minimal
element not in the (finite) set $\\{f(k)\;|\;f\in E\\}$. Finally we get that
$(s^{\prime\prime},E\cup\\{f\\})\leq(s,E)$, which is what we needed to show. ∎

Given this we can now prove Theorem 2.4.

###### Proof of Theorem 2.4.

Assume $\mathsf{MA}(\sigma{\rm-centered})$. Enumerate all $\omega$-sequences
of subtrees of $\omega^{<\omega}$ as
$\\{\vec{T}_{\alpha}\;|\;\alpha<2^{\aleph_{0}}\\}$. For each
$\alpha<2^{\aleph_{0}}$ and $n<\omega$ denote by $T_{\alpha}(n)$ the $n^{\rm
th}$ tree in $\vec{T}_{\alpha}$. Fix an eventually different family
$\mathcal{E}_{0}$ of size ${<}2^{\aleph_{0}}$. We will recursively build a
continuous $\subseteq$-increasing sequence of eventually different families
$\\{\mathcal{E}_{\alpha}\;|\;\alpha<2^{\aleph_{0}}\\}$ so that for each
$\alpha<2^{\aleph_{0}}$ $\mathcal{E}_{\alpha}$ has cardinality
${<}2^{\aleph_{0}}$ and if
$\vec{T}_{\alpha}\in\mathcal{I}_{T}(\mathcal{E}_{\alpha})^{+}$ then there is a
real in $\mathcal{E}_{\alpha+1}$ densely diagonalizing $T_{\alpha}(n)$ for
each $n<\omega$. If such a sequence can be constructed then clearly
$\mathcal{E}=\bigcup_{\alpha<2^{\aleph_{0}}}\mathcal{E}_{\alpha}$ will be the
desired tight eventually different family.

We already have $\mathcal{E}_{0}$. Now suppose that we have constructed
$\mathcal{E}_{\alpha}$. If
$\vec{T}_{\alpha}\nsubseteq\mathcal{I}_{T}(\mathcal{E}_{\alpha})^{+}$ then let
$g_{\alpha}$ be any function eventually different from every element of
$\mathcal{E}_{\alpha}$. That such an element exists is guaranteed by
$\mathsf{MA}(\sigma{\rm-centered})$ (using eventually different forcing). If
$\vec{T}_{\alpha}\subseteq\mathcal{I}_{T}(\mathcal{E}_{\alpha})^{+}$ then, by
applying $\mathsf{MA}(\sigma{\rm-centered})$ to
${\mathord{\mathbb{P}}}_{\mathcal{E}_{\alpha}}$ and noting that we only need
to meet $|\mathcal{E}_{\alpha}|+\aleph_{0}$ dense sets, find a $g_{\alpha}$
eventually different from each element of $\mathcal{E}_{\alpha}$ and densely
diagonalizing every $T_{\alpha}(n)$ for $n<\omega$. In either case let
$\mathcal{E}_{\alpha+1}=\mathcal{E}_{\alpha}\cup\\{g_{\alpha}\\}$. Either way
it’s clear that we have fulfilled the requisite conditions. ∎

Of course a natural question is whether tight eventually different families
exist in $\mathsf{ZFC}$. We do not know the answer to this. However the
analogous question for tight MAD families, namely if there is a tight MAD
family in $\mathsf{ZFC}$, remains one of the most stubbornly open problems in
this area so it’s reasonable to expect that finding a $\mathsf{ZFC}$ example
of a tight eventually different family, or showing one does not exist may be
difficult as well.

######  Question 1.

Is it consistent with $\mathsf{ZFC}$ that there are no tight maximal
eventually different families?

##  3\. Cohen Indestructibility and Definability

Let us investigate the analogy between tight eventually different families and
tight MAD families further. As mentioned in the introduction, in [10,
Corollary 3.2] it is shown that tight MAD families are Cohen indestructible.
Here, recall that if ${\mathord{\mathbb{P}}}$ is a forcing notion and
$\mathcal{A}$ is MAD then it is said to
${\mathord{\mathbb{P}}}$-indestructible if $\Vdash_{\mathord{\mathbb{P}}}$
“$\check{\mathcal{A}}$ is MAD”. Here, we show that the same holds true of
tight eventually different families. We then use this to show that there are
no analytic tight eventually different families and, under large cardinals, no
definable such families in a strong sense.

######  Theorem 3.1.

Suppose $\kappa$ is a cardinal and $\mathbb{C}_{\kappa}=add(\omega,\kappa)$ is
the forcing to add $\kappa$-many Cohen reals. If $\mathcal{E}$ is tight then
$\Vdash_{\mathbb{C}_{\kappa}}$ “$\check{\mathcal{E}}$ is tight”.

###### Proof.

Since every new real in the Cohen extension is added by a single Cohen real,
it suffices to prove the theorem in the case $\kappa=1$. Denote
$\mathbb{C}_{1}$ by $\mathbb{C}$. Let $\mathcal{E}$ be tight and let
$\\{\dot{T}_{n}\;|\;n<\omega\\}$ be a countable set of $\mathbb{C}$-names for
subtrees of $\omega^{<\omega}$. We will show that for each $p\in\mathbb{C}$
either there is a $q\leq p$, an $n<\omega$ so that
$q\Vdash\dot{T}_{n}\in\mathcal{I}_{T}(\mathcal{E})\check{}$ or there is a
$g\in\mathcal{E}$ so that $p\Vdash$ “For each $n<\omega$, $\check{g}$ densely
diagonalizes $\dot{T}_{n}$”. Clearly the theorem will follow from this.

Fix $p\in\mathbb{C}$ and enumerate the set of conditions below $p$ as
$\\{p_{j}\;|\;j<\omega\\}$. For each $n,j<\omega$ let
$T_{n,j}=\\{t\in\omega^{<\omega}\;|\;\exists q\leq
p_{j}\,q\Vdash\check{t}\in\dot{T}_{n}\\}$. For each $n,j<\omega$ the set
$T_{n,j}$ is a tree since if some $t\in T_{n,j}$ as witnessed by some $q$ then
the same $q$ witnesses that $s\in T_{n,j}$ for each $s\subseteq t$. There are
two cases.

Case 1: There are $n,j<\omega$ and $t\in T_{n,j}$ so that $(T_{n,j})_{t}$ is
almost covered by finitely many functions from $\mathcal{E}$, i.e.
$T_{n,j}\in\mathcal{I}_{T}(\mathcal{E})$. Fix $q\leq p_{n}$ so that $q$
witnesses that $t\in T_{n}$ and suppose that $f_{0},...,f_{n-1}\in\mathcal{E}$
are such that $(T_{n})_{t}\subseteq^{*}\bigcup\\{f_{0},...,f_{n-1}\\}$.
Unwinding the definition of $T_{n,j}$ we get that if $q=p_{m}$ then in fact
$T_{n,m}\subseteq^{*}\bigcup\\{f_{0},...,f_{n-1}\\}$. Suppose that $k>\omega$
is large enough that for all $l>k$, if $s\in T_{n,m}$ and $l\in{\rm dom}(s)$
then $s(l)\in\\{f_{0}(l),...,f_{n-1}(l)\\}$. This means that if $r\leq p_{m}$
forces some $s\in\dot{T}_{n}$ of length at least $k$ then for all $l>k$ we
have
$r\Vdash\check{s}(\check{l})\in\\{\check{f}_{0}(\check{l}),...,\check{f}_{n-1}(\check{l})\\}$.
In otherwords, $p_{m}$ must force that $\dot{T}_{n}$ is in
$\mathcal{I}_{T}(\mathcal{E})$.

Case 2: Case 1 fails. In other words this means that each $T_{n,j}$ is not in
$\mathcal{I}_{T}(\mathcal{E})$. By the assumption that $\mathcal{E}$ is tight,
this means that there is a single $g\in\mathcal{E}$ which densely diagonalizes
all of the $T_{n,j}$’s. We claim that $p\Vdash$ “For all $n<\omega$ $g$
densely diagonalizes $\dot{T}_{n}$”. To see this, fix $n<\omega$ and let
$q=p_{m}<p$ for some $m$. Suppose $\dot{t}$ is a name for an element of
$\dot{T}_{n}$. By strengthening $q$ if necessary we can assume that $t$ is
decided by $q$ to be some $t\in T_{n,m}$. Moreover, since $g$ densely
diagonalizes $T_{n,m}$ there is an $s\supseteq t$ which is in $T_{n,m}$ as
witnessed by some $r\leq q$ and an $l\in{\rm dom}(s)\setminus{\rm dom}(t)$ and
$s(l)=g(l)$. It follows that $r$ forces there to be a node of $\dot{T}_{n}$
strengthening $\dot{t}$ and agreeing with $g$ above the domain of $\dot{t}$.
Since $q$ was arbitrary it follows that $p$ actually forces this for each
$\dot{t}$ and $n$ and hence $p$ forces $g$ to densely diagonalize each
$\dot{T}_{n}$ as needed. ∎

It follows from this result and Theorem 10.1 below that there is a coanalytic
Cohen indestructible maximal eventually different family. This was known, [8],
but only by a more complicated construction. We also have the following
result.

######  Corollary 3.2.

In the Cohen model there is a tight eventually different family of size
$\aleph_{1}$.

We can use this theorem to show that not every maximal eventually different
family is tight. In fact something stronger is true.

######  Theorem 3.3.

If $\mathcal{E}$ is an analytic maximal eventually different family and
${\mathord{\mathbb{P}}}$ is any forcing notion adding a real then
$\Vdash_{\mathord{\mathbb{P}}}$ “$\check{\mathcal{E}}$ is no longer maximal”.
In particular, tight eventually different families are never analytic.

Since there are Borel maximal eventually different families, see [12], these
ones are not tight.

###### Proof of Theorem 3.3.

Let $\mathcal{A}$ be an analytic, maximal eventually different family. Note
that $\mathcal{A}$ is uncountable and therefore, by the perfect set property
for analytic sets, it contains a perfect set $P\subseteq\mathcal{A}$.
Moreover, since every perfect set contains a copy of Cantor space, we can find
a continuous injection $f:2^{\omega}\to\mathcal{A}$. Fix such an $f$.

Observe that “$\mathcal{A}$ is an eventually different family” is
$\Pi^{1}_{1}$: namely we have $\forall x,y[x\notin\mathcal{A}\lor
y\notin\mathcal{A}\lor x\neq^{*}y]$. Since $\mathcal{A}$ is analytic, and
hence its complement is co-analytic the observation follows. In particular,
the interpretation of $\mathcal{A}$ remains an eventually different family in
any forcing extension. Now let ${\mathord{\mathbb{P}}}$ be a forcing notion
adding a real, $x$. Working in $V^{\mathord{\mathbb{P}}}$ we consider the real
$f(x)\in\mathcal{A}^{V^{\mathord{\mathbb{P}}}}$. This real is eventually
different from every element of $\mathcal{A}\cap V$ and hence $\mathcal{A}\cap
V$ is no longer maximal. ∎

Observe the theorem only used the perfect set property for $\mathcal{A}$
alongside some generic absoluteness. As a result the same proof gives the
following.

######  Theorem 3.4.

  1. (1)

If $E$ is a co-analytic maximal eventually different family which is
indestructible with respect to some forcing adding a real then $E$ does not
contain a perfect set.

  2. (2)

In the presence of sufficient large cardinals there are no
${\mathord{\mathbb{P}}}$-indestructible maximal eventually different families
in $L(\mathbb{R})$ for any forcing notion ${\mathord{\mathbb{P}}}$ adding a
real.

Note also that all of these results apply equally in the case of cofinitary
groups and maximal eventually different sets of permutations. In particular,
there is no analytic, maximal cofinitary group which is indestructible for any
forcing notion adding a real.

##  4\. Strong Preservation of Tightness

The purpose of this section is to prove a preservation theorem for tight,
eventually different families, akin to [11, Corollary 32] which showed the
same for tight MAD families. The preservation theorem we prove concerns the
notion of strong preservation.

######  Definition 4.1.

Let ${\mathord{\mathbb{P}}}$ be a proper forcing notion and $\mathcal{E}$ a
tight eventually different family. We say that ${\mathord{\mathbb{P}}}$
strongly preserves the tightness of $\mathcal{E}$ if for every sufficiently
large $\theta$, every condition $p$ and every $M\prec H_{\theta}$ countable
with $p,{\mathord{\mathbb{P}}},\mathcal{E}\in M$, if $g\in\mathcal{E}$
strongly diagonalizes every element of $M\cap\mathcal{I}_{T}(\mathcal{E})^{+}$
then there is an $(M,{\mathord{\mathbb{P}}})$-generic $q\leq p$ so that $q$
forces that $g$ densely diagonalizes every element of
$M[\dot{G}]\cap\mathcal{I}_{T}(\mathcal{E})^{+}$. Such a $q$ is called an
$(M,{\mathord{\mathbb{P}}},\mathcal{E},g)$-generic condition.

Clearly if ${\mathord{\mathbb{P}}}$ strongly preserves the tightness of
$\mathcal{E}$ then in particular $\Vdash_{\mathord{\mathbb{P}}}$“
$\check{\mathcal{E}}$ is tight”. The point of this section is to prove the
following.

######  Theorem 4.2.

Suppose that $\mathcal{E}$ is a tight eventually different family. If
$\langle{\mathord{\mathbb{P}}}_{\alpha},\dot{\mathbb{Q}}_{\alpha}\;|\;\alpha<\gamma\rangle$
is a countable support iteration of proper forcing notions so that for all
$\alpha$ we have $\Vdash_{\alpha}$ “$\dot{\mathbb{Q}}_{\alpha}$ strongly
preserves the tightness of $\check{\mathcal{E}}$” then
${\mathord{\mathbb{P}}}_{\gamma}$ strongly preserves the tightness of
$\mathcal{E}$.

This theorem follows immediately from the following two lemmas. From now on
fix a tight eventually different family $\mathcal{E}$.

######  Lemma 4.3.

Suppose ${\mathord{\mathbb{P}}}$ strongly preserves the tightness of
$\mathcal{E}$ and $\dot{\mathbb{Q}}$ is a ${\mathord{\mathbb{P}}}$-name for a
poset which strongly preserves the tightness of $\mathcal{E}$. Then
${\mathord{\mathbb{P}}}*\dot{\mathbb{Q}}$ strongly preserves the tightness of
$\mathcal{E}$. Moreover if $p$ is
$(M,{\mathord{\mathbb{P}}},\mathcal{E},g)$-generic and forces $\dot{q}$ to be
$(M[\dot{G}],{\mathord{\mathbb{P}}},\mathcal{E},g)$ generic then $(p,\dot{q})$
is $(M,{\mathord{\mathbb{P}}},\mathcal{E},g)$-generic.

###### Proof.

Suppose $p$ is $(M,{\mathord{\mathbb{P}}},\mathcal{E},g)$-generic and forces
$\dot{q}$ to be $(M[\dot{G}],{\mathord{\mathbb{P}}},\mathcal{E},g)$-generic.
Then obviously $(p,\dot{q})$ is $(M,{\mathord{\mathbb{P}}})$-generic.
Moreover, by definition it forces that for every ${\mathord{\mathbb{P}}}$-name
for a $\dot{\mathbb{Q}}$-name for an element of
$\mathcal{I}_{T}(\mathcal{E})^{+}$ in $M$ $g$ densely diagonalizes it. But
therefore $(p,\dot{q})$ forces that for every
${\mathord{\mathbb{P}}}*\dot{\mathbb{Q}}$-name for an element of
$\mathcal{I}_{T}(\mathcal{E})^{+}$ in $M$ is densely diagonalized by $g$ as
needed. ∎

######  Lemma 4.4.

Let
$\langle{\mathord{\mathbb{P}}}_{\alpha},\dot{\mathbb{Q}}_{\alpha}\;|\;\alpha<\gamma\rangle$
be a countable support iteration of proper forcing notions so that for all
$\alpha$ we have $\Vdash_{\alpha}$ “$\dot{\mathbb{Q}}_{\alpha}$ strongly
preserves the tightness of $\check{\mathcal{E}}$”, let $\theta$ be
sufficiently large, $M\prec H_{\theta}$ countable containing
${\mathord{\mathbb{P}}}_{\gamma},\gamma,\mathcal{E}$. For each $\alpha\in
M\cap\gamma$ and every
$(M,{\mathord{\mathbb{P}}}_{\alpha},\mathcal{E},g)$-generic condition
$p\in{\mathord{\mathbb{P}}}_{\alpha}$ the following holds:

If $\dot{q}$ is a ${\mathord{\mathbb{P}}}_{\alpha}$-name
$p\Vdash_{\alpha}\dot{q}\in{\mathord{\mathbb{P}}}_{\gamma}\cap M$ and
$p\Vdash_{\alpha}\dot{q}\upharpoonright\alpha\in\dot{G}_{\alpha}$ then there
is an $(M,{\mathord{\mathbb{P}}}_{\gamma},\mathcal{E},g)$-generic condition
$\bar{p}\in{\mathord{\mathbb{P}}}_{\gamma}$ so that
$\bar{p}\upharpoonright\alpha=p$ and
$\bar{p}\Vdash_{\gamma}\dot{q}\in\dot{G}$.

Before proving this lemma, let us fix ahead of time a convention regarding the
enumeration of trees $T\subseteq\omega^{<\omega}$. First fix a computable
enumeration of $\omega^{<\omega}$ so that shorter sequences appear first. Now,
given a tree $T\subseteq\omega^{<\omega}$ we can push that enumeration forward
onto $T$ in the sense that the $i^{\rm th}$ node of $T$ is the $i^{\rm th}$
node in $T$ relative to the computable enumeration of $\omega^{<\omega}$ we
fixed. In other words if our computable enumeration of $\omega^{<\omega}$ is,
say $\\{t_{k}\;|\;k<\omega\\}$ then the $0^{\rm th}$ node of $T$ is $t_{i}$ so
that $i$ is least with $t_{i}\in T$ and so on. In this way we can refer
unambiguously to an enumeration of the nodes of a tree in a forcing extension.
This convention will also be used in the remaining sections of the paper as
well.

###### Proof.

The proof is by induction on $\gamma$. The case where $\gamma$ is a successor
ordinal follows from Lemma 4.3 so we focus on the limit case. Fix $\alpha,p,M$
etc as in the statement of the lemma and let $\\{\alpha_{n}\;|\;n<\omega\\}$
be a strictly increasing sequence in $M\cap\gamma$ with supremum $\gamma$ so
that $\alpha_{0}=\alpha$. Fix a bijection $\varphi:\omega\to\omega^{2}$ with
coordinate functions $\varphi_{0}$ and $\varphi_{1}$. Enumerate the dense open
subsets of $\mathbb{P}_{\gamma}$ in $M$ as $\\{D_{n}\;|\;n<\omega\\}$.
Enumerate the ${\mathord{\mathbb{P}}}_{\gamma}$-names for
$\mathcal{I}_{T}(\mathcal{E})^{+}$ trees in $M$ as
$\\{\dot{T}_{n}\;|\;n<\omega\\}$. We will recursively define sequences
$\\{p_{n}\;|\;n<\omega\\}$, $\\{\dot{q}_{n}\;|\;n<\omega\\}$,
$\\{\dot{t}_{n}\;|\;n<\omega\\}$ and $\\{\dot{k}\;|\;k<\omega\\}$ as follows.

  1. (1)

$p_{0}=p$ and $\dot{q}_{0}=\dot{q}$

  2. (2)

$p_{n}$ is an $(M,{\mathord{\mathbb{P}}}_{\alpha_{n}},\mathcal{E},g)$-generic
condition

  3. (3)

$p_{n+1}\upharpoonright\alpha_{n}=p_{n}$

  4. (4)

$\dot{q}_{n}$ is a ${\mathord{\mathbb{P}}}_{\alpha_{n}}$ name so that
$p_{n}\Vdash_{\alpha_{n}}\dot{q}_{n}\in{\mathord{\mathbb{P}}}_{\gamma}\cap M$
and
$p_{n}\Vdash_{\alpha_{n}}\dot{q}_{n}\upharpoonright\alpha_{n}\in\dot{G}_{\alpha_{n}}$

  5. (5)

$p_{n+1}\Vdash_{\alpha_{n+1}}\dot{q}_{n+1}\leq\dot{q}_{n}$ and
$p_{n+1}\Vdash_{\alpha_{n+1}}\dot{q}_{n+1}\in D_{n}$

  6. (6)

$\dot{t}_{n}$ is a ${\mathord{\mathbb{P}}}_{\gamma}$ name for a node in
$\dot{T}_{\varphi_{0}(n)}$ strictly above the $\varphi_{1}(n)^{\rm th}$ node
in $\dot{T}_{\varphi_{0}(n)}$

  7. (7)

$\dot{k}_{n}$ is a name for an element of $\omega$ forced to be in the domain
of $\dot{t}_{n}$ above the domain of the $\varphi_{1}(n)^{\rm th}$ node in
$\dot{T}_{\varphi_{0}(n)}$ and
$p_{n}\Vdash_{\alpha_{n}}$“$\dot{q}_{n}\Vdash_{\gamma}\check{g}(\dot{k}_{n})=\dot{t}_{n}(\dot{k}_{n})$”

Assuming that such a sequence can be constructed we let
$\bar{p}=\bigcup_{n<\omega}p_{n}$. Clearly this condition is as required.
Therefore it remains to see that we can construct such a quadruple of
sequences. This is done by induction. The base case is given. Suppose that we
have constructed for some $n<\omega$ sequences $\\{p_{m}\;|\;m<n+1\\}$,
$\\{\dot{q}_{m}\;|\;m<n+1\\}$, $\\{\dot{t}_{m}\;|\;m<n+1\\}$ and
$\\{\dot{k}_{m}\;|\;m<n+1\\}$ satisfying the above conditions and we construct
$p_{n+1}$, $\dot{q}_{n+1}$, $\dot{t}_{n+1}$ and $\dot{k}_{n+1}$.

Let $p_{n}\in G_{\alpha_{n}}$ be generic and work in $V[G_{\alpha_{n}}]$. By
assumption there is a $q_{n}\in{\mathord{\mathbb{P}}}_{\gamma}\cap M$ which is
the evaluation of the name $\dot{q}_{n}$ by $G_{\alpha_{n}}$ so that
$q_{n}\upharpoonright\alpha_{n}\in G_{\alpha_{n}}$. Since $p_{n}$ is
$(M,{\mathord{\mathbb{P}}}_{\alpha_{n}})$ generic there is an $r\in D_{n}\cap
M$ so that $r\leq q_{n}$ and $r\upharpoonright\alpha_{n}\in G_{\alpha_{n}}$.
Without loss we may assume that $r$ decides that the $\varphi_{1}(n)^{\rm th}$
node in $\dot{T}_{\varphi_{0}(n)}$ to be some $s_{n}\in\omega^{<\omega}$. Now
let $W$ be the set of all $t\in{\omega}^{<\omega}$ so that there is an
$\bar{r}\leq r$ so that $\bar{r}\in{\mathord{\mathbb{P}}}_{\gamma}$ and the
following hold.

  1. (1)

$\bar{r}\upharpoonright\alpha_{n}\in G_{\alpha_{n}}$

  2. (2)

$\bar{r}\Vdash_{\gamma}$ “$\check{t}\in\dot{T}_{\varphi_{0}(n)}$ and
$\check{t}\supsetneq\check{s}_{n}$”

Observe that, by appending $s_{n}$ and its predecessors to $W$ we have a tree
with stem (including) $s_{n}$. This is because if some $\bar{r}$ forces
$t\in\dot{T}_{\varphi_{0}(n)}$ to be strictly above $s_{n}$ and
$s_{n}\supsetneq t^{\prime}\supsetneq t$ then trivially the same $\bar{r}$
forces the same of $t^{\prime}$ so $W$ is closed downwards under sequences
extending $s_{n}$. Moreover $W\in M$ by construction (since
$\dot{T}_{\varphi_{0}(n)}$ and $r$ are) and, crucially
$W\in\mathcal{I}_{T}(\mathcal{E})^{+}$. To see this last point observe that if
$r\in G$ is ${\mathord{\mathbb{P}}}_{\gamma}$ generic then, in $V[G]$ we must
have that the evaluation of $\dot{T}_{\varphi_{0}(n)}$ is contained in $W$ so
if $W$ could be covered by finitely many functions from $\mathcal{E}$ then so
could $\dot{T}_{\varphi_{0}(n)}$ contradicting the fact that it was forced to
be a tree in $\mathcal{I}_{T}(\mathcal{E})^{+}$.

We can therefore apply our inductive assumption to $\alpha_{n}$ and conclude
that $g$ densely diagonalizes $W$. It follows that there is a node $t\in W$ so
that $t\supsetneq s_{n}$ and there is a $k\in{\rm dom}(t)\setminus{\rm
dom}(s_{n})$ so that $g(k)=t(k)$. Let $\dot{q}_{n+1}$ be a name for the
$\bar{r}$ witnessing that $t\in W$ and let $\dot{t}_{n}$ and $\dot{k}_{n}$ be
names for $t$ and $k$ back in $V$. Finally, by our inductive hypothesis we can
find a $(M,{\mathord{\mathbb{P}}}_{\alpha_{n+1}},\mathcal{E},g)$-generic
condition $p_{n+1}\leq p_{n}$ forcing all of this as needed. ∎

Sacks forcing strongly preserves the tightness of any tight eventually
different family. This gives an alternative proof of the fact that
$\mathfrak{a}_{e}=\aleph_{1}$ in the Sacks model. The argument for this is
very similar to the analogous one we give for Miller forcing in Section 5 so
we leave the details of this to the reader. In Sections 6-8 we also show that
partition forcing, infinitely often equal forcing and Shelah’s
$\mathbb{Q}_{\mathcal{I}}$ forcing all strongly preserves the tightness of any
given tight eventually different family. Before giving these arguments we turn
to a slightly different type of eventually different family.

##  5\. Tight Eventually Different Permutations

The foregoing discussion of tight eventually different families works mutatis
mutandis for eventually different sets of permutations of $\omega$. Recall
that $\mathfrak{a}_{p}$ is the least size of a maximal eventually different
family of permutations. It remains an intriguing open question if
$\mathsf{ZFC}$ proves $\mathfrak{a}_{e}=\mathfrak{a}_{p}$. Let us give the
“permutation” version of the above definitions and state the analogous
results. Since the proofs are almost verbatim the same we simply indicate the
necessary changes and leave the details to the reader.

Call a tree $T\subseteq\omega^{<\omega}$ injective if each $t\in T$ is
injective as a finite function. Let $\mathcal{P}$ be a family of eventually
different permutations.

######  Definition 5.1.

The tree ideal generated by $\mathcal{P}$, denoted
$\mathcal{I}_{T}(\mathcal{P})$, is the set of all injective trees
$T\subseteq\omega^{<\omega}$ so that there is a $t\in T$ and a finite set
$X\in[\mathcal{E}]^{<\omega}$ so that $\bigcup T_{t}\subseteq^{*}\bigcup X$.

Dually an injective tree $T\subseteq\omega^{<\omega}$ is in
$\mathcal{I}_{T}(\mathcal{P})^{+}$ if for each $t\in T$ it’s not the case that
$\bigcup T_{t}$ can be almost covered by finitely many functions from
$\mathcal{P}$.

The definition of tightness for sets of permutations is now identical to that
of functions.

######  Definition 5.2.

An eventually different set of permutations $\mathcal{P}$ is tight if given
any sequence of countably many injective trees $\\{T_{n}\;|\;n<\omega\\}$ so
that $T_{n}\in\mathcal{I}_{T}(\mathcal{P})^{+}$ for all $n<\omega$ there is a
single $g\in\mathcal{P}$ which densely diagonalizes all the $T_{n}$’s.

The same proof as before shows that:

######  Proposition 5.3.

If $\mathcal{P}$ is tight then it is maximal.

We also have the analogue of Theorem 2.4

######  Theorem 5.4.

Assume $\mathsf{MA}(\sigma{\rm-centered})$. Every eventually different family
$\mathcal{P}_{0}$ of size ${<}2^{\aleph_{0}}$ is contained in a tight
eventually different set of permutations. In particular $\mathsf{CH}$ implies
that tight eventually different sets of permutations exist.

###### Proof.

The proof is almost the same as that of Theorem 2.4. We just indicate what the
right poset is and the necessary modifications. Let $\mathcal{P}$ be an
eventually different set of permutations (not necessarily maximal). Define the
forcing notion ${\mathord{\mathbb{Q}}}_{\mathcal{P}}$ to be the set of all
pairs $(s,E)$ so that the following hold.

  1. (1)

$s\in\omega^{<\omega}$ is injective

  2. (2)

$E\in[\mathcal{E}]^{<\omega}$

The order on ${\mathord{\mathbb{Q}}}_{\mathcal{P}}$ is defined exactly the
same as for ${\mathord{\mathbb{P}}}_{\mathcal{E}}$. Mimicking the proof of
Theorem 2.4, it’s enough to show that the following sets are dense.

  1. (1)

For each $k<\omega$ the set of conditions $(s,E)$ so that $k\in{\rm dom}(s)$.

  2. (2)

For each $k<\omega$ the set of conditions $(s,E)$ so that $k\in{\rm
range}(s)$.

  3. (3)

For each $T\in\mathcal{I}_{T}(\mathcal{P})^{+}$, and $t\in T$ the set of
conditions $(s,E)$ so that there is an $t^{\prime}\in T$ which extends $t$ and
a $k\in{\rm dom}(t^{\prime})\setminus{\rm dom}(t)$ so that
$s(k)=t^{\prime}(k)$.

  4. (4)

For each $f\in\mathcal{E}$ the set of conditions $(s,E)$ so that $f\in E$.

This is shown exactly as for eventually different families, the one caveat
being that we need to be able ensure that when we strengthen some $(s,E)$ we
can guarantee that if $T\in\mathcal{I}_{T}(\mathcal{P})^{+}$ then we can find
an $t^{\prime}\supsetneq t$ so that $t^{\prime}\in T$ and a $k\in{\rm
dom}(t^{\prime})\setminus{\rm dom}(t)\cap{\rm dom}(s)$ so that we can add
$(k,t^{\prime}(k))$ to $s$ without wrecking injectivity. This is where we use
that fact that $T$ is injective. Namely, we can find a level in $T$ above
which no element of any sequence is in the range of $s$ (since $s$ is finite)
and therefore we can find the needed $k$. Everything else in the proof is
exactly as in the case of eventually different families of functions. ∎

We also have the same indestructibility results.

######  Theorem 5.5.

Suppose $\kappa$ is a cardinal and $\mathbb{C}_{\kappa}=add(\omega,\kappa)$ is
the forcing to add $\kappa$-many Cohen reals. If $\mathcal{P}$ is a tight
eventually different set of permutations then $\Vdash_{\mathbb{C}_{\kappa}}$
“$\check{\mathcal{P}}$ is tight”.

Similarly, there is an analogue of Theorem 4.2 for tight sets of permutations.
We give the definition and theorem below.

######  Definition 5.6.

Let ${\mathord{\mathbb{P}}}$ be a proper forcing notion and $\mathcal{P}$ a
tight eventually different set of permutations. We say that
${\mathord{\mathbb{P}}}$ strongly preserves the tightness of $\mathcal{P}$ if
for every sufficiently large $\theta$, every condition $p$ and every $M\prec
H_{\theta}$ countable with $p,{\mathord{\mathbb{P}}},\mathcal{P}\in M$, if
$g\in\mathcal{P}$ strongly diagonalizes every element of
$M\cap\mathcal{I}_{T}(\mathcal{P})^{+}$ then there is an
$(M,{\mathord{\mathbb{P}}})$-generic $q\leq p$ so that $q$ forces that $g$
densely diagonalizes every element of
$M[\dot{G}]\cap\mathcal{I}_{T}(\mathcal{P})^{+}$. Such a $q$ is called an
$(M,{\mathord{\mathbb{P}}},\mathcal{P},g)$-generic condition.

######  Theorem 5.7.

Suppose that $\mathcal{P}$ is a tight eventually different family. If
$\langle{\mathord{\mathbb{P}}}_{\alpha},\dot{\mathbb{Q}}_{\alpha}\;|\;\alpha<\gamma\rangle$
is a countable support iteration of proper forcing notions so that for all
$\alpha$ we have $\Vdash_{\alpha}$ “$\dot{\mathbb{Q}}_{\alpha}$ strongly
preserves the tightness of $\check{\mathcal{P}}$” then
${\mathord{\mathbb{P}}}_{\gamma}$ strongly preserves the tightness of
$\mathcal{P}$.

The proof of Theorems 5.5 and 5.7 are identical to those of Theorems 3.1 and
4.2 so we leave the details to the reader. The one point to note is that the
$T_{n,j}$’s of Theorem 3.1 and the $W$ found in the inductive proof of Lemma
4.4 are both injective since $\dot{T}_{n}$ (respectively
$\dot{T}_{\varphi_{0}(n)}$) is forced to be injective hence if some
$r\Vdash\check{t}\in\dot{T}_{n}$ (respectively $\dot{T}_{\varphi_{0}(n)}$)
then it must be the case that $t$ is injective.

As in this section every further one the proofs involving tight eventually
different families and tight eventually different permutations are almost
identical and, largely we prove the case of tight eventually different
families in detail and leave the the case of tight eventually different
permutations to the reader. This situation adds fuel to the idea that perhaps
$\mathsf{ZFC}$ proves that $\mathfrak{a}_{e}=\mathfrak{a}_{p}$ and in any case
shows how difficult it could be to separate these two invariants even if this
ends up being possible.

##  6\. Miller Forcing and the Consistency of
$\mathfrak{a}_{e}=\mathfrak{a}_{p}<\mathfrak{d}=\mathfrak{a}_{T}$

In this section we show that Miller forcing strongly preserves the tightness
of any tight eventually different family of functions and any tight eventually
different set of permutations. As a result we obtain the consistency of
$\mathfrak{a}_{e}=\mathfrak{a}_{p}<\mathfrak{d}=\mathfrak{a}_{T}$. Recall that
Miller forcing, also called rational perfect set forcing and denoted
$\mathbb{PT}$ consists of all trees $T\subseteq\omega^{<\omega}$ so that for
every node $t\in T$ there is an $s\in T$ extending $t$ with infinitely many
immediate successors. The order is inclusion. As is often done we work with
the dense subset of trees in which every node has either one or infinitely
many immediate successors. It’s well known that this forcing is proper and,
when iterated $\omega_{2}$-many times with countable support produces a model
of $non(\mathcal{M})=cov(\mathcal{M})=\aleph_{1}<\mathfrak{d}=\aleph_{2}$ (and
$\mathfrak{a}_{T}=\aleph_{2}$ as well). More information about Miller forcing
can be found in [1], see in particular Definition 7.3.43.

######  Theorem 6.1.

Let $\mathcal{E}$ be a tight eventually different family. Miller forcing
$\mathbb{PT}$ strongly preserves the tightness of $\mathcal{E}$.

Before proving this theorem we recall some basic terminology. Recall that if
$p\in\mathbb{PT}$ is a Miller tree and $n<\omega$ then a node $t\in p$ is an
${n}^{\rm th}$-splitting node if it has infinitely many immediate successors
and it has the $n-1$ predecessors with this property. Denote by ${\rm
Split}_{n}(p)$ the set of $n$-splitting nodes. We say that for two Miller
trees $p,q\in\mathbb{PT}$ that $q\leq_{n}p$ if $q\leq p$ and ${\rm
Split}_{n}(p)={\rm Split}_{n}(q)$.

###### Proof.

Let $p\in{\mathord{\mathbb{P}}}(\mathcal{K})$ be a condition, let $M\prec
H_{\theta}$ countable with $\theta$ sufficiently large,
$p,\mathbb{PT},\mathcal{E}\in M$. Let $g\in\mathcal{E}$ densely diagonalize
every $T\in M\cap\mathcal{I}_{T}(\mathcal{E})^{+}$. Let
$\\{T_{n}\;|\;n<\omega\\}$ be an enumeration of all $\mathbb{PT}$ names for
trees in $\mathcal{I}_{T}(\mathcal{E})^{+}$. Let $\\{D_{n}\;|\;n<\omega\\}$
enumerate all dense open subsets of $\mathbb{PT}$ in $M$. Let
$\varphi:\omega\to\omega^{2}$ be a bijection with coordinate maps
$\varphi_{0}$ and $\varphi_{1}$. Inductively we will construct sequences
$\\{p_{n}\;|\;n<\omega\\}$, $\\{\dot{t}_{n}\;|\;n<\omega\\}$,
$\\{\dot{k}_{n}\;|\;n<\omega\\}$ so that the following hold.

  1. (1)

$p_{0}=p$

  2. (2)

$\\{p_{n}\;|\;n<\omega\\}\subseteq M$ and for all $n<\omega$
$p_{n+1}\leq_{n+1}p_{n}$

  3. (3)

For each $n$ $\dot{t}_{n}$ is a $\mathbb{PT}$ name for a node in
$\dot{T}_{\varphi_{0}(n)}$ extending the $\varphi_{1}(n)^{\rm th}$-node in
$\dot{T}_{\varphi_{0}(n)}$

  4. (4)

For each $n<\omega$ $\dot{k}_{n}$ is a name for an element of $\omega$ in $M$

  5. (5)

For each ${n+1}^{\rm st}$-splitting node $t$ of $p_{n+1}$ we have that
$(p_{n+1})_{t}\in D_{n}$, and forces for some $s_{n}\in\omega^{<\omega}$ that
the $\varphi_{1}(n)^{\rm th}$-node in $\dot{T}_{\varphi_{0}(n)}$ is
$\check{s}_{n}$ and $(p_{n+1})_{t}\Vdash\dot{k}_{n}\in{\rm
dom}(\dot{t}_{n})\setminus{\rm
dom}(\check{s}_{n})\land\dot{t}_{n}(\dot{k}_{n})=\check{g}(k)$.

Suppose first that we can construct such a sequence. Let
$q=\bigcap_{n<\omega}p_{n}$. It follows almost immediately that $q$ is
$(M,\mathbb{PT},\mathcal{E},g)$-generic as needed.

Therefore it remains to construct the requisite sequence. This is done by
induction. Let $p_{0}=p$. Now assume that $\\{p_{j}\;|\;j<n+1\\}$,
$\\{\dot{t}_{j}\;|\;j<n\\}$ and $\\{\dot{k}_{j}\;|\;j<n\\}$ have been defined.
We will define $p_{n+1}$, $\dot{t}_{n}$ and $\dot{k}_{n}$. For each
${n+1}^{\rm st}$-splitting node $t$ of $p_{n}$, let $q_{t}\leq(p_{n})_{t}$ be
an element of $M\cap D_{n}$ which forces for some $s_{n}\in\omega^{<\omega}$
that the $\varphi_{1}(n)^{\rm th}$-node in $\dot{T}_{\varphi_{0}(n)}$ is
$\check{s}_{n}$. Let $W_{t}=\\{u\supsetneq s_{n}\;|\;q_{t}\nVdash
u\notin\dot{T}_{\varphi_{0}(n)}\\}$. As in the proof of Lemma 4.2 $W_{t}$
alongside $s_{n}$ and its predecessors is a tree in
$\mathcal{I}_{T}(\mathcal{E})^{+}\cap M$ hence it is densely diagonalized by
$g$. Therefore we can find a $u_{t}\in W_{t}$ and a $k_{t}\in{\rm
dom}(u)\setminus{\rm dom}(s_{n})$ so that $u_{t}(k_{t})=g(k_{t})$. Let
$r_{t}\leq q_{t}$ force that $\check{u_{t}}\in\dot{T}_{\varphi_{0}(n)}$. Now
let $p_{n+1}=\bigcup_{t\in{\rm Split}_{n+1}(p_{n})}r_{t}$. Let
$\dot{t}_{n}=\\{(\check{u}_{t},r_{t})\;|\;t\in{\rm Split}_{n+1}(p_{n})\\}$ and
$\dot{k}_{n}=\\{(\check{k}_{t},r_{t})\;|\;t\in{\rm Split}_{n+1}(p_{n})\\}$.
Clearly these suffice.

∎

Essentially the same proof gives the analogous result for tight eventually
different families of permutations. We state the theorem below and leave the
(primarily cosmetic) modifications of the above proof to the reader.

######  Theorem 6.2.

Let $\mathcal{P}$ be a tight eventually different family of permutations.
Miller forcing strongly preserves the tightness of $\mathcal{P}$.

As a result we get the following.

######  Theorem 6.3.

In the iterated Miller model the inequality
$\mathfrak{a}_{e}=\mathfrak{a}_{p}<\mathfrak{d}=\mathfrak{a}_{T}$ holds.

Note that this inequality also holds in the Cohen model, however unlike the
Cohen model, this model $cov(\mathcal{M})=\aleph_{1}$ so we can in fact also
gain freedom over $cov(\mathcal{M})$. Also, as mentioned in the introduction,
$\mathfrak{a}_{p}=\aleph_{1}$ in the Miller model was originally shown by
Kastermans and Zhang using parametrized diamonds in [14].

##  7\. Partition Forcing and the Consistency of
$\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{d}<\mathfrak{a}_{T}$

In this section we show that Miller’s partition forcing strongly preserves the
tightness of any tight eventually different family of functions and any tight
eventually different family of permutations. As a result we obtain the
consistency of
$\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{d}<\mathfrak{a}_{T}$. The proof
of the preservation result mirrors the analogous one for tight MAD families
given in [11, Proposition 38]. We recall some terminology used there.

Recall from the introduction that if $\mathcal{K}$ is an uncountable partition
of $2^{\omega}$ into closed sets then ${\mathord{\mathbb{P}}}(\mathcal{K})$,
the partition forcing, is the set of perfect trees $p$ so that for all
$C\in\mathcal{K}$ we have $[p]\cap C$ is nowhere dense in $[p]$. This forcing
was introduced in [15] to increase $\mathfrak{a}_{T}$. In order to ensure that
a perfect tree is in ${\mathord{\mathbb{P}}}(\mathcal{K})$ we need the
definition of a nice set of reals.

######  Definition 7.1 (Definition 36 of [11]).

Fix a partition of $2^{\omega}$ into closed sets
$\mathcal{K}=\\{C_{\alpha}\;|\;\alpha<\omega_{1}\\}$. We say that
$X=\\{x_{s}\;|\;s\in\omega^{<\omega}\\}\subseteq 2^{\omega}$ is nice (for
$\mathcal{K}$) if the following conditions holds:

  1. (1)

For every $s\in\omega^{<\omega}$ the sequence $\langle
x_{s^{\frown}n}\rangle_{n<\omega}$ converges to $x_{s}$ and
$\Delta(x_{s},x_{s^{\frown}n})<\Delta(x_{s},x_{s^{\frown}n+1})$ for all
$n<\omega$222Recall that if $x\neq y\in 2^{\omega}$ then $\Delta(x,y)$ is the
least $k\in\omega$ so that $x(k)\neq y(k)$..

  2. (2)

For all $s,t,z\in\omega^{<\omega}$ if $s\subseteq t\subseteq z$ then
$\Delta(x_{s},x_{z})<\Delta(x_{t},x_{z})$.

  3. (3)

For every $s\in\omega^{<\omega}$ let $\alpha_{s}<\omega_{1}$ so that $x_{s}\in
C_{\alpha_{s}}$. If $s\subseteq t$ then $\alpha_{s}\neq\alpha_{t}$.

The point is the following Lemma.

######  Lemma 7.2 (Lemma 37 of [11]).

Let $p$ be a Sacks tree and let $\mathcal{K}$ be an uncountable partition of
$2^{\omega}$ into closed sets. If there is an
$X=\\{x_{s}\;|\;s\in\omega^{<\omega}\\}$ which is nice for $\mathcal{K}$ and
dense in $[p]$ then $p\in{\mathord{\mathbb{P}}}(\mathcal{K})$.

Armed with these facts we can prove the main theorem of this section. Our
proof is extremely similar to the analogous proof for tight MAD families, [11,
Proposition 38]. The modifications follow those of Theorem 6.1.

######  Theorem 7.3.

Let $\mathcal{K}\subseteq P(2^{\omega})$ be an uncountable partition of
$2^{\omega}$ into closed sets and let $\mathcal{E}$ be a tight eventually
different family. The forcing notion ${\mathord{\mathbb{P}}}(\mathcal{K})$
strongly preserves the tightness of $\mathcal{E}$.

###### Proof.

Let $p\in{\mathord{\mathbb{P}}}(\mathcal{K})$ be a condition, let $M\prec
H_{\theta}$ countable with $\theta$ sufficiently large,
$p,{\mathord{\mathbb{P}}}(\mathcal{K}),\mathcal{K},\mathcal{E}\in M$. Let
$g\in\mathcal{E}$ densely diagonalize every $T\in
M\cap\mathcal{I}_{T}(\mathcal{E})^{+}$. Let $\\{T_{n}\;|\;n<\omega\\}$ be an
enumeration of all ${\mathord{\mathbb{P}}}(\mathcal{K})$ names for trees in
$\mathcal{I}_{T}(\mathcal{E})^{+}$. Let $\\{D_{n}\;|\;n<\omega\\}$ enumerate
all dense open subsets of ${\mathord{\mathbb{P}}}(\mathcal{K})$ in $M$. Let
$\varphi:\omega\to\omega^{2}$ be a bijection with coordinate maps
$\varphi_{0}$ and $\varphi_{1}$. Inductively we will construct sequences
$\\{p_{n}\;|\;n<\omega\\}$, $\\{\dot{t}_{n}\;|\;n<\omega\\}$,
$\\{\dot{k}_{n}\;|\;n<\omega\\}$ and a set
$X=\\{x_{s}\;|\;s\in\omega^{<\omega}\\}$ so that the following hold.

  1. (1)

$p_{0}=p$

  2. (2)

$\\{p_{n}\;|\;n<\omega\\}\subseteq M$ and for all $n<\omega$ $p_{n+1}\leq
p_{n}$

  3. (3)

$X\subseteq 2^{\omega}\cap M$ is nice for $\mathcal{K}$

  4. (4)

$X\subseteq[p_{n}]$ for every $n<\omega$

  5. (5)

For each $n$ $\dot{t}_{n}$ is a ${\mathord{\mathbb{P}}}(\mathcal{K})$ name for
a node in $\dot{T}_{\varphi_{0}(n)}$ extending the $\varphi_{1}(n)^{\rm
th}$-node in $\dot{T}_{\varphi_{0}(n)}$

  6. (6)

For each $n<\omega$ $\dot{k}_{n}$ is a name for an element of $\omega$ in $M$

  7. (7)

For each $s\in\omega^{n}$ and $i,m\in\omega$ if
$m=\Delta(x_{s},x_{s^{\frown}i})$ and $t=(x_{s^{\frown}i})\upharpoonright m$
then $(p_{n+1})_{t}\in D_{n}$, forces for some $s_{n}\in\omega^{<\omega}$ that
the $\varphi_{1}(n)^{\rm th}$-node in $\dot{T}_{\varphi_{0}(n)}$ is
$\check{s}_{n}$ and $(p_{n+1})_{t}\Vdash\dot{k}_{n}\in{\rm
dom}(\dot{t}_{n})\setminus{\rm
dom}(\check{s}_{n})\land\dot{t}_{n}(\dot{k}_{n})=\check{g}(k)$.

Suppose first that we can construct such a sequence. Let
$q=\bigcap_{n<\omega}p_{n}$. Clearly $q$ is a condition since it’s a perfect
tree in which $X$ is dense. From this it follows almost immediately that it is
$(M,{\mathord{\mathbb{P}}}(\mathcal{K}),\mathcal{E},g)$-generic as needed.

Therefore it remains to construct the sequence described above. This is done
by induction. Let $p_{0}=p$, $x_{\emptyset}$ be any element of $[p_{0}]\cap
M$. Now assume that $\\{p_{j}\;|\;j<n+1\\}$, $\\{x_{s}\;|\;s\in\omega^{\leq
n}\\}$, $\\{\dot{t}_{j}\;|\;j<n\\}$ and $\\{\dot{k}_{j}\;|\;j<n\\}$ have been
defined. We will define $p_{n+1}$, $\\{x_{s}\;|\;s\in\omega^{n+1}\\}$,
$\dot{t}_{n}$ and $\dot{k}_{n}$. For each $s\in\omega^{n}$ let $l\in\omega$ be
so that $l>\Delta(x_{s},x_{s}^{\prime})$ for all $s^{\prime}\subsetneq s$.
Define $Y_{s}$ be the set of all $m>l$ so that $x_{s}\upharpoonright m\in
p_{n}$ is a splitting node. For each $m\in Y_{s}$ let
$t_{m}=(x_{s}\upharpoonright m)^{\frown}(1-x_{s}(m))$ (which is a node of
$p_{n}$ since $x_{s}\upharpoonright m$ is splitting). Let
$p^{s}_{m}=(p_{n})_{t_{m}}$. Note that $p^{s}_{m}\in M$. By strengthening if
necessary we may also assume that there is an $u^{s}_{m}\in{\omega}^{\omega}$
so that $p^{s}_{m}\Vdash$“$\check{u}^{s}_{m}$ is the $\varphi_{1}(n)^{\rm th}$
node in $T_{\varphi_{0}(n)}$”. Now let
$W^{s}_{m}=\\{t\in\omega^{<\omega}\;|\;t\supsetneq u^{s}_{m}\;{\rm
and}\;p^{s}_{m}\nVdash t\notin\dot{T}_{\varphi_{0}(n)}\\}$. As in the proof of
Theorem 4.2, $W^{s}_{m}$, alongside $u^{s}_{m}$ and its predecessors is a tree
in $\mathcal{I}_{T}(\mathcal{E})^{+}$ (in $M$) so $g$ densely diagonalizes it.
Let $r^{s}_{m}\leq p^{s}_{m}$ so that $r^{s}_{m}\in D_{n}$, $[r^{s}_{m}]\cap
C_{\alpha_{z}}=\emptyset$ for every $z\subseteq s$ and there is a
$t^{s}_{m}\in W^{s}_{m}$ and a $k^{s}_{m}<\omega$ so that $k^{s}_{m}\in{\rm
dom}(t)\setminus{\rm dom}(u^{s}_{m})$ and
$r^{s}_{m}\Vdash\check{t}^{s}_{m}\in\dot{T}_{\varphi_{0}(n)}$ and
$t^{s}_{m}(k^{s}_{m})=g(k^{s}_{m})$. Enumerate $Y_{s}$ as
$\\{m^{s}_{i}\;|\;i<\omega\\}$. For each $i<\omega$ choose $x_{s^{\frown}i}$
to be any branch in $r^{s}_{m^{s}_{i}}$ and let
$p_{n+1}=\bigcup\\{r^{s}_{m^{s}_{i}}\;|\;s\in\omega^{n}\land i\in\omega\\}$.
Finally let
$\dot{t}_{n}=\\{(\check{t}^{s}_{m^{s}_{i}},r^{s}_{m^{s}_{i}})\;|\;s\in\omega^{n}\land
i\in\omega\\}$ and
$\dot{k}_{n}=\\{(\check{k}^{s}_{m^{s}_{i}},r^{s}_{m^{s}_{i}})\;|\;s\in\omega^{n}\land
i\in\omega\\}$. Clearly these suffice.

∎

As before essentially the same proof gives the analogous result for tight
eventually different families of permutations. We state the theorem below and
leave the (primarily cosmetic) modifications of the above proof to the reader.

######  Theorem 7.4.

Let $\mathcal{K}\subseteq P(2^{\omega})$ be an uncountable partition of
$2^{\omega}$ into closed sets and let $\mathcal{P}$ be a tight eventually
different family of permutations. The forcing notion
${\mathord{\mathbb{P}}}(\mathcal{K})$ strongly preserves the tightness of
$\mathcal{P}$.

As a result of these theorems we have the following.

######  Theorem 7.5.

In the iterated partition forcing model the inequality
$\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{d}<\mathfrak{a}_{T}$ holds.

###### Proof.

Using a bookkeeping device to keep track with the partitions, let
${\mathord{\mathbb{P}}}$ be the $\omega_{2}$ length countable support of
partition forcing. Let $G\subseteq{\mathord{\mathbb{P}}}$ be generic. In
$V[G]$ we have $\mathfrak{a}_{e}=\aleph_{1}$ by Theorem 7.3 and
$\mathfrak{a}_{p}=\aleph_{1}$ by Theorem 7.3. Moreover this forcing is known
to be $\omega^{\omega}$-bounding ([21]) so $\mathfrak{d}=\aleph_{1}$. Finally
that $\mathfrak{a}_{T}=\aleph_{2}$ in this model is by [15, Theorem 6] (indeed
this is what it was designed to do). ∎

######  Remark 3.

In [6] it was shown that in the iterated partition forcing model that
$\mathfrak{a}=\mathfrak{i}=\mathfrak{u}=\aleph_{1}$. Alongside Theorem 7.5
this result gives credence to the idea that the iterated partition forcing
model is a model where every relative of $\mathfrak{a}$ other than
$\mathfrak{a}_{T}$ is $\aleph_{1}$. It remains open what happens in this model
to $\mathfrak{a}_{g}$.

Putting together the results of this section, the previous one, Fact 1.3 and
Proposition 1.4 we get the following which encapsulates one of the main
results of this paper.

######  Corollary 7.6.

Any $\\{\aleph_{1},\aleph_{2}\\}$-valued assignment of
$\\{\mathfrak{a}_{e},\mathfrak{a}_{p},\mathfrak{d},\mathfrak{a}_{T}\\}$
respecting $\mathfrak{d}\leq\mathfrak{a}_{T}$ and
$\mathfrak{a}_{e}=\mathfrak{a}_{p}$ is consistent. In particular,
$\mathfrak{a}_{p}$ and $\mathfrak{a}_{e}$ are both independent of both
$\mathfrak{d}$ and $\mathfrak{a}_{T}$.

##  8\. Infinitely Often Equal Forcing and the consistency of
$\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{u}=\mathfrak{d}<non(\mathcal{N})=cof(\mathcal{N})$

Recall that $\mathbb{EE}$, the infinitely often equal forcing consists of
functions $p$ with domain a co-infinite subset of $\omega$ so that for all
$n\in{\rm dom}(p)$ we have $p(n)\in 2^{n}$, see [1, Definition 7.4.11]. The
extension relation is reverse inclusion. The forcing is
$\omega^{\omega}$-bounding and makes the ground model reals null, see [1,
Section 7.4.C]. It is also Axiom A as witnessed by the relations $p\leq_{n}q$
defined by $p\supseteq q$ and the first $n$ elements of $\omega\setminus{\rm
dom}(p)$ are the same as the first $n$ elements $\omega\setminus{\rm dom}(q)$.
We will show that this forcing strongly preserves the tightness of eventually
different families of functions and sets of permutations. Towards this we need
the following definition.

######  Definition 8.1.

Let $p\in\mathbb{EE}$, $n<\omega$ and let $k_{n}\in\omega\setminus{\rm
dom}(p)$ be the $n^{\rm th}$ element of $\omega\setminus{\rm dom}(p)$. An
$n$-gap closure of $p$ is a function $t$ with domain $k_{n}$ so that for all
$l<k_{n}$, $t(l)\in 2^{l}$ and if $l\in{\rm dom}(p)$ then $t(l)=p(l)$.

The point of the definition, as the name suggests, is that if $t$ is an
$n$-gap closure of $p$ then $p\cup t$ is a condition and the entire interval
$[0,k_{n}]$ is in its domain. Given an $n$-gap closure $t$ of $p$ let
$p_{t}=p\cup t$. Note that for each $n$ and $p$ there are only finitely many
$n$-gap closures for $p$ (though the exact number depends on $p$).

######  Lemma 8.2.

Suppose $p\in\mathbb{EE}$ is a condition and $D$ is a dense open subset. For
any $n<\omega$ there is a $q\leq_{n}p$ so that for each $n$-gap closure $t$ of
$q$ we have $q_{t}\in D$.

###### Proof.

Enumerate the finitely many $n$-gap closures of $p$ as
$\\{t_{0},...,t_{k-1}\\}$. Inductively define a decreasing sequence
$p_{k-1}\leq_{n}p_{k-2}\leq_{n}...\leq_{n}p_{1}\leq_{n}p_{0}\leq_{n}p$ so that
for all $l<k$ we have $(p_{l})_{t_{l}}\in D$. Namely, for each $l<k$ pick a
strengthening of $p^{\prime}_{l+1}\leq p_{l}\cup t_{l+1}$ so that
$p^{\prime}_{l+1}\in D$ and let $p_{l+1}$ be $p^{\prime}_{l+1}$ with the
elements of $t_{l+1}\setminus p_{l}$ deleted. It follows that $q=p_{k-1}$ is
as needed since for any $l<k$ $q_{t_{l}}$ strengthens $p^{\prime}_{l}$ and
hence is in $D$. ∎

With Lemma 8.2 in hand the proof of strong preservation of tightness for
$\mathbb{EE}$ resembles that for Miller Forcing.

######  Theorem 8.3.

Let $\mathcal{E}$ be a tight eventually different family. The forcing
$\mathbb{EE}$ strongly preserves $\mathcal{E}$.

###### Proof.

Let $p\in\mathbb{EE}$ be a condition, let $M\prec H_{\theta}$ countable with
$\theta$ sufficiently large, $p,\mathbb{EE},\mathcal{E}\in M$. Let
$g\in\mathcal{E}$ densely diagonalize every $T\in
M\cap\mathcal{I}_{T}(\mathcal{E})^{+}$. Let $\\{\dot{T}_{n}\;|\;n<\omega\\}$
be an enumeration of all $\mathbb{EE}$ names for trees in
$\mathcal{I}_{T}(\mathcal{E})^{+}$. Let $\\{D_{n}\;|\;n<\omega\\}$ enumerate
all dense open subsets of $\mathbb{EE}$ in $M$. Let
$\varphi:\omega\to\omega^{2}$ be a bijection with coordinate maps
$\varphi_{0}$ and $\varphi_{1}$. Inductively we will construct sequences
$\\{p_{n}\;|\;n<\omega\\}$, $\\{\dot{t}_{n}\;|\;n<\omega\\}$,
$\\{\dot{k}_{n}\;|\;n<\omega\\}$ so that the following hold.

  1. (1)

$p_{0}=p$

  2. (2)

$\\{p_{n}\;|\;n<\omega\\}\subseteq M$ and for all $n<\omega$
$p_{n+1}\leq_{n+1}p_{n}$

  3. (3)

For each $n$ $\dot{t}_{n}$ is a $\mathbb{EE}$ name for a node in
$\dot{T}_{\varphi_{0}(n)}$ extending the $\varphi_{1}(n)^{\rm th}$-node in
$\dot{T}_{\varphi_{0}(n)}$

  4. (4)

For each $n<\omega$ $\dot{k}_{n}$ is a name for an element of $\omega$ in $M$

  5. (5)

For each ${n+1}^{\rm st}$-gap closure $t$ of $p_{n+1}$ we have that
$(p_{n+1})_{t}\in D_{n}$, and forces for some $s_{n}\in\omega^{<\omega}$ that
the $\varphi_{1}(n)^{\rm th}$-node in $\dot{T}_{\varphi_{0}(n)}$ is
$\check{s}_{n}$ and $(p_{n+1})_{t}\Vdash\dot{k}_{n}\in{\rm
dom}(\dot{t}_{n})\setminus{\rm
dom}(\check{s}_{n})\land\dot{t}_{n}(\dot{k}_{n})=\check{g}(\dot{k}_{n})$.

Suppose first that we can construct such a sequence. Let
$q=\bigcup_{n<\omega}p_{n}$. It follows almost immediately that $q$ is
$(M,\mathbb{PT},\mathcal{E},g)$-generic as needed.

The proof of the construction is almost verbatim the same as that for Miller
forcing using Lemma 8.2. The only difference is the last point on the list
above. Specifically, given $p_{n}$ and an $n+1$-gap closure $t$ (of which
there are only finitely many) we can strengthen $(p_{n})_{t}$ to decide all of
the (finitely much) information needed for the final point. Repeating this
process for each of the finitely many gap closures gives the required
strengthening $p_{n+1}$. ∎

Using essentially the same proof as above we can show that $\mathbb{EE}$
preserves tight eventually different sets of permutations and tight MAD
families.

######  Theorem 8.4.

Let $\mathcal{P}$ be a tight eventually different set of permutations and let
$\mathcal{A}$ be a tight MAD family. $\mathbb{EE}$ strongly preserves the
tightness of $\mathcal{P}$ and the strongly preserves the tightness of
$\mathcal{A}$.

Given this we can show as in the previous sections the following. Note that
$\mathbb{EE}$ preserves P-points ([1, Lemma 7.4.15]).

######  Theorem 8.5.

Let ${\mathord{\mathbb{P}}}$ be the $\omega_{2}$-length countable support
iteration of $\mathbb{EE}$ and let $G\subseteq{\mathord{\mathbb{P}}}$ be
generic over $V$. In $V[G]$ we have
$\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{a}=\mathfrak{d}=\mathfrak{u}<non(\mathcal{N})=cof(\mathcal{N})$.

##  9\. Shelah’s Forcing ${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ and the
consistency of $\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{i}<\mathfrak{u}$

In this section we show that Shelah’s forcing
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ for proving the consistency of
$\mathfrak{i}<\mathfrak{u}$ strongly preserves tight eventually different
families and tight eventually different sets of permutations. As a result we
obtain the consistency of
$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{d}=\mathfrak{i}=cof(\mathcal{N})<\mathfrak{u}$.
First we recall the poset ${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ from [20] and
some of its properties. The exposition in this section strongly mirrors that
of [6, Section 5], where it is shown that
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ strongly preserves tight MAD families.

We start, by recalling some terminology from [20], see also [6]. Given an
ideal $\mathcal{I}$ on $\omega$, we say that an equivalence relation $E$ on a
subset of $\omega$ is an $\mathcal{I}$-equivalence relation if ${\rm
dom}(E)\in\mathcal{I}^{*}$ (the dual filter) and each $E$-equivalence class is
in $\mathcal{I}$. For $\mathcal{I}$-equivalence relations $E_{1}$ and $E_{2}$,
we define $E_{q}\leq_{\mathcal{I}}E_{2}$ if ${\rm dom}(E_{1})\subseteq{\rm
dom}(E_{2})$ and every $E_{1}$ equivalence class is the union of a set of
$E_{2}$ equivalence classes. Moreover, we will make use of the notion of a
$A$-$n$-determined function. More precisely: Given a subset $A$ of $\omega$,
we say that a function $g:2^{A}\to 2$ is $A$-$n$-determined if there is a set
$a\subseteq A\cap n+1$ such that whenever $\eta\upharpoonright
a=\nu\upharpoonright a$ for $\eta,\nu$ in ${{}^{A}\omega}$, we have
$g(\eta)=g(\nu)$. For each $i\in A$, $g_{i}$ is the function mapping
$\eta\in{{}^{A}2}$ to $\eta(i)$. The following claim appears in [20]: If $g$
is a $A$-$n$-determined function, then $g=\varphi(g_{0},...,g_{n})$, where
$\varphi(g_{0},...,g_{n})$ which is obtained as a maximum, minimum and
complement (e.g. $1-g_{i}$) of $g_{0},...,g_{n}$ and the constant functions on
$0$ and $1$. Again following [20], given an $\mathcal{I}$-equivalence relation
$E$, we denote by $A=A(E)=\\{x\;|\;x\in{\rm dom}(E)\;{\rm and}\;x={\rm
min}[x]_{E}\\}$.

######  Definition 9.1 (The conditions of
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$).

Let $\mathcal{I}$ be an ideal on $\omega$. Define
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ to be the set of $p=(H^{p},E^{p})$
where

  1. (1)

$E$ is an $\mathcal{I}$-equivalence relation,

  2. (2)

$H$ is a function with domain $\omega$ so that for each $n$ $H(n)$ is
$A(E)$-$n$-determined,

  3. (3)

if $n\in A(E)$ then $H(n)=g_{n}$,

  4. (4)

if $n\in{\rm dom}(E)\setminus A(E)$ and $nEi$ for an $i\in A(E)$ then $H(n)$
is either $g_{i}$ or $1-g_{i}$.

For a condition $p\in{\mathord{\mathbb{Q}}}_{\mathcal{I}}$ let
$A^{p}=A(E^{p})$. Before we can define the order on
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ we need one more definition, again
appearing in [6]: For $p,q\in{\mathord{\mathbb{Q}}}_{\mathcal{I}}$ with
$A^{p}\subseteq A^{q}$, we write $H^{p}(n)=^{**}H^{q}(n)$ if for each $\eta\in
2^{A^{p}}$ we have $H^{p}(n)(\eta)=H^{q}(n)(\eta^{\prime})$ where

$\eta^{\prime}(j)=\begin{cases}\eta(j)&j\in A^{p}\\\ H^{p}(j)(\eta)&j\in
A^{q}\setminus A^{p}\\\ \end{cases}$

Now we can define the order on ${\mathord{\mathbb{Q}}}_{\mathcal{I}}$.

######  Definition 9.2.

Let $p,q\in{\mathord{\mathbb{Q}}}_{\mathcal{I}}$. We let $p\leq q$ if

  1. (1)

$E^{p}\leq_{\mathcal{I}}E^{q}$

  2. (2)

if $H^{q}(n)=g_{i}$ for $n\in{\rm dom}(E^{q})$ then $H^{p}(n)=H^{p}(i)$,

  3. (3)

if $H^{q}(n)=1-g_{i}$ for $n\in{\rm dom}(E^{q})$ then
$H^{p}(n)=^{**}H^{q}(n)$,

  4. (4)

if $n\in\omega\setminus{\rm dom}(E^{q})$ then $H^{p}(n)=^{**}H^{q}(n)$.

Finally let $p\leq_{n}q$ if $p\leq q$ and $A^{p}$ contains the first $n$
elements of $A^{q}$.

If $\mathcal{I}$ is a maximal ideal $\mathcal{I}$ then the forcing notion
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ is proper [20, Claim 1.13]. Moreover it
has the Sacks property [20, Claim 1.12] and hence is
$\omega^{\omega}$-bounding. Forcing with
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ kills the maximality of
$\mathcal{I}$[20, Claim 1.5]. As a result, using a bookkeeping device to
ensure we cover all possible $\mathcal{I}$’s, iteratively forcing with
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ makes $\mathfrak{u}=\aleph_{2}$.

######  Lemma 9.3.

Let $p\in{\mathord{\mathbb{Q}}}_{\mathcal{I}}$. For an initial segment $u$ of
$A^{p}$ and $h:u\to 2$ let $p^{[h]}$ be the pair $q=(H^{q},E^{q})$ defined by
letting, for each $n\in\omega$
$H^{q}(n)=\varphi(g_{0},...,g_{i}/h(i),...,g(n))$ where
$H^{p}(n)=\varphi(g_{0},...,g_{n})$ and
$E^{q}=E^{p}\upharpoonright\bigcup\\{[i]_{E^{p}}\;|\;i\in A^{p}\setminus
u\\}$.

  1. (1)

[20, Claim 1.7, (2)] $p^{[h]}$ is a condition in
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ extending $p$ and the set of
$\\{p^{[h]}\;|\;h\in 2^{u}\\}$ is predense below $p$.

  2. (2)

[20, Claim 1.8]. If $u$ is a the set of the first $n$ elements of $A^{p}$ and
$D$ is a dense subset of ${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ then there is
a $q\in{\mathord{\mathbb{Q}}}_{\mathcal{I}}$ so that $q\leq_{n}p$ and
$q^{[h]}\in D$ for any $h\in 2^{u}$.

Finally we will need the following game to analyze fusions of conditions from
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$.

######  Definition 9.4 (The Game $\mathsf{GM}_{\mathcal{I}}(E)$).

The game $\mathsf{GM}_{\mathcal{I}}(E)$ is played as follows. On the $n^{\rm
th}$-move, the first player chooses an $\mathcal{I}$-equivalence relation
$E^{1}_{n}\leq_{\mathcal{I}}E^{2}_{n-1}$ ($E^{1}_{0}=E$), and the second
played chooses an $\mathcal{I}$-equivalence relation
$E^{2}_{n}\leq_{\mathcal{I}}E^{1}_{n}$. After $\omega$ many moves the second
player wins if and only if $\bigcup_{n>0}{\rm dom}(E^{1}_{n})\setminus{\rm
dom}(E^{2}_{n})\in\mathcal{I}$.

######  Remark 4.

If the second player wins in the game $\mathsf{GM}_{\mathcal{I}}(E)$ then the
game is invariant to taking subsets. That is, the game is invariant to taking
$\leq_{\mathcal{I}}$-extensions $\\{E^{2,*}_{n}\\}_{n\in\omega}$ with ${\rm
dom}(E^{2,*}_{n})\subseteq{\rm dom}(E^{2}_{n})$.

######  Lemma 9.5 (Claim 1.10 (1) of [20]).

The Game $\mathsf{GM}_{\mathcal{I}}(E)$ is not determined for a maximal ideal
$\mathcal{I}$.

Putting all of this together we can now show the following.

######  Theorem 9.6.

For any maximal ideal $\mathcal{I}$ and any tight eventually different family
$\mathcal{E}$ the forcing notion ${\mathord{\mathbb{Q}}}_{\mathcal{I}}$
strongly preserves the tightness of $\mathcal{E}$.

###### Proof.

Let $p\in{\mathord{\mathbb{Q}}}_{\mathcal{I}}$, $M$ a countable elementary
submodel of $H_{\theta}$ for $\theta$ sufficiently large such that
$\mathcal{I},\mathcal{E},p\in M$ and let $g\in\mathcal{E}$ densely diagonalize
every $T\in\mathcal{I}_{T}(\mathcal{E})^{+}\cap M$. We fix an enumeration
$\\{D_{n}\;|\;n\in\omega\\}$ of all open dense subsets of
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ that are in $M$, and an enumeration
$\\{\dot{T}_{n}\;|\;n\in\omega\\}$ of all
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$-names for elements of
$\mathcal{I}_{T}(\mathcal{E})^{+}$ that are in $M$. Let
$\varphi:\omega\to\omega^{2}$ with coordinate functions $\varphi_{0}$ and
$\varphi_{1}$.

We define a strategy for the first player in the game
$\mathsf{GM}_{\mathcal{I}}(E)$, which cannot be winning in all rounds.

We set $p_{0}=q_{0}=p$ and $u_{0}=\emptyset$. We assume that the first player
has chosen $E^{1}_{n}$, $q_{n}$, $p_{n}$, $u_{n}$, and the second one an
$E^{2}_{n}$. We give instructions to choose $E^{1}_{n+1}$, $q_{n+1}$,
$p_{n+1}$, $u_{n+1}$. We begin with $q_{n+1}$:

  1. (1)

${\rm dom}E^{q_{n+1}}={\rm dom}E^{p_{n}}$,

  2. (2)

$xE^{q_{n+1}}y$ if and only if one of the following holds:

    1. (a)

$xE^{2}_{n}y$.

    2. (b)

There is $k\in u_{n}$ with $x,y\in[k]_{E^{p_{n}}}$ and $x,y\not\in{\rm
dom}E^{2}_{n}$.

    3. (c)

There are $k_{0},k_{1}\not\in\bigcup\\{[i]_{E^{p_{n}}}\;|\;i\in u_{n}\\}$ with
$x\in[k_{0}]_{E^{p_{n}}}$, $y\in[k_{1}]_{E^{p_{n}}}$ and
$k_{0},k_{1}\not\in{\rm dom}E^{2}_{n}$.

  3. (3)

$H^{q_{n+1}}$ is chosen such that:

    1. (a)

If $l\in\omega\setminus{\rm dom}E^{p_{n}}$ then
$H^{q_{n+1}}(l)=^{**}H^{p_{n}}(l)$.

    2. (b)

If $l\in{\rm dom}E^{p_{n}}\setminus A^{q_{n+1}}$, $H^{p_{n}}(l)=g_{i}$ then
$H^{q_{n+1}}(l)=H^{q_{n+1}}(i)$.

    3. (c)

If $l\in{\rm dom}E^{p_{n}}\setminus A^{q_{n+1}}$, $H^{p_{n}}(l)=1-g_{i}$ then
$H^{q_{n+1}}(l)=1-H^{q_{n+1}}(i)$.

    4. (d)

If $l\in A^{p_{n}}\setminus A^{q_{n+1}}$ then
$H^{q_{n+1}}(l)=^{**}H^{p_{n}}({\rm min}[l]_{E^{q_{n+1}}})$.

Note that for the already defined condition $q_{n+1}$ we have
$q_{n+1}\leq_{n}p_{n}$. Take $u_{n+1}=u_{n}\cup\\{\min(A^{q_{n+1}}\setminus
u_{n})\\}$. Let $D^{\prime}_{n}$ be the set of all $r\leq p$ so that

  1. (1)

$r$ decides the $\varphi_{1}(n)^{\rm th}$-node of $T_{\varphi_{0}(n)}$ to be
some $\check{s}_{n}$

  2. (2)

There is a $k<\omega$ and a $t\supseteq s_{n}$ so that $k\in{\rm
dom}(t)\setminus{\rm dom}(s_{n})$ and
$r\Vdash\check{t}\in\dot{T}_{\varphi_{0}(n)}$ and $g(k)=t(k)$.

As in the previous proofs, the set $D^{\prime}_{n}$ is open dense below $p$
(and also below $q_{n+1}$). Then $D^{\prime}_{n}\cap D_{n}$ is dense below
$q_{n+1}$. Therefore we can apply Lemma 9.3 to obtain
$p_{n+1}\leq_{n+1}q_{n+1}$ such that for each $h\in{}^{u_{n+1}}\\{0,1\\}$, the
condition $p^{[h]}_{n+1}\in D^{\prime}_{n}\cap D_{n}\cap M$. In particular, if
$h\in 2^{u_{n+1}}$ then $p_{n+1}^{[h]}$ decides the $\varphi_{1}(n)^{\rm th}$
node of $\dot{T}_{phi_{0}(n)}$ to be some $s_{n}$, forces that there is a a
$k<\omega$ and a $t\supseteq s_{n}$ so that $k\in{\rm dom}(t)\setminus{\rm
dom}(s_{n})$ and $g(k)=t(k)$ and $p_{n+1}^{[h]}\in D_{n}\cap M$. It follows
that $p_{n+1}\Vdash$“there is a $t$ in $\dot{T}_{\varphi_{0}(n)}$ extending
the $\varphi_{1}(n)^{\rm th}$ node and a $k\in{\rm dom}(t)$ above the domain
of the $\varphi_{1}(n)^{\rm th}$ node of $\dot{T}_{\varphi_{0}(n)}$ so that
$t(k)=g(k)$”. Finally, we set

$E^{1}_{n+1}=E^{p_{n+1}}\upharpoonright({\rm
dom}E^{p_{n+1}}\setminus\bigcup\\{[i]_{E^{p_{n+1}}}\;|\;i\in u_{n+1}\\}).$

We define a fusion $q$ of a sequence $\langle p_{n}\;|\;n\in\omega\rangle$.
Relation $E^{q}$ has ${\rm dom}E^{q}=\bigcap\\{{\rm
dom}E^{p_{n}}\;|\;n\in\omega\\}$, and $xE^{q}y$ if for every $n$ large enough,
$xE^{p_{n}}y$. Function $H^{q}$ is equal to $H^{p_{n}}$ for large enough $n$.
In order to guarantee $q\in{\mathord{\mathbb{Q}}}_{\mathcal{I}}$, it is
necessary to choose a play with the first player using the described strategy,
but he looses. Thus the second player wins and by Remark 4, we can assume that
${\rm min}({\rm dom}E^{2}_{n})>\max u_{n+1}$. Consequently, ${\rm
dom}E^{p_{n}}\setminus{\rm
dom}E^{2}_{n}\subseteq\bigcup\\{[k]_{E^{q_{n+1}}}\;|\;k\in u_{n+1}\\}$, and
thus ${\rm dom}E^{q}\in\mathcal{I}^{*}$. One can check that other properties
for $q\in{\mathord{\mathbb{Q}}}_{\mathcal{I}}$ are satisfied by the definition
of $q$.

Finally, condition $q$ is ($M$,
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$)-generic, and $q\leq_{n}p_{n}$ for each
$n$. Hence, we have $q$ forces that $g$ densely diagonalizes every
$\dot{T}_{n}$.

∎

As always we have the same result for families of permutations.

######  Theorem 9.7.

For any maximal ideal $\mathcal{I}$ and any tight eventually different set of
permutations $\mathcal{P}$ the forcing notion
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ strongly preserves the tightness of
$\mathcal{P}$.

Putting all of this together with the fact that
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ has the Sacks property and iterating
${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ with countable support $\omega_{2}$
many times alongside some bookkeeping device to handle the ideals
$\mathcal{I}$ forces $\mathfrak{i}=\mathfrak{a}<\mathfrak{u}$ we get the
following.

######  Theorem 9.8.

In the iterated ${\mathord{\mathbb{Q}}}_{\mathcal{I}}$ model we have
$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{i}=cof(\mathcal{N})<\mathfrak{u}$.

##  10\. Definability

In this section we show that if $V=L$ there is a $\Pi^{1}_{1}$ tight
eventually different family and a $\Pi^{1}_{1}$ tight eventually set of
permutations. The construction is extremely similar to Miller’s seminal proof
that there is a $\Pi^{1}_{1}$ MAD family in $L$, [16, Theorem 8.23]. From our
construction we are able to conclude that in all of the models considered
above if the ground model is $L$ then there is are $\Pi^{1}_{1}$ tight
witnesses to $\mathfrak{a}_{e}=\aleph_{1}$ and $\mathfrak{a}_{p}=\aleph_{1}$.
This strengthens the main result of [7] where it is shown that if $V=L$ then
there is a $\Pi^{1}_{1}$ Sacks indestructible maximal eventually different
family.

######  Theorem 10.1.

If $V=L$ then there is a $\Pi^{1}_{1}$ tight eventually different family.

As noted above the proof of this theorem is extremely similar to [16, Theorem
8.23]. In our opinion this highlights part of the appeal of tightness for sets
of functions: it provides a uniform, relatively simple construction of a
$\Pi^{1}_{1}$ maximal eventually different family which remains maximal (and
$\Pi^{1}_{1}$) in an extension of $L$ by a large number of posets.

As in [16, Lemma 8.24] the proof relies on the following coding lemma. For
this section for each $n<\omega$ let $c_{n}\in\omega^{\omega}$ be the function
with constant value $n$.

######  Lemma 10.2.

Let $\mathcal{E}_{0}$ be a countable, eventually different family which
contains each $c_{n}$. Let $\\{T_{n}\;|\;n<\omega\\}$ be a countable
collection of trees in $\mathcal{I}_{T}(\mathcal{E}_{0})^{+}$. Let
$z\in[\omega]^{\omega}$ be arbitrary. There is a function
$f\in\omega^{\omega}$ which is eventually different from each
$g\in\mathcal{E}_{0}$, densely diagonalizes $T_{n}$ for each $n$ and computes
$z$. Moreover such an $f$ can be found computably from $\mathcal{E}_{0}$,
$\\{T_{n}\;|\;n<\omega\\}$ and $z$.

###### Proof.

Fix $\mathcal{E}_{0}$, $\\{T_{n}\;|\;n<\omega\\}$ and $z\in[\omega]^{\omega}$
as in the statement of the lemma. Enumerate
$\mathcal{E}_{0}=\\{g_{k}\;|\;k<\omega\\}$. Fix a bijection
$\varphi:\omega\to\omega^{2}$ with coordinate functions $\varphi_{0}$ and
$\varphi_{1}$. We define $f$ in stages. Namely we define a
$\subseteq$-increasing sequence $\langle f_{i}\;|\;i<\omega\\}$ so that the
following hold.

  1. (1)

For all $i<\omega$ $f_{i}$ is a finite partial function from $\omega$ to
$\omega$ so that $i\in{\rm dom}(f_{i})$.

  2. (2)

For all $i<\omega$ there is an $s\in T_{\varphi_{0}(i)}$ extending the
$\varphi_{1}(i)^{\rm th}$ node of $T_{\varphi_{0}(i)}$, call it $t_{i}$ and a
$k\in{\rm dom}(s)\setminus{\rm dom}(t_{i})$ so that $f_{i+1}(k)=s(k)$ (and in
particular is defined).

  3. (3)

If $i<j<\omega$ then $f_{i}\supseteq f_{j}\cap g_{i}$.

  4. (4)

For every $n<\omega$ for all $k>n$ $n\notin{\rm range}(f_{k})$ and $n\in z$ if
and only if $|c_{n}\cap f_{n}|$ is even.

Assuming we can define such a sequence, we let $f=\bigcup_{i<\omega}f_{i}$.
This $f$ is as needed since, by 1 it is a function, by 2 it densely
diagonalizes every $T_{n}$, by 3 it is eventually different from every element
of $\mathcal{E}_{0}$ and by $4$ it computes $z$.

We define $\\{f_{i}\;|\;i<\omega\\}$ by recursion as follows. At stage $0$,
let $f_{0}=\\{(0,0)\\}$ if $0\notin z$ and let $f_{0}=\\{(0,0),(1,0)\\}$ if
$0\in z$. Now suppose we have defined $f_{n}$. Let $k$ be the least number not
in the set $\\{g_{0}(n+1),...,g_{n}(n+1),0,...,n\\}$. First, if $n+1\notin{\rm
dom}(f_{n})$ let $f_{n}^{\prime}=f_{n}\cup\\{(n+1,k)\\}$. Now, let $t_{n}$ be
the $\varphi_{1}(n)^{\rm st}$ node of $T_{\varphi_{0}(n)}$. Let $s_{n}$ be
least in the lexicographic ordering extending $t_{n}$ in $T_{\varphi_{0}(n)}$
so that there is a $k\in{\rm dom}(s_{n})\setminus{\rm dom}(t_{n})$ so that
$k\notin{\rm dom}(f_{n}^{\prime})$, and
$s_{n}(k)\notin\\{g_{0}(n+1),...,g_{n}(n+1),0,...,n\\}$. That such an $s_{n}$
and $k$ exists follows by the fact that
$T_{\varphi_{0}(n)}\in\mathcal{I}_{T}(\mathcal{E}_{0})^{+}$. Let
$f_{n}^{\prime\prime}=f_{n}^{\prime}\cup\\{(k,s_{n}(k))\\}$. If $n+1\in z$ and
$|c_{n+1}\cap f^{\prime\prime}_{n}|$ is even or $n+1\notin z$ and
$|c_{n+1}\cap f^{\prime\prime}_{n}|$ is odd then we let
$f_{n+1}=f_{n}^{\prime\prime}$. Otherwise, let $l$ be the least element so
that $l\notin{\rm dom}(f^{\prime\prime}_{n})$ and $g_{i}(l)\neq n+1$ for all
$i<n+1$. That such an $l$ exists follow from the fact that $\mathcal{E}_{0}$
is an eventually different family containing every constant function. Now let
$f_{n+1}=f_{n}^{\prime\prime}\cup\\{(l,n+1)\\}$. This completes the
construction and hence the lemma. ∎

The proof of Theorem 10.1 follows along the lines described in [16, p.194]
given this coding lemma. We give the details below for completeness.

###### Proof of Theorem 10.1.

Assume $V=L$. We will build a tight eventually different family
$\mathcal{E}=\\{f_{\alpha}\;|\;\alpha<\omega_{1}\\}$ recursively. For each
finite $n<\omega$ let $f_{n}=c_{n}$. Now suppose we have constructed
$\mathcal{E}_{\alpha}:=\\{f_{\xi}\;|\;\xi<\alpha\\}$ for some
$\alpha<\omega_{1}$. Let $\vec{T}_{\alpha}$ be the $\leq_{L}$-least countable
sequence of trees in $\mathcal{I}_{T}(\mathcal{E}_{\alpha})^{+}$. We construct
$f_{\alpha+1}$ as in the coding lemma with $\mathcal{E}_{\alpha}$ our
countable eventually different family, $\vec{T}_{\alpha}$ our sequence of
trees and $z$ a real coding an extensional, wellfounded relation
$E\subseteq\omega\times\omega$ and $\alpha$ so that $(\omega,E)\cong
L_{\alpha}$. This completes the construction. It just remains to check that it
works.

It’s not hard to show that the family $\mathcal{E}$ is tight. To see this,
suppose $\mathcal{E}$ were not tight and let $\vec{T}$ be the $\leq_{L}$-least
countable sequence of trees in $\mathcal{I}_{T}(\mathcal{E})^{+}$ which are
not densely diagonalized by any element of $\mathcal{E}$. Let $M\prec
L_{\omega_{2}}$ be a countable elementary submodel containing $\mathcal{E}$
and $\vec{T}$. Let $L_{\delta}$ be its transitive collapse. It follows that
$L_{\delta}\models$“$\vec{T}$ is the $\leq_{L}$-least counter example to
tightness of $\mathcal{E}$” which means that at stage
$(\omega_{1})^{L_{\delta}}$ we added a real to $\mathcal{E}$ densely
diagonalizing $\vec{T}$, contradiction.

What is less clear is that $\mathcal{E}$ is $\Pi^{1}_{1}$. To see this observe
that $X\in\mathcal{E}$ if and only if $X$ codes an extensional, wellfounded
relation $E\subseteq\omega\times\omega$ and a ordinal $\alpha$ so that
$(\omega,E)\cong L_{\alpha}$ in the way described in the coding lemma and
$L_{\alpha+\omega}\models X\in\mathcal{E}$ where the expression
“$X\in\mathcal{E}$” is shorthand for the definition of $\mathcal{E}$ given
recursively above (relativized to $L_{\alpha+\omega}$). For more details on
this argument the reader is referred to [16]. ∎

Note that in contrast to the case of MAD families this is not the best
possible complexity for a maximal eventually different family since
$\mathsf{ZF}$ proves that there is a Borel maximal eventually different family
([12]) and in fact even a closed one ([19]). However, we will apply this
result to obtain models where the continuum hypothesis fails and there is
still a $\Pi^{1}_{1}$ maximal eventually different family of size
$\aleph_{1}$, which will be the best possible complexity for that cardinality.
Moreover, in light of Theorem 3.3, this is the best possible complexity for a
tight maximal eventually different family (of functions or permutations).

The same argument works in the case of tight eventually different sets of
permutations and tight eventually different families. The only point that is
different is that in place of the $c_{n}$’s we need a computable set of
disjoint permutations covering $\omega_{2}$ (in the case of permutations) or
an infinite computable partition of $\omega$ into infinite sets.

######  Theorem 10.3.

If $V=L$ there are a $\Pi^{1}_{1}$ tight eventually different set of
permutations and a tight MAD family.

As a result of Theorems 10.1 and 10.3 we have the following.

######  Theorem 10.4.

The following constellations of cardinal characteristics are compatible with
the existence of a $\Pi^{1}_{1}$ tight maximal eventually different family of
functions, a $\Pi^{1}_{1}$ tight eventually different family of permutations
and a $\Pi^{1}_{1}$ tight MAD family.

  1. (1)

$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{d}=\mathfrak{a}_{T}<2^{\aleph_{0}}$

  2. (2)

$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}<\mathfrak{d}=\mathfrak{a}_{T}=2^{\aleph_{0}}$

  3. (3)

$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{d}<\mathfrak{a}_{T}=2^{\aleph_{0}}$

  4. (4)

$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{d}=\mathfrak{u}<non(\mathcal{N})=cof(\mathcal{N})=2^{\aleph_{0}}$.
In this case we can also arrange that the witness to $\mathfrak{u}$ is
$\Pi^{1}_{1}$.

  5. (5)

$\mathfrak{a}=\mathfrak{a}_{e}=\mathfrak{a}_{p}=\mathfrak{i}=cof(\mathcal{N})<\mathfrak{u}$.
In this case we can also arrange that the witness to
$\mathfrak{i}=\aleph_{1}$.

###### Proof.

For each case we simply add $\omega_{2}$ many reals over $L$ with countable
support of a given type. For item 1 we use Sacks forcing, for item 2 we use
Miller forcing, for item 3 we use partition forcing, for item 4 we use
$\mathbb{EE}$ and for item 5 we use ${\mathord{\mathbb{Q}}}_{\mathcal{I}}$.
For the second to final model note that there is a $\Pi^{1}_{1}$ basis for a
P-point in $L$ ([18]). For the final model note that there is a $\Pi^{1}_{1}$
maximal independent family in $L$ which is preserved by countable support
iterations of the form ${\mathord{\mathbb{P}}}(\mathcal{K})$, see [4]. ∎

##  11\. Concluding remarks

The results of the previous sections show that tightness for eventually
different family and an eventually different set of permutations is a robust
notion and can provide a lot of inside into studying these types of maximal
sets of reals in much the same way as tight MAD families provide insight into
questions about $\mathfrak{a}$.

A natural line of inquiry is whether we can define natural notions of
tightness for other relatives of $\mathfrak{a}$. An obvious question in this
respect is the following.

######  Question 2.

Is there an analogy of tightness for $\mathfrak{a}_{T}$ or for
$\mathfrak{a}_{g}$?

A positive answer to this problem for $\mathfrak{a}_{g}$ would presumably
strengthen the main result of [8] in much the same way that Theorem 10.1
strengthens the main result of [7].

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