# Characterizing existence of certain ultrafilters

Rafał Filipów<EMAIL_ADDRESS>http://mat.ug.edu.pl/ rfilipow ,
Krzysztof Kowitz<EMAIL_ADDRESS>and  Adam Kwela
<EMAIL_ADDRESS>http://kwela.strony.ug.edu.pl/

###### Abstract.

Following Baumgartner [J. Symb. Log. 60 (1995), no. 2], for an ideal
$\mathcal{I}$ on $\omega$, we say that an ultrafilter $\mathcal{U}$ on
$\omega$ is an _$\mathcal{I}$ -ultrafilter_ if for every function
$f:\omega\to\omega$ there is $A\in\mathcal{U}$ with $f[A]\in\mathcal{I}$.

If there is an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-ultrafilter, then $\mathcal{I}$ is not below $\mathcal{J}$ in
the Katětov order $\leq_{\operatorname{K}}$ (i.e. for every function
$f:\omega\to\omega$ there is $A\in\mathcal{I}$ with
$f^{-1}[A]\notin\mathcal{J}$). On the other hand, in general
$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$ does not imply that
existence of an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-ultrafilter is consistent.

We provide some sufficient conditions on ideals to obtain the equivalence:
$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$ if and only if it is
consistent that there exists an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-ultrafilter. In some cases when the Katětov order is not enough
for the above equivalence, we provide other conditions for which a similar
equivalence holds. We are mainly interested in the cases when the family of
all $\mathcal{I}$-ultrafilters or $\mathcal{J}$-ultrafilters coincides with
some known family of ultrafilters: P-points, Q-points or selective
ultrafilters (a.k.a. Ramsey ultrafilters). In particular, our results provide
a characterization of Borel ideals $\mathcal{I}$ which can be used to
characterize P-points as $\mathcal{I}$-ultrafilters.

Moreover, we introduce a cardinal invariant which is used to obtain a
sufficient condition for the existence of an $\mathcal{I}$-ultrafilter which
is not a $\mathcal{J}$-ultrafilter. Finally, we prove some new results
concerning existence of certain ultrafilters under various set-theoretic
assumptions.

###### Key words and phrases:

ultrafilter, I-ultrafilter, P-point, Q-point, Ramsey ultrafilter, selective
ultrafilter, Ramsey ultrafilter, ideal, summable ideal, Borel ideal, Katetov
order, cardinal invariant of the continuum, cardinal characteristic of the
continuum

###### 2010 Mathematics Subject Classification:

Primary: 03E17; Secondary: 03E05, 03E35, 03E50, 03E75

##  1\. Introduction

All notions and notations used in the introduction are defined in Section 2.

If there is an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-ultrafilter, then
$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$. On the other hand, in
general $\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$ does not imply
that existence of an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-ultrafilter is consistent. For instance,
$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathrm{conv}$, but
$\mathrm{Fin}^{2}$-ultrafilters and $\mathrm{conv}$-ultrafilters coincide. We
provide some sufficient conditions on ideals to obtain equivalence:
$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$ if and only if it is
consistent that there exists an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-ultrafilter. In some cases when the Katětov order is not enough
for the above equivalence, we provide other conditions for which a similar
equivalence holds. We are mainly interested in the cases when the family of
all $\mathcal{I}$-ultrafilters or $\mathcal{J}$-ultrafilters coincides with
some known family of ultrafilters: P-points, Q-points or selective
ultrafilters.

In a series of papers [15, 17, 19, 20], Flašková constructed
$\mathcal{I}$-ultrafilters which are not $\mathcal{J}$-ultrafilters for
various pairs of ideals $\mathcal{I}$ and $\mathcal{J}$ under various set-
theoretic assumptions. A closer analysis of her constructions allows us to
define a combinatorial “core” of her proofs. Namely, we define a new cardinal
characteristic and show how to use it to prove existence of
$\mathcal{I}$-ultrafilters which are not $\mathcal{J}$-ultrafilters. We prove
some new results of this type.

It is known that some well known families of ultrafilters can be characterized
as $\mathcal{I}$-ultrafilters with the aid of definable (usually Borel) ideals
$\mathcal{I}$. For instance, an ultrafilter is a P-point if and only if it is
a $\mathrm{Fin}^{2}$-ultrafilter. The ideal $\mathrm{Fin}^{2}$ is Borel, in
fact it is $F_{\sigma\delta\sigma}$ and it is known that P-points cannot be
characterized as $\mathcal{I}$-ultrafilters for any $F_{\sigma}$ ideal
$\mathcal{I}$ (see [19, Theorem 3.1]). Our results provide a characterization
of Borel ideals $\mathcal{I}$ which can be used to characterize P-points as
$\mathcal{I}$-ultrafilters.

The paper is organized in the following way. In Section 2, we review the
notions and notations used in the rest of the paper. We also provide some
basic facts needed in the sequel, and we prove those of them that we could not
find in the literature. In Section 3 we introduce a cardinal invariant and an
order which provide us with the main tools that we use in the rest of the
paper. Moreover, we show how those tools can be used to construct specific
$\mathcal{I}$-ultrafilters and characterize existence of certain classes of
$\mathcal{I}$-ultrafilters. In Section 4 we find a lower bound for the above
mentioned invariant in terms of some other cardinal characteristics known from
the literature. Moreover, we discover a class of ideals for which the
following equivalence holds:
$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$ if and only if it is
consistent that there exists an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-ultrafilter. In Section 5 we introduce a game that is designed
to provide one more lower bound for our cardinal invariant in the case when
$\mathcal{I}$ contains a tall summable ideal. This enables us to relax the
set-theoretic assumptions in some of our results. The remaining sections of
the paper are devoted to three well know classes of ultrafilters: P-points,
Q-points and selective ultrafilters. Among others, we characterize existence
of P-points (Q-points, selective ultrafilters, resp.) which are not
$\mathcal{J}$-ultrafilters. It turns out, that in the case of Q-points this
characterization can be expressed solely in terms of the Katětov order. On the
other hand, in the case of P-points (selective ultrafilters, resp.), the
Katětov order is not enough to obtain such a characterization. However, we
managed to show that extendability to $P^{+}$-ideals (selective ideals, resp.)
are enough in these cases. The characterization mentioned in the previous
paragraph is stated in Section 6. Finally, in our last Section 10 we present
some by-products of our studies: we show that every $G_{\delta\sigma}$ ideal
is extendable to an $F_{\sigma}$ ideal and that neither the ideals
$\mathcal{L}$ (generated by lacunary sets) nor the ideal $\mathcal{R}$
(generated by homogeneous sets in some fixed instance of the random graph on
$\omega$) is $\leq_{\operatorname{K}}$-homogeneous.

##  2\. Preliminaries

All the notions and notations used in the paper, but not defined in this
section seem to be quite standard and can be found for instance in [26, 27,
6].

By $\omega$ we denote the set of all natural numbers. We identify a natural
number $n$ with the set $\\{0,1,\dots,n-1\\}$ (for instance, $n\setminus k$
means the set $\\{i\in\omega:k\leq i<n\\}$).

###  2.1. Ideals

A family $\mathcal{I}\subseteq\mathcal{P}(X)$ is called an _ideal on $X$_ if
it satisfies the following conditions:

  1. (1)

if $A,B\in\mathcal{I}$, then $A\cup B\in\mathcal{I}$,

  2. (2)

if $A\subseteq B$ and $B\in\mathcal{I}$, then $A\in\mathcal{I}$,

  3. (3)

$\mathcal{I}$ contains all finite subsets of $X$,

  4. (4)

$X\notin\mathcal{I}$.

A family $\mathcal{F}\subseteq\mathcal{P}(X)$ is called a _filter on $X$_ if
it satisfies the following conditions:

  1. (1)

if $A,B\in\mathcal{F}$, then $A\cap B\in\mathcal{F}$,

  2. (2)

if $A\subseteq B$ and $A\in\mathcal{F}$, then $B\in\mathcal{F}$,

  3. (3)

$\mathcal{F}$ contains all cofinite subsets of $X$,

  4. (4)

$\emptyset\notin\mathcal{F}$.

If $\mathcal{A}\subseteq\mathcal{P}(X)$ then we write
$\mathcal{A}^{*}=\\{B\subseteq X:X\setminus B\in\mathcal{A}\\}$. It is easy to
see that if $\mathcal{I}$ is an ideal then $\mathcal{I}^{*}$ is a filter
(called the _dual filter of $\mathcal{I}$_). Conversely, if $\mathcal{F}$ is a
filter then $\mathcal{F}^{*}$ is an ideal (called the _dual ideal of
$\mathcal{F}$_).

We say that an ideal $\mathcal{I}$ on $X$ is generated by a family
$\mathcal{A}\subseteq\mathcal{P}(X)$ whenever $\mathcal{I}=\\{A\subseteq
X:\exists F\in[\mathcal{A}]^{<\omega}(A\subseteq\bigcup F)\\}$. For an ideal
$\mathcal{I}$ on $X$, we write $\mathcal{I}^{+}=\\{A\subseteq
X:A\notin\mathcal{I}\\}$ and call it the _coideal of $\mathcal{I}$_. The ideal
of all finite subsets of an infinite set $X$ is denoted by $\mathrm{Fin}(X)$
(we write $\mathrm{Fin}$ instead of $\mathrm{Fin}(\omega)$ for short). For an
ideal $\mathcal{I}$ on $X$ and $A\subseteq X$ we define
$\mathcal{I}\restriction A=\\{B\cap A:B\in\mathcal{I}\\}$. It is easy to see
that $\mathcal{I}\restriction A$ is an ideal on $A$ if and only if
$A\notin\mathcal{I}$. An ideal $\mathcal{I}$ on $X$ is _tall_ if for every
infinite $A\subseteq X$ there is an infinite $B\in\mathcal{I}$ such that
$B\subseteq A$ (some authors use the name _dense ideal_ in this case). An
ideal $\mathcal{I}$ on $X$ is a _P-ideal_ (_weak P-ideal_ , resp.) if for any
countable family $\mathcal{A}\subseteq\mathcal{I}$ there is
$B\in\mathcal{I}^{*}$ ($B\in\mathcal{I}^{+}$, resp.) such that $A\cap B$ is
finite for every $A\in\mathcal{A}$. An ideal $\mathcal{I}$ on $X$ is a
_$P^{+}$ -ideal_ (_selective ideal_ , resp.) if for any decreasing sequence
$\\{A_{n}:n\in\omega\\}\subseteq\mathcal{I}^{+}$ there is
$A\in\mathcal{I}^{+}$, $A\subseteq A_{0}$ such that $A\cap(A_{n}\setminus
A_{n+1})$ is finite ($|A\cap(A_{n}\setminus A_{n+1})|\leq 1$, resp.) for every
$n$.

For $A\subseteq\omega\times\omega$ and $n\in\omega$, we write
$A_{(n)}=\\{k\in\omega:(n,k)\in A\\}$ i.e. $A_{(n)}$ is the vertical section
of $A$ at the point $n$.

In the sequel, we will use the following ideals.

  * •

$\mathrm{Fin}^{2}=\\{A\subseteq\omega\times\omega:\forall^{\infty}n\,(|A_{(n)}|<\omega)\\}$.

  * •

$\mathcal{ED}=\\{A\subseteq\omega\times\omega:\exists
m\,\forall^{\infty}n\,(|A_{(n)}|<m)\\}$.

  * •

$\mathcal{ED}_{\mathrm{Fin}}=\\{A\subseteq\Delta:\exists m\,\forall
n\,(|A_{(n)}|<m)\\}$, where $\Delta=\\{(n,k)\in\omega^{2}:n\geq k\\}$.

  * •

The _summable ideals_ : $\mathcal{I}_{g}=\\{A\subseteq\omega:\sum_{n\in
A}g(n)<\infty\\}$, where $g:\omega\to[0,\infty)$ satisfies
$\sum_{n\in\omega}g(n)=\infty$.

  * •

$\mathrm{nwd}=\\{A\subseteq\mathbb{Q}\cap[0,1]:\text{$A$ is nowhere
dense}\\}$.

  * •

$\mathrm{conv}=\\{A\subseteq\mathbb{Q}\cap[0,1]:\text{$A$ has at most finitely
many limit points}\\}$.

  * •

$\mathcal{R}$ is the ideal generated by homogeneous sets in some fixed
instance of the random graph on $\omega$ (see e.g. [33, p.29] for details).

  * •

$\mathcal{W}$ is the ideal of all subsets of $\omega$ which does not contain
arbitrary long finite arithmetic progression.

Let $\mathcal{I},\mathcal{J}$ be ideals on $X$ and $Y$ respectively. We write
$\mathcal{J}\approx\mathcal{I}$ if there is a bijection $f:X\to Y$ such that
$f^{-1}[B]\in\mathcal{I}\iff B\in\mathcal{J}$, and say that $\mathcal{I}$ and
$\mathcal{J}$ are _isomorphic_.

One can think about all of the above ideals as ideals on $\omega$ by
identifying the domain set of the ideal (for instance, $\omega\times\omega$ or
$\mathbb{Q}\cap[0,1]$) with $\omega$ via a fixed bijection.

If $\mathcal{I}\not\approx\mathrm{Fin}$ and
$\mathcal{I}\not\approx\mathrm{Fin}\oplus\mathcal{P}(\omega)=\\{A\subseteq\omega\times
2:(\omega\times\\{0\\})\cap A\text{ is finite}\\}$, then we define
$\operatorname{\operatorname{add}^{*}}(\mathcal{I})=\min\\{|\mathcal{A}|:\mathcal{A}\subseteq\mathcal{I}\land\neg(\exists
B\in\mathcal{I}\,\forall A\in\mathcal{A}\,(|A\setminus B|<\omega)\\}$. It is
easy to see that $\operatorname{\operatorname{add}^{*}}(\mathcal{I})=\omega$
for non P-ideals and
$\operatorname{\operatorname{add}^{*}}(\mathcal{I})\geq\omega_{1}$ for
P-ideals.

###  2.2. Borel ideals

An ideal on $\omega$ is called _$F_{\sigma}$_ ($\mathbf{\Sigma}^{0}_{\alpha}$,
Borel, analytic and so on, resp.) if it is an $F_{\sigma}$
($\mathbf{\Sigma}^{0}_{\alpha}$, Borel, analytic and so on, resp.) subset of
$\mathcal{P}(\omega)$ with a topology induced from the Cantor space
$\\{0,1\\}^{\omega}$ by identifying subsets of $\omega$ with their
characteristic functions.

A map $\phi:\mathcal{P}(\omega)\to[0,+\infty]$ is a _submeasure on $\omega$_
if for every $A,B\subseteq\omega$ we have: $\phi(\emptyset)=0$ and
$\phi(A)\leq\phi(A\cup B)\leq\phi(A)+\phi(B)$. A submeasure $\phi$ is _lower
semicontinuous_ (in short: lsc) if $\phi(A)=\lim_{n\to\infty}\phi(A\cap n)$
for all $A\subseteq\omega$. For any lsc submeasure $\phi$, we define the
following families: $\mathrm{Fin}(\phi)=\\{A\subseteq\omega:\phi(A)<\infty\\}$
and $\mathrm{Exh}(\phi)=\\{A\subseteq\omega:\lim_{n\to\infty}\phi(A\setminus
n)=0\\}$.

######  Theorem 2.1 ([32] and [34]).

  1. (1)

If $\mathcal{I}$ is an $F_{\sigma}$-ideal, then there is a lsc submeasure
$\phi$ such that $\mathcal{I}=\mathrm{Fin}(\phi)$.

  2. (2)

If $\mathcal{I}$ is an analytic P-ideal, then there is a lsc submeasure $\phi$
such that $\mathcal{I}=\mathrm{Exh}(\phi)$.

  3. (3)

If $\mathcal{I}$ is an $F_{\sigma}$ P-ideal, then there is a lsc submeasure
$\phi$ such that $\mathcal{I}=\mathrm{Exh}(\phi)=\mathrm{Fin}(\phi)$.

It is not difficult to see that each summable ideal $\mathcal{I}_{g}$ is an
$F_{\sigma}$ P-ideal, and it is tall iff $\lim_{n\to\infty}g(n)=0$.

######  Proposition 2.2 ([25, Proposition 4.5]).

Each tall analytic P-ideal contains a tall summable ideal. In particular, an
ideal contains a tall summable ideal if and only if it contains a tall
analytic P-ideal.

###  2.3. Katětov order

By $Y^{X}$ we denote the family of all functions from $X$ into $Y$. A function
$f\in Y^{X}$ is _finite-to-one_ if the inverse image $f^{-1}[\\{y\\}]$ is
finite for every $y\in Y$.

Let $\mathcal{I},\mathcal{J}$ be ideals on $X$ and $Y$ respectively. For
$\mathcal{F}\subseteq Y^{X}$, we write
$\mathcal{J}\leq_{\mathcal{F}}\mathcal{I}$ if there is a function
$f\in\mathcal{F}$ such that $f^{-1}[B]\in\mathcal{I}$ for each
$B\in\mathcal{J}$. In particular, we write

  1. (1)

$\mathcal{J}\leq_{\operatorname{K}}\mathcal{I}$ if $\mathcal{F}=Y^{X}$
($\leq_{\operatorname{K}}$ is called the _Katětov order_),

  2. (2)

$\mathcal{J}\leq_{\operatorname{KB}}\mathcal{I}$ if $\mathcal{F}$ is the
family of all finite-to-one functions from $X$ to $Y$
($\leq_{\operatorname{KB}}$ is called the _Katětov-Blass order_),

  3. (3)

$\mathcal{J}\leq_{\operatorname{P}}\mathcal{I}$ if $\mathcal{F}$ is the family
of all one-to-one functions from $X$ to $Y$,

  4. (4)

$\mathcal{J}\sqsubseteq\mathcal{I}$ if $\mathcal{F}$ is the family of all
bijections from $X$ to $Y$ (in this case we say that $\mathcal{I}$ contains an
isomorphic copy of $\mathcal{J}$).

In the sequel, we will be particularly interested in three families of
functions:

  * •

$\omega^{\omega}$ – the family of all functions $f:\omega\to\omega$

  * •

$\mathrm{Fin}-1$ – the family of all finite-to-one functions
$f:\omega\to\omega$,

  * •

$1-1$ – the family of all one-to-one functions $f:\omega\to\omega$.

Moreover, as we can think about ideals on countable sets as ideals on
$\omega$, we will sometimes write
$\mathcal{I}\leq_{\omega^{\omega}}\mathcal{J}$,
$\mathcal{I}\leq_{\mathrm{Fin}-1}\mathcal{J}$ or
$\mathcal{I}\leq_{1-1}\mathcal{J}$ even if $\mathcal{I}$ and $\mathcal{J}$ are
defined on other countable sets than $\omega$.

######  Proposition 2.3.

  1. (1)

$\mathcal{ED}_{\mathrm{Fin}}\not\leq_{\operatorname{K}}\mathrm{Fin}^{2}$.

  2. (2)

$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathcal{I}$ for each P-ideal
$\mathcal{I}$.

  3. (3)

$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathcal{I}$ for each
$P^{+}$-ideal $\mathcal{I}$.

  4. (4)

$\mathcal{ED}\not\leq_{\operatorname{K}}\mathcal{I}$ for each selective ideal
$\mathcal{I}$.

  5. (5)

$\mathcal{I}\not\leq_{\operatorname{K}}\mathrm{Fin}^{2}$ for each tall P-ideal
$\mathcal{I}$.

  6. (6)

$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{ED}_{\mathrm{Fin}}$ for each
tall P-ideal $\mathcal{I}$.

###### Proof.

(1) See [24]. (2) See [11, Observation 2.3] (3) See [10, Corollary 6.7].

(4) Take any $f:\omega\to\omega\times\omega$ and define
$A_{n}=f^{-1}[(\omega\setminus n)\times\omega]$ for each $n$. If
$A_{n}\in\mathcal{I}$ for some $n$, then
$f^{-1}[n\times\omega]\notin\mathcal{I}$ and the proof is finished. Suppose
that $A_{n}\notin\mathcal{I}$ for each $n$. Since $\mathcal{I}$ is selective,
there is $A\notin\mathcal{I}$ such that $|A\cap(A_{n}\setminus A_{n+1})|\leq
1$ for each $n$. Then $B=f[A]\in\mathcal{ED}$ and
$f^{-1}[B]\notin\mathcal{I}$, so the proof is finished.

(5) Suppose that $\mathcal{I}\leq_{\operatorname{K}}\mathrm{Fin}^{2}$. Let
$f:\omega\times\omega\to\omega$ be such that $f^{-1}[B]\in\mathrm{Fin}^{2}$
for each $B\in\mathcal{I}$. We have 2 cases.

Case (1). The set $f[\\{n\\}\times\omega]$ is finite for infinitely many $n$.
Let $C=\\{n\in\omega:f[\\{n\\}\times\omega]\text{\ is finite}\\}$. Then for
each $n\in C$ there is $k_{n}\in f[\\{n\\}\times\omega]$ such that
$(f^{-1}[\\{k_{n}\\}])_{(n)}$ is infinite. Let $A=\\{k_{0},k_{1},\dots\\}$.
Since $f^{-1}[A]\notin\mathrm{Fin}^{2}$, we have $A\notin\mathcal{I}$. In
particular, $A$ has to be infinite. Since $\mathcal{I}$ is tall, there is an
infinite set $B\in\mathcal{I}$ with $B\subseteq A$. But then
$f^{-1}[B]\notin\mathrm{Fin}^{2}$, a contradiction.

Case (2). The set $f[\\{n\\}\times\omega]$ is infinite for all but finitely
many $n$. Let $C=\\{n\in\omega:f[\\{n\\}\times\omega]\text{\ is infinite}\\}$.
Since $\mathcal{I}$ is tall, for each $n\in C$ there is an infinite set
$B_{n}\in\mathcal{I}$, with $B_{n}\subseteq f[\\{n\\}\times\omega]$. Observe
that the set $(f^{-1}[B_{n}])_{(n)}$ is infinite for each $n\in C$.

Since $\mathcal{I}$ is a P-ideal, there is $B\in\mathcal{I}$ such that
$B_{n}\setminus B$ is finite for each $n$.

As $f^{-1}[B]\in\mathrm{Fin}^{2}$, the set $(f^{-1}[B])_{(n)}$ is finite for
all but finitely many $n$, say for all $n\in D$.

All in all, the set $(f^{-1}[B_{n}\setminus
B])_{(n)}=(f^{-1}[B_{n}])_{(n)}\setminus(f^{-1}[B])_{(n)}$ is infinite for all
$n\in C\cap D$.

Since $B_{n}\setminus B$ is finite, for each $n\in C\cap D$ there is $c_{n}\in
B_{n}\setminus B$ such that $(f^{-1}[\\{c_{n}\\}])_{(n)}$ is infinite.

Let $A=\\{c_{n}:n\in C\cap D\\}$. Note that $C\cap D$ is infinite. Thus,
$f^{-1}[A]\notin\mathrm{Fin}^{2}$ and consequently $A\notin\mathcal{I}$. In
particular, $A$ is infinite. Since $\mathcal{I}$ is tall, there is an infinite
set $A^{\prime}\in\mathcal{I}$ with $A^{\prime}\subseteq A$. But then
$f^{-1}[A^{\prime}]\notin\mathrm{Fin}^{2}$, a contradiction.

(6) Fix any $f:\Delta\to\omega$, where $\Delta=\\{(i,j)\in\omega^{2}:\ j\leq
i\\}$ (i.e., $\mathcal{ED}_{\mathrm{Fin}}=\mathcal{ED}\restriction\Delta$).
Suppose to the contrary that $f$ witnesses
$\mathcal{I}\leq_{\operatorname{K}}\mathcal{ED}_{\mathrm{Fin}}$.

Observe that for each $n\in\omega$ and an infinite $A\subseteq\omega$ one can
find $k>n$ such that $f[(A\times\\{k\\})\cap\Delta]$ is infinite. Indeed,
otherwise we can inductively pick sets $D_{i}\in[\omega]^{\omega}$, for $i>n$,
such that for each $i>n$ we have:

  * •

$D_{i+1}\subseteq D_{i}$;

  * •

$D_{i}\times\\{i\\}\subseteq\Delta$;

  * •

$f[D_{i}\times\\{i\\}]=\\{x_{i}\\}$ for some $x_{i}\in\omega$.

Then either $\\{x_{i}:\ i>n\\}\in\mathrm{Fin}\subseteq\mathcal{I}$ and
$f^{-1}[\\{x_{i}:\
i>n\\}]\supseteq\bigcup_{i>n}D_{i}\times\\{i\\}\notin\mathcal{ED}_{\mathrm{Fin}}$
or using tallness of $\mathcal{I}$ we could find an infinite
$C\subseteq\\{x_{i}:\ i>n\\}$, $C\in\mathcal{I}$ such that

$f^{-1}[C]\supseteq\bigcup\\{D_{i}\times\\{i\\}:\ x_{i}\in
C\\}\notin\mathcal{ED}_{\mathrm{Fin}}.$

We will inductively define sequences $\\{k_{n}:n\in\omega\\}\subseteq\omega$
and $\\{E_{n}:n\in\omega\\}\subseteq[\omega]^{\omega}$ such that for each
$n\in\omega$:

  * •

$E_{n+1}\subseteq E_{n}$;

  * •

$k_{n+1}>k_{n}$;

  * •

$E_{n}\times\\{k_{n}\\}\subseteq\Delta$;

  * •

$f\restriction E_{n}\times\\{k_{n}\\}$ is one-to-one;

  * •

$f[E_{n}\times\\{k_{n}\\}]\in\mathcal{I}$.

Start by choosing any $k_{0}\in\omega$ such that
$f[(\omega\times\\{k_{0}\\})\cap\Delta]$ is infinite (such $k_{0}$ exists by
the above observation) and any infinite $E^{\prime}_{0}\subseteq\omega$ such
that $E^{\prime}_{0}\times\\{k_{0}\\}\subseteq\Delta$ and $f\restriction
E^{\prime}_{0}\times\\{k_{0}\\}$ is one-to-one. As $\mathcal{I}$ is tall,
there is $E_{0}\subseteq E^{\prime}_{0}$ such that
$f[E_{0}\times\\{k_{0}\\}]\in\mathcal{I}$. At step $n+1$, find any
$k_{n+1}>k_{n}$ such that $f[E_{n}\times\\{k_{n+1}\\}\cap\Delta]$ is infinite
(such $k_{n+1}$ exists by the observation from the second paragraph of this
proof) and pick an infinite $E^{\prime}_{n+1}\subseteq E_{n}$ such that
$E^{\prime}_{n+1}\times\\{k_{n+1}\\}\subseteq\Delta$ and $f\restriction
E^{\prime}_{n+1}\times\\{k_{n+1}\\}$ is one-to-one. Again, using tallness of
$\mathcal{I}$ find $E_{n+1}\subseteq E^{\prime}_{n+1}$ such that
$f[E_{n+1}\times\\{k_{n+1}\\}]\in\mathcal{I}$.

Once the induction is completed, as $\mathcal{I}$ is a P-ideal, there is
$B\in\mathcal{I}$ such that $f[E_{n}\times\\{k_{n}\\}]\setminus B$ is finite
for all $n\in\omega$. Then $F_{n}=E_{n}\setminus\\{i\in E_{n}:\ f(i,k_{n})\in
B\\}\in\mathrm{Fin}$ as $f\restriction E_{n}\times\\{k_{n}\\}$ is one-to-one.
Thus

$f^{-1}[B]\supseteq\bigcup_{n\in\omega}\left((E_{n}\setminus
F_{n})\times\\{k_{n}\\}\right)\notin\mathcal{ED}_{\mathrm{Fin}}$

as for each $m\in\omega$ we have

$\left|(\\{l\\}\times\omega)\cap\bigcup_{n\in\omega}\left((E_{n}\setminus
F_{n})\times\\{k_{n}\\}\right)\right|\geq m,$

where $l=\min(E_{m}\setminus\bigcup_{j\leq m}F_{j})$. ∎

###  2.4. Ultrafilters and $\mathcal{I}$-ultrafilters

Recall that an ultrafilter $\mathcal{U}$ on $\omega$ is

  1. (1)

a _P-point_ if $\mathcal{U}^{*}$ is a P-ideal (equivalently, if
$\mathcal{U}^{*}$ is a $P^{+}$-ideal),

  2. (2)

a _selective ultrafilter_ (a.k.a.  _Ramsey ultrafilter_) if $\mathcal{U}^{*}$
is a selective ideal,

  3. (3)

a _Q-point_ if for each partition $\\{A_{n}:n\in\omega\\}$ of $\omega$ into
finite sets there is $A\in\mathcal{U}$ such that $|A_{n}\cap A|=1$ for each
$n$.

It is known that an ultrafilter is selective if and only if it is both a
P-point and a Q-point (see e.g. [6]).

Following Baumgartner, Brendle and Flašková [4, 11], for an ideal
$\mathcal{I}$ on $\omega$, we say that an ultrafilter $\mathcal{U}$ on
$\omega$ is

  1. (1)

an _$\mathcal{I}$ -ultrafilter_ if
$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{U}^{*}$ i.e. there is no
function $f:\omega\to\omega$ such that $f^{-1}[B]\in\mathcal{U}^{*}$ for each
$B\in\mathcal{I}$,

  2. (2)

a _weak $\mathcal{I}$-ultrafilter_ if
$\mathcal{I}\not\leq_{\operatorname{KB}}\mathcal{U}^{*}$ i.e. there is no
finite-to-one function $f:\omega\to\omega$ such that
$f^{-1}[B]\in\mathcal{U}^{*}$ for each $B\in\mathcal{I}$,

  3. (3)

an _$\mathcal{I}$ -point_ if
$\mathcal{I}\not\leq_{\operatorname{P}}\mathcal{U}^{*}$ i.e. there is no one-
to-one function $f:\omega\to\omega$ such that
$f^{-1}[A]\in\mathcal{U}^{\star}$ for all $A\in\mathcal{I}$. If $\mathcal{I}$
is tall, then, using [2, Lemma 3.3], one can see that in the definition of
$\mathcal{I}$-point one can consider only bijections instead of one-to-one
functions functions i.e.
$\mathcal{I}\not\leq_{\operatorname{P}}\mathcal{U}^{*}\iff\mathcal{I}\not\sqsubseteq\mathcal{U}^{*}$.

It was already mentioned in the Introduction that if there is an
$\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-ultrafilter, then
necessary $\mathcal{I}\not\leq_{K}\mathcal{J}$. Similarly, for instance, if
there is an $\mathcal{I}$-ultrafilter (weak $\mathcal{I}$-ultrafilter or
$\mathcal{I}$-point, resp.) which is not a $\mathcal{J}$-point, then
$\mathcal{I}\not\leq_{K}\mathcal{J}$ ($\mathcal{I}\not\leq_{KB}\mathcal{J}$ or
$\mathcal{I}\not\leq_{P}\mathcal{J}$, resp.).

######  Theorem 2.4 (see e.g. [11, Observation 2.1]).

  1. (1)

$\mathcal{U}$ is a P-point $\iff$ $\mathcal{U}$ is a
$\mathrm{Fin}^{2}$-ultrafilter $\iff$ $\mathcal{U}$ is a weak
$\mathrm{Fin}^{2}$-ultrafilter $\iff$ $\mathcal{U}$ is a
$\mathrm{Fin}^{2}$-point $\iff$ $\mathcal{U}$ is a $\mathrm{conv}$-ultrafilter
$\iff$ $\mathcal{U}$ is a weak $\mathrm{conv}$-ultrafilter $\iff$
$\mathcal{U}$ is a $\mathrm{conv}$-point.

  2. (2)

$\mathcal{U}$ is a Q-point $\iff$ $\mathcal{U}$ is a weak
$\mathcal{ED}_{\mathrm{Fin}}$-ultrafilter $\iff$ $\mathcal{U}$ is a
$\mathcal{ED}_{\mathrm{Fin}}$-point.

  3. (3)

$\mathcal{U}$ is a selective ultrafilter $\iff$ $\mathcal{U}$ is an
$\mathcal{ED}$-ultrafilter $\iff$ $\mathcal{U}$ is a weak
$\mathcal{ED}$-ultrafilter $\iff$ $\mathcal{U}$ is an $\mathcal{ED}$-point
$\iff$ $\mathcal{U}$ is a $\mathcal{R}$-ultrafilter $\iff$ $\mathcal{U}$ is a
weak $\mathcal{R}$-ultrafilter $\iff$ $\mathcal{U}$ is a $\mathcal{R}$-point.

###  2.5. Homogeneous ideals

An ideal $\mathcal{I}$ is called _homogeneous_ if $\mathcal{I}\restriction
A\approx\mathcal{I}$ for each $A\in\mathcal{I}^{+}$. Similarly, an ideal
$\mathcal{I}$ is called _$\leq_{\mathcal{F}}$ -homogeneous_ if
$\mathcal{I}\restriction A\approx_{\mathcal{F}}\mathcal{I}$ (i.e.
$\mathcal{I}\restriction A\leq_{\mathcal{F}}\mathcal{I}$ and
$\mathcal{I}\leq_{\mathcal{F}}\mathcal{I}\restriction A$) for each
$A\in\mathcal{I}^{+}$. Homogeneous ideals were defined and examined in [29],
whereas $\leq_{\operatorname{K}}$-homogeneous ideals were defined and examined
in [23, 33] where the authors used the name _$\operatorname{K}$ -uniform_
ideals in this case.

Obviously, homogeneity implies $\leq_{\operatorname{P}}$-homogeneity which
implies $\leq_{\operatorname{KB}}$-homogeneity, and the latter implies
$\leq_{\operatorname{K}}$-homogeneity in turn. Moreover, it is easy to see
that if $\mathcal{F}$ contains the identity function, then $\mathcal{I}$ is
$\leq_{\mathcal{F}}$-homogeneous if and only if $\mathcal{I}\restriction
A\leq_{\mathcal{F}}\mathcal{I}$ for each $A\in\mathcal{I}^{+}$.

######  Proposition 2.5.

The ideals $\mathcal{ED}_{\mathrm{Fin}}$, $\mathrm{Fin}^{2}$ and $\mathcal{W}$
are homogeneous.

###### Proof.

See [29, Example 2.4], [29, Remark below Proposition 2.9] and [29, Example
2.6], respectively. ∎

###  2.6. $P^{+}(\mathcal{I})$-ideal

Let $\mathcal{I}$ be an ideal on $X$. We write $B\subseteq^{\mathcal{I}}A$
when $B\setminus A\in\mathcal{I}$ (if $\mathcal{I}=\mathrm{Fin}$, we write
$B\subseteq^{*}A$). A set $B\in\mathcal{I}^{+}$ is an _$\mathcal{I}^{+}$
-pseudointersection_ of a family $\mathcal{A}$ if $B\subseteq^{\mathcal{I}}A$
for each $A\in\mathcal{A}$ (if $\mathcal{I}=\mathrm{Fin}$, we just say
_pseudointersection_). An ideal $\mathcal{I}$ is a _$P^{+}(\mathcal{I})$
-ideal_ if for each decreasing sequence $A_{0}\supseteq A_{1}\supseteq\ldots$
of sets from $\mathcal{I}^{+}$ there is an
$\mathcal{I}^{+}$-pseudointersection of the family $\\{A_{n}:n\in\omega\\}$.
Obviously, if $\mathcal{I}$ is a $P^{+}$-ideal, then it is
$P^{+}(\mathcal{I})$-ideal as well.

######  Proposition 2.6.

  1. (1)

Every $F_{\sigma}$ ideal is a $P^{+}$-ideal.

  2. (2)

$\mathrm{Fin}^{2}$ is a $P^{+}(\mathrm{Fin}^{2})$-ideal, but it is not a
$P^{+}$-ideal.

###### Proof.

(1) See [33, Lemma 3.2.4].

(2) $\mathrm{Fin}^{2}$ is not a $P^{+}$-ideal, because the sequence
$A_{n}=(\omega\setminus n)\times\omega$ does not have a
$(\mathrm{Fin})^{+}$-pseudointersection. Below, we show that
$\mathrm{Fin}^{2}$ is a $P^{+}(\mathrm{Fin}^{2})$-ideal.

Let $A_{n}\in(\mathrm{Fin}^{2})^{+}$ be a decreasing sequence. Then we can
pick $k_{0}<k_{1}<\dots$ such that $B_{n}=(A_{n})_{(k_{n})}$ is infinite for
each $n$. Then

$B=\bigcup_{n\in\omega}\\{k_{n}\\}\times B_{n}\in(\mathrm{Fin}^{2})^{+}$

and $B\setminus A_{n}\in\mathrm{Fin}^{2}$ for each $n$. ∎

###  2.7. Pseudointersection numbers

A _nonempty_ family $\mathcal{A}\subseteq\mathcal{P}(\omega)$ has
_$\mathcal{I}^{+}$ -FIP_ if $\bigcap F\in\mathcal{I}^{+}$ for each finite
$F\subseteq\mathcal{A}$ (if $\mathcal{I}=\mathrm{Fin}$, we say _SFIP_ instead
of $\mathrm{Fin}^{+}$-FIP). The _pseudointersection number_ of an ideal
$\mathcal{I}$ on $\omega$ is defined by:

$\begin{split}\mathfrak{p}(\mathcal{I})=\min(\\{|\mathcal{A}|:\text{
$\mathcal{A}$ has $\mathcal{I}^{+}$-FIP,}&\text{ but $\mathcal{A}$ does not
have }\\\ &\text{an
$\mathcal{I}^{+}$-pseudointersection}\\}\cup\\{\mathfrak{c}^{+}\\}).\end{split}$

The following easy proposition summarizes a few basic properties of
$\mathfrak{p}(\mathcal{I})$.

######  Proposition 2.7.

  1. (1)

$\mathfrak{p}(\mathrm{Fin})=\mathfrak{p}$.

  2. (2)

$\mathfrak{p}(\mathcal{I})=\mathfrak{c}^{+}$ for each maximal ideal
$\mathcal{I}$.

  3. (3)

$\mathfrak{p}(\mathcal{I})\geq\omega_{1}$ $\iff$ $\mathcal{I}$ is a
$P^{+}(\mathcal{I})$-ideal.

######  Theorem 2.8 ([35], see also [23, Proposition 8.19] or [8, p. 897]).

$\mathfrak{p}(\mathrm{Fin}^{2})=\omega_{1}$.

The following theorem seems to be known (see [9] or [16]), however, we were
unable to find a written proof of it anywhere, so we decided to include it for
completeness.

######  Theorem 2.9.

$\mathfrak{p}\leq\mathfrak{p}(\mathcal{I})$ for each $F_{\sigma}$ ideal
$\mathcal{I}$.

###### Proof.

Let $\phi$ be a lsc submeasure such that $\mathcal{I}=\mathrm{Fin}(\phi)$ (see
Theorem 2.1). Fix $\kappa<\mathfrak{p}$ and any
$\mathcal{A}\in[\mathcal{I}^{+}]^{\kappa}$ having $\mathcal{I}^{+}$-FIP. We
will find a set $B\in\mathcal{I}^{+}$ such that $B\setminus A\in\mathcal{I}$
for each $A\in\mathcal{A}$.

We define a poset $(\mathbb{P},\leq)$ as follows:
$\mathbb{P}=[\omega]^{<\omega}\times[\mathcal{A}]^{<\omega}$ and
$(s,F)\leq(t,G)\iff$ $s\supseteq t$, $F\supseteq G$ and $s\setminus
t\subseteq\bigcap G$. Now we define dense subsets of the poset:

  1. (1)

$D_{A}=\\{(s,F)\in\mathbb{P}:A\in F\\}$ for $A\in\mathcal{A}$,

  2. (2)

$E_{n}=\\{(s,F)\in\mathbb{P}:\phi(s)>n\\}$ for every $n\in\omega$.

Let $\mathcal{D}=\\{D_{A}:A\in\mathcal{A}\\}\cup\\{E_{n}:n\in\omega\\}$. Since
$\mathbb{P}$ is $\sigma$-centered and $|\mathcal{D}|<\mathfrak{p}$, using
Bell’s Theorem ([5], see also [6, Theorem 7.12] or [3, Theorem 1.4.22]), there
exists a filter $\mathcal{G}\subseteq\mathbb{P}$ such that $\mathcal{G}\cap
D\neq\emptyset$ for each $D\in\mathcal{D}$.

We define $B=\bigcup\\{s\in[\omega]^{<\omega}:\text{$(s,F)\in\mathcal{G}$ for
some finite $F\subseteq\mathcal{A}$}\\}$. Then $B\in\mathcal{I}^{+}$, because
for each $n$ there is $(s,F)\in\mathcal{G}\cap E_{n}$, so
$\phi(B)\geq\phi(s)>n$.

Finally, $B\setminus A$ is finite for each $A\in\mathcal{A}$. Indeed, if
$A\in\mathcal{A}$, then there is $(s,F)\in\mathcal{G}\cap D_{A}$ and we will
show that $B\setminus A\subseteq s$.

For $k\in B\setminus A$, we find $(s^{\prime},F^{\prime})\in\mathcal{G}$ such
that $k\in s^{\prime}$. Since $\mathcal{G}$ is a filter, there exists
$(s^{\prime\prime},F^{\prime\prime})\in\mathcal{G}$ such that
$(s^{\prime\prime},F^{\prime\prime})\leq(s,F)$ and
$(s^{\prime\prime},F^{\prime\prime})\leq(s^{\prime},F^{\prime})$. Then
$s^{\prime\prime}\setminus s\subseteq\bigcap F\subseteq A$ and $k\notin A$, so
$k\notin s^{\prime\prime}\setminus s$. Taking into account that $k\in
s^{\prime\prime}$, we obtain $k\in s$. ∎

##  3\. The cardinal characteristics and the order associated with
$\mathcal{I}$-ultrafilters

In this section we develop tools which will be used in our further
considerations – we introduce two cardinal invariants associated to pairs of
ideals and families of functions and show how they can be used in our studies.

###  3.1. The cardinal $\operatorname{\mathfrak{fip}}$

######  Definition 3.1.

For ideals $\mathcal{I},\mathcal{J}\subseteq\mathcal{P}(\omega)$ and a family
$\mathcal{F}\subseteq\omega^{\omega}$ we define

$\begin{split}\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})=&\min(\\{|\mathcal{A}|:\text{$\mathcal{A}$
has $\mathcal{J}^{+}$-FIP and }\\\ &\neg\left(\forall f\in\mathcal{F}\,\exists
B\in\mathcal{I}\,(\mathcal{A}\cup\\{f^{-1}[B]\\}\text{ has
$\mathcal{J}^{+}$-FIP})\right)\\}\cup\\{\mathfrak{c}^{+}\\}).\end{split}$

The following theorem provides us a basic tool for constructing
$\mathcal{I}$-ultrafilters which are not $\mathcal{J}$-ultrafilters.

######  Theorem 3.2.

If
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})\geq|\mathcal{F}|$,
then there exists an ultrafilter $\mathcal{U}$ such that
$\mathcal{I}\not\leq_{\mathcal{F}}\mathcal{U}^{*}$ and
$\mathcal{J}\subseteq\mathcal{U}^{*}$ .

In particular, if $\mathcal{F}=\omega^{\omega}$ ($\mathcal{F}$ is the family
of all finite-to-one functions or $\mathcal{F}$ is the family of all one-to-
one functions, resp.), we obtain an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not a
$\mathcal{J}$-point.

###### Proof.

Let $\mathcal{F}=\\{f_{\alpha}:\alpha<|\mathcal{F}|\\}$. We will construct a
sequence $\\{A_{\alpha}:\alpha<|\mathcal{F}|\\}\subseteq\mathcal{P}(\omega)$
such that for each $\alpha$ we have:

  1. (1)

$\\{A_{\beta}:\beta<\alpha\\}$ has $\mathcal{J}^{+}$-FIP,

  2. (2)

$f_{\alpha}[A_{\alpha}]\in\mathcal{I}$.

Suppose $A_{\beta}$ has been constructed for $\beta<\alpha$. We have 2 cases.

Case 1. If there is $F\in[\alpha]^{<\omega}$ with
$f_{\alpha}[\bigcap\\{A_{\beta}:\beta\in F\\}]\in\mathcal{I}$, we put
$A_{\alpha}=\bigcap\\{A_{\beta}:\beta\in F\\}$.

Case 2. Suppose that $f_{\alpha}[\bigcap\\{A_{\beta}:\beta\in
F\\}]\notin\mathcal{I}$ for each $F\in[\alpha]^{<\omega}$. Since
$|\alpha|<|\mathcal{F}|\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$
and $\mathcal{A}=\\{A_{\beta}:\beta<\alpha\\}$ has $\mathcal{J}^{+}$-FIP,
there is $B\in\mathcal{I}$ such that $\mathcal{A}\cup\\{f_{\alpha}^{-1}[B]\\}$
has $\mathcal{J}^{+}$-FIP. Then we put $A_{\alpha}=f_{\alpha}^{-1}[B]$.

The construction of $\langle A_{\alpha}:\alpha<|\mathcal{F}|\rangle$ is
finished.

Since $\\{A_{\alpha}:\alpha<|\mathcal{F}|\\}$ has $\mathcal{J}^{+}$-FIP, the
family $\mathcal{J}^{*}\cup\\{A_{\alpha}:\alpha<|\mathcal{F}|\\}$ has
$\mathcal{J}^{+}$-FIP as well (in particular it has SFIP). Thus, there is an
ultrafilter $\mathcal{U}$ such that
$\mathcal{J}^{*}\cup\\{A_{\alpha}:\alpha<|\mathcal{F}|\\}\subseteq\mathcal{U}$.
It is easy to see that $\mathcal{U}$ is the required ultrafilter. ∎

###  3.2. The new order and the cardinal $\mathfrak{u}$

For an ideal $\mathcal{I}$ on $\omega$ and
$\mathcal{A}\subseteq\mathcal{P}(\omega)$ we write

$\mathcal{I}(\mathcal{A})=\left\\{B\subseteq\omega:\exists
F\in[\mathcal{A}]^{<\omega}(B\setminus\bigcup F\in\mathcal{I})\right\\}.$

Notice that either $\mathcal{I}(\mathcal{A})=\mathcal{P}(\omega)$ or
$\mathcal{I}(\mathcal{A})$ is the ideal generated by
$\mathcal{I}\cup\mathcal{A}$. The latter case holds exactly when
$\omega\notin\mathcal{I}(\mathcal{A})$ (or, equivalently, when the family
$\\{\omega\setminus A:A\in\mathcal{A}\\}$ has $\mathcal{I}^{+}$-FIP).

######  Definition 3.3.

Let $\mathcal{I}$ and $\mathcal{J}$ be ideals on $\omega$,
$\mathcal{F},\mathcal{G}\subseteq\omega^{\omega}$ and
$\lambda\leq\mathfrak{c}$.

  1. (1)

We write $\mathcal{I}\preceq^{\lambda}_{\mathcal{F},\mathcal{G}}\mathcal{J}$
if for each ideal $\mathcal{J}^{\prime}$ with
$\mathcal{J}\leq_{\mathcal{G}}\mathcal{J}^{\prime}$ there exists
$\mathcal{A}\in[\mathcal{P}(\omega)]^{<\lambda}$ such that
$\omega\notin\mathcal{J}^{\prime}(\mathcal{A})$ and
$\mathcal{I}\leq_{\mathcal{F}}\mathcal{J}^{\prime}(\mathcal{A})$.

  2. (2)

We define

$\mathfrak{u}(\mathcal{I},\mathcal{F},\mathcal{J},\mathcal{G})=\min\left(\left\\{\lambda\leq\mathfrak{c}:\
\mathcal{I}\preceq^{\lambda}_{\mathcal{F},\mathcal{G}}\mathcal{J}\right\\}\cup\left\\{\mathfrak{c}^{+}\right\\}\right).$

The following theorem provides us a basic tool for proving when the existence
of $\mathcal{I}$-ultrafilters which are not $\mathcal{J}$-ultrafilters is
equivalent to $\mathcal{I}\nleq_{\operatorname{K}}\mathcal{J}$.

######  Theorem 3.4.

If $\mathcal{G}$ contains the identity function, then

$\begin{split}\mathfrak{u}(\mathcal{I},\mathcal{F},\mathcal{J},\mathcal{G})>|\mathcal{F}|\iff&\text{
there exists an ultrafilter $\mathcal{U}$ such that}\\\ &\text{
$\mathcal{I}\not\leq_{\mathcal{F}}\mathcal{U}^{*}$ and
$\mathcal{J}\leq_{\mathcal{G}}\mathcal{U}^{*}$.}\end{split}$

###### Proof.

$(\Leftarrow)$ If $\mathcal{U}$ is an ultrafilter such that
$\mathcal{I}\not\leq_{\mathcal{F}}\mathcal{U}^{*}$ and
$\mathcal{J}\leq_{\mathcal{G}}\mathcal{U}^{*}$, then
$\mathcal{J}^{\prime}=\mathcal{U}^{*}$ is a maximal ideal
$\leq_{\mathcal{G}}$-above $\mathcal{J}$ which witnesses
$\mathcal{I}\not\preceq^{\lambda}_{\mathcal{F},\mathcal{G}}\mathcal{J}$ for
every $\lambda\leq\mathfrak{c}$. Hence
$\mathfrak{u}(\mathcal{I},\mathcal{F},\mathcal{J},\mathcal{G})=\mathfrak{c}^{+}>|\mathcal{F}|$.

$(\Rightarrow)$ If
$\mathfrak{u}(\mathcal{I},\mathcal{F},\mathcal{J},\mathcal{G})>|\mathcal{F}|$,
then we have
$\mathcal{I}\not\preceq^{|\mathcal{F}|}_{\mathcal{F},\mathcal{G}}\mathcal{J}$.
Thus, there is an ideal $\mathcal{J}^{\prime}$ such that
$\mathcal{J}\leq_{\mathcal{G}}\mathcal{J}^{\prime}$ and
$\mathcal{I}\not\leq_{\mathcal{F}}\mathcal{J}^{\prime}(\mathcal{A})$ for every
$\mathcal{A}\in[\mathcal{P}(\omega)]^{<|\mathcal{F}|}$ with
$\omega\notin\mathcal{J}^{\prime}(\mathcal{A})$ .

Let $\mathcal{F}=\\{f_{\alpha}:\alpha<|\mathcal{F}|\\}$. We will construct a
sequence $\langle A_{\alpha}:\alpha<|\mathcal{F}|\rangle$ of subsets of
$\omega$ such that for each $\alpha$ we have:

  1. (1)

$\\{\omega\setminus A_{\beta}:\beta<\alpha\\}$ has
$(\mathcal{J}^{\prime})^{+}$-FIP,

  2. (2)

$f_{\alpha}[\omega\setminus A_{\alpha}]\in\mathcal{I}$.

Suppose $A_{\beta}$ has been constructed for $\beta<\alpha$. Let
$\mathcal{A}=\\{A_{\beta}:\beta<\alpha\\}$. Since $\\{\omega\setminus
A_{\beta}:\beta<\alpha\\}$ has $(\mathcal{J}^{\prime})^{+}$-FIP,
$\omega\notin\mathcal{J}^{\prime}(\mathcal{A})$. Since
$|\mathcal{A}|\leq|\alpha|<|\mathcal{F}|$,
$\mathcal{I}\not\leq_{\mathcal{F}}\mathcal{J}^{\prime}(\mathcal{A})$. Thus,
there is $B\in(\mathcal{J}^{\prime}(\mathcal{A}))^{+}$ such that
$f_{\alpha}[B]\in\mathcal{I}$. Then we put $A_{\alpha}=\omega\setminus B$, and
the construction of $\langle A_{\alpha}:\alpha<|\mathcal{F}|\rangle$ is
finished.

Since $\\{\omega\setminus A_{\alpha}:\alpha<|\mathcal{F}|\\}$ has
$(\mathcal{J}^{\prime})^{+}$-FIP, the family
$(\mathcal{J}^{\prime})^{*}\cup\\{\omega\setminus
A_{\alpha}:\alpha<|\mathcal{F}|\\}$ has $(\mathcal{J}^{\prime})^{+}$-FIP as
well (in particular it has SFIP). Thus, there is an ultrafilter $\mathcal{U}$
such that $(\mathcal{J}^{\prime})^{*}\cup\\{\omega\setminus
A_{\alpha}:\alpha<|\mathcal{F}|\\}\subseteq\mathcal{U}$. It is easy to see
that $\mathcal{U}$ is the required ultrafilter. ∎

######  Corollary 3.5.

Let $\mathcal{I}$ and $\mathcal{J}$ be ideals on $\omega$.

  1. (1)

$\mathcal{I}\not\preceq_{\omega^{\omega},\omega^{\omega}}^{\mathfrak{c}}\mathcal{J}$
($\mathcal{I}\not\preceq_{\omega^{\omega},\mathrm{Fin}-1}^{\mathfrak{c}}\mathcal{J}$
or $\mathcal{I}\not\preceq^{\mathfrak{c}}_{\omega^{\omega},1-1}\mathcal{J}$,
resp.) $\iff$ there exists an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-ultrafilter (weak $\mathcal{J}$-ultrafilter or
$\mathcal{J}$-point, resp.).

  2. (2)

$\mathcal{I}\not\preceq_{\mathrm{Fin}-1,\mathrm{Fin}-1}^{\mathfrak{c}}\mathcal{J}$
($\mathcal{I}\not\preceq_{\mathrm{Fin}-1,1-1}^{\mathfrak{c}}\mathcal{J}$,
resp.) $\iff$ there exists a weak $\mathcal{I}$-ultrafilter which is not a
weak $\mathcal{J}$-ultrafilter ($\mathcal{J}$-point, resp.).

  3. (3)

$\mathcal{I}\not\preceq_{1-1,1-1}^{\mathfrak{c}}\mathcal{J}$ $\iff$ there
exists an $\mathcal{I}$-point which is not a $\mathcal{J}$-point.

######  Remark.

It is easy to see that $\mathcal{I}\leq_{\operatorname{K}}\mathcal{J}$ implies
$\mathcal{I}\preceq_{\omega^{\omega},\omega^{\omega}}^{\mathfrak{c}}\mathcal{J}$.
However, the inverse implication is not true in general. Indeed, it is known
that $\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathrm{conv}$ [24], but
$\mathrm{Fin}^{2}\preceq_{\omega^{\omega},\omega^{\omega}}^{\mathfrak{c}}\mathrm{conv}$
holds as $\mathrm{Fin}^{2}$-ultrafilters and $\mathrm{conv}$-ultrafilters and
P-points all are the same notion (see Theorem 2.4(1)).

We end this section with a result connecting the two cardinal invariants
introduced above.

######  Theorem 3.6.

If $\mathcal{G}$ contains the identity function, then either
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})=\mathfrak{u}(\mathcal{I},\mathcal{F},\mathcal{J},\mathcal{G})=\mathfrak{c}^{+}$
or

$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})<\mathfrak{u}(\mathcal{I},\mathcal{F},\mathcal{J},\mathcal{G}).$

###### Proof.

Assume first that
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})<\mathfrak{c}^{+}$.
We will show that
$\mathcal{I}\not\preceq_{\mathcal{F},\mathcal{G}}^{\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})}\mathcal{J}$.
Let $\mathcal{J}^{\prime}=\mathcal{J}$ and take any
$\mathcal{A}\subseteq\mathcal{P}(\omega)$ such that
$|\mathcal{A}|<\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$
and $\omega\notin\mathcal{J}^{\prime}(\mathcal{A})$. Let
$\mathcal{B}=\\{\omega\setminus A:A\in\mathcal{A}\\}$. Then $\mathcal{B}$ has
$\mathcal{J}^{+}$-FIP and
$|\mathcal{B}|<\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$,
so for each $f\in\mathcal{F}$ there is $C\in\mathcal{I}$ such that
$\mathcal{B}\cup\\{f^{-1}[C]\\}$ has $\mathcal{J}^{+}$-FIP. In particular,
$f^{-1}[C]\notin\mathcal{J}^{\prime}(\mathcal{A})$. Consequently,
$\mathcal{I}\not\leq_{\mathcal{F}}\mathcal{J}^{\prime}(\mathcal{A})$.

If
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})=\mathfrak{c}^{+}$,
then it is not difficult to see that
$\mathcal{I}\npreceq^{\lambda}_{\mathcal{F},\mathcal{G}}\mathcal{J}$ for each
$\lambda\leq\mathfrak{c}$, so
$\mathfrak{u}(\mathcal{I},\mathcal{F},\mathcal{J},\mathcal{G})=\mathfrak{c}^{+}$.
∎

##  4\. More on the cardinals $\operatorname{\mathfrak{fip}}$ and
$\mathfrak{u}$

###  4.1. Some lower bounds

######  Theorem 4.1.

Let $\mathcal{F}\subseteq\omega^{\omega}$ and $\mathcal{I}$, $\mathcal{J}$ be
ideals on $\omega$ such that $\mathcal{I}$ is tall and
$\mathcal{I}\not\leq_{\mathcal{F}}\mathcal{J}\restriction A$ for all
$A\notin\mathcal{J}$. We have:

  1. (1)

$\mathfrak{p}(\mathcal{J})\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$.

  2. (2)

$\operatorname{\operatorname{add}^{*}}(\mathcal{I})\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$.

###### Proof.

(1) Let $\mathcal{A}$ be a family having $\mathcal{J}^{+}$-FIP and such that
$|\mathcal{A}|<\mathfrak{p}(\mathcal{J})$. Then there is $C\notin\mathcal{J}$
such that $C\subseteq^{\mathcal{J}}A$ for each $A\in\mathcal{A}$. In
particular, $C\subseteq^{\mathcal{J}}\bigcap F$ for each
$F\in[\mathcal{A}]^{<\omega}$. Let $f\in\mathcal{F}$. Since
$\mathcal{I}\not\leq_{\mathcal{F}}\mathcal{J}\restriction C$, there is
$D\subseteq C$, $D\notin\mathcal{J}$ with $f[D]\in\mathcal{I}$. We put
$B=f[D]$ and notice that for each $F\in[\mathcal{A}]^{<\omega}$, $\bigcap
F\cap f^{-1}[B]\supseteq\bigcap\mathcal{F}\cap D\notin\mathcal{J}$. It means
that $\mathcal{A}\cup\\{f^{-1}[B]\\}$, has $\mathcal{J}^{+}$-FIP, so
$|\mathcal{A}|<\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$.

(2) Let $\mathcal{A}$ be a family having $\mathcal{J}^{+}$-FIP and such that
$|\mathcal{A}|<\operatorname{\operatorname{add}^{*}}(\mathcal{I})$. Fix any
$f\in\mathcal{F}$.

If there is $n\in\omega$ such that $\\{f^{-1}[\\{n\\}]\\}\cup\mathcal{A}$ has
$\mathcal{J}^{+}$-FIP, then
$|\mathcal{A}|<\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$
and the proof is finished. Otherwise, $\\{f^{-1}[\\{n\\}]\\}\cup\mathcal{A}$
does not have $\mathcal{J}^{+}$-FIP for any $n\in\omega$. Then, for each
$F\in[\mathcal{A}]^{<\omega}$, we define

$Y_{1}^{F}=\\{n\in\omega:f^{-1}[\\{n\\}]\cap\bigcap F\in\mathcal{J}\\}\text{\
\ \ and \ \ \ }Y_{2}^{F}=\omega\setminus Y_{1}^{F},$

and pick a set $B_{F}\in\mathcal{I}$ in the following manner:

  * •

If $\\{f^{-1}[Y_{1}^{F}]\\}\cup\mathcal{A}$ has $\mathcal{J}^{+}$-FIP, then
$A=f^{-1}[Y_{1}^{F}]\cap\bigcap F\notin\mathcal{J}$. Since
$\mathcal{I}\not\leq_{\mathcal{F}}\mathcal{J}\restriction A$, there is
$B_{F}\in\mathcal{I}$ such that $B_{F}\subseteq f[f^{-1}[Y_{1}^{F}]\cap\bigcap
F]$ and $f^{-1}[B_{F}]\cap(f^{-1}[Y_{1}^{F}]\cap\bigcap F)\notin\mathcal{J}$.

  * •

If $\\{f^{-1}[Y_{2}^{F}]\\}\cup\mathcal{A}$ has $\mathcal{J}^{+}$-FIP, then
$Y_{2}^{F}$ is infinite (because $\\{f^{-1}[\\{n\\}]\\}\cup\mathcal{A}$ does
not have $\mathcal{J}^{+}$-FIP for each $n$), so there is infinite
$B_{F}\subseteq Y_{2}^{F}$ such that $B_{F}\in\mathcal{I}$ (because
$\mathcal{I}$ is tall).

Since
$|[\mathcal{A}]^{<\omega}|<\operatorname{\operatorname{add}^{*}}(\mathcal{I})$,
there is $B\in\mathcal{I}$ such that $B_{F}\setminus B$ is finite for each
$F\in[\mathcal{A}]^{<\omega}$. We claim that $\\{f^{-1}[B]\\}\cup\mathcal{A}$
has $\mathcal{J}^{+}$-FIP (if so, then
$|\mathcal{A}|<\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$
and the proof will be finished).

Take any $F\in[\mathcal{A}]^{<\omega}$. We have two cases.

_Case 1._ The family $\\{f^{-1}[Y_{1}^{F}]\\}\cup\mathcal{A}$ has
$\mathcal{J}^{+}$-FIP.

Then $f^{-1}[B_{F}]\cap\bigcap F\notin\mathcal{J}$. Moreover, $B_{F}\setminus
B$ is finite and $f^{-1}[\\{n\\}]\cap\bigcap F\in\mathcal{J}$ for each $n\in
B_{F}\setminus B$ (as $B_{F}\subseteq f[f^{-1}[Y_{1}^{F}]\subseteq
Y_{1}^{F}$), so $f^{-1}[B_{F}\setminus B]\cap\bigcap F\in\mathcal{J}$. Thus,
$f^{-1}[B]\cap\bigcap F\supseteq f^{-1}[B_{F}\cap B]\cap\bigcap
F\notin\mathcal{J}$.

_Case 2._ The family $\\{f^{-1}[Y_{2}^{F}]\\}\cup\mathcal{A}$ has
$\mathcal{J}^{+}$-FIP.

Then $B_{F}$ is infinite, so $B_{F}\cap B\neq\emptyset$ as $B_{F}\setminus B$
is finite. Take any $n\in B_{F}\cap B$. Then $f^{-1}[\\{n\\}]\cap\bigcap
F\notin\mathcal{J}$ (as $n\in B_{F}\subseteq Y_{2}^{F}$), and
$f^{-1}[B]\supseteq f^{-1}[\\{n\\}]$ (as $n\in B$). Thus,
$f^{-1}[B]\cap\bigcap F\notin\mathcal{J}$. ∎

As a simple corollary, we obtain a result of Flašková from [19].

######  Corollary 4.2 ([19, Proposition 2.3]).

Assume $\mathfrak{p}=\mathfrak{c}$. There exists an $\mathcal{I}$-ultrafilter
for each tall ideal $\mathcal{I}$.

###### Proof.

Since $\mathfrak{p}(\mathrm{Fin})=\mathfrak{p}$ and
$\mathcal{I}\not\leq_{\operatorname{K}}\mathrm{Fin}\approx\mathrm{Fin}\restriction
A$ for each $A\notin\mathrm{Fin}$, it is enough to apply Theorems 3.2 and
4.1(1). ∎

###  4.2. Connection with the Katětov order

The following result, which was already mentioned in the Introduction,
provides sufficient conditions on ideals to obtain the equivalence:
$\mathcal{I}\not\leq_{K}\mathcal{J}$ if and only if it is consistent that
there exists an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-ultrafilter.

######  Theorem 4.3.

Let $\mathcal{I}$ be a tall ideal and $\mathcal{J}$ be a
$\text{P}^{+}(\mathcal{J})$-ideal.

  1. (1)

If $\mathcal{J}$ is $\leq_{\operatorname{K}}$-homogeneous, then under CH, the
following conditions are equivalent.

    1. (a)

$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$.

    2. (b)

$\mathcal{I}\not\preceq_{\omega^{\omega},\omega^{\omega}}^{\mathfrak{c}}\mathcal{J}$.

    3. (c)

$\mathcal{I}\not\preceq_{\omega^{\omega},\mathrm{Fin}-1}^{\mathfrak{c}}\mathcal{J}$.

    4. (d)

$\mathcal{I}\not\preceq_{\omega^{\omega},1-1}^{\mathfrak{c}}\mathcal{J}$.

    5. (e)

There exists an $\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-point.

    6. (f)

There exists an $\mathcal{I}$-ultrafilter which is not a weak
$\mathcal{J}$-ultrafilter.

    7. (g)

There exists an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-ultrafilter.

  2. (2)

If $\mathcal{J}$ is $\leq_{\operatorname{KB}}$-homogeneous, then under CH, the
following conditions are equivalent.

    1. (a)

$\mathcal{I}\not\leq_{\operatorname{KB}}\mathcal{J}$.

    2. (b)

$\mathcal{I}\not\preceq_{\mathrm{Fin}-1,\mathrm{Fin}-1}^{\mathfrak{c}}\mathcal{J}$.

    3. (c)

$\mathcal{I}\not\preceq_{\mathrm{Fin}-1,1-1}^{\mathfrak{c}}\mathcal{J}$.

    4. (d)

There exists a weak $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-point.

    5. (e)

There exists a weak $\mathcal{I}$-ultrafilter which is not a weak
$\mathcal{J}$-ultrafilter.

  3. (3)

If $\mathcal{J}$ is $\leq_{\operatorname{P}}$-homogeneous, then under CH, the
following conditions are equivalent.

    1. (a)

$\mathcal{I}\not\leq_{\operatorname{P}}\mathcal{J}$.

    2. (b)

$\mathcal{I}\not\preceq_{1-1,1-1}^{\mathfrak{c}}\mathcal{J}$.

    3. (c)

There exists an $\mathcal{I}$-point which is not a $\mathcal{J}$-point.

  4. (4)

If $\mathcal{J}$ is an $F_{\sigma}$ ideal, then CH can be relaxed to the
assumption $\mathfrak{p}=\mathfrak{c}$ in (1), (2) and (3).

  5. (5)

If $\mathcal{I}$ and $\mathcal{J}$ are Borel and $\mathcal{J}$ is
$\leq_{\operatorname{K}}$-homogeneous ($\leq_{\operatorname{KB}}$-homogeneous
or $\leq_{\operatorname{P}}$-homogeneous, resp.), then the following
conditions are equivalent.

    1. (a)

It is consistent that there exists an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not a
$\mathcal{J}$-point.

    2. (b)

Under CH, there exists an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not a
$\mathcal{J}$-point.

In particular, if one can find an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not a
$\mathcal{J}$-point in some sophisticated model of ZFC, then one could do it
already under CH.

  6. (6)

Equivalences similar to item (5) (which concerns items (1e), (2d), and (3c)
above) hold for counterparts of items (1f), (1g) and (2e).

###### Proof.

(1) Taking into account the obvious implications, using Corollary 3.5 and
knowing that
$\mathcal{I}\not\preceq_{\omega^{\omega},\omega^{\omega}}^{\mathfrak{c}}\mathcal{J}$
implies $\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$, the proof will be
finished once we show that $\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$
implies that there is an $\mathcal{I}$-ultrafilter which is not a
$\mathcal{J}$-point.

Assume that $\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$. Since
$\mathcal{J}$ is $\leq_{\operatorname{K}}$-homogeneous, we obtain
$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}\restriction A$ for all
$A\notin\mathcal{J}$. Now, Theorem 4.1 implies
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\omega^{\omega})\geq\mathfrak{p}(\mathcal{J})$.
Since $\mathcal{J}$ is a $P^{+}(\mathcal{J})$-ideal,
$\mathfrak{p}(\mathcal{J})\geq\omega_{1}$ by Proposition 2.7(3). Thus, using
CH and applying Theorem 3.2, we obtain an $\mathcal{I}$-ultrafilter which is
not a $\mathcal{J}$-point.

The proofs of items (2) and (3) are very similar to the proof of item (1) as
one can easily check that
$\mathcal{I}\not\preceq_{\mathrm{Fin}-1,\mathrm{Fin}-1}^{\mathfrak{c}}\mathcal{J}$
($\mathcal{I}\not\preceq_{1-1,1-1}^{\mathfrak{c}}\mathcal{J}$, resp.) implies
$\mathcal{I}\not\leq_{KB}\mathcal{J}$ ($\mathcal{I}\not\leq_{P}\mathcal{J}$,
resp.).

(4) It follows from the proofs of (1), (2), (3) and the fact that
$\mathfrak{p}(\mathcal{J})\geq\mathfrak{p}$ for $F_{\sigma}$ ideals (Theorem
2.9).

(5) The implication “(5b)$\implies$(5a)” is obvious, so below we show only the
implication “(5a)$\implies$(5b)”. Assume that it is consistent that there
exists an $\mathcal{I}$-ultrafilter (weak $\mathcal{I}$-ultrafilter or
$\mathcal{I}$-point, resp.) which is not a $\mathcal{J}$-point. Then,
obviously, it is consistent that
$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$
($\mathcal{I}\not\leq_{\operatorname{KB}}\mathcal{J}$,
$\mathcal{I}\not\leq_{\operatorname{P}}\mathcal{J}$, resp.) – see Subsection
2.4. However, if $\mathcal{I}$ and $\mathcal{J}$ are Borel ideals, then the
sentence $\mathcal{I}\nleq_{\operatorname{K}}\mathcal{J}$
($\mathcal{I}\not\leq_{\operatorname{KB}}\mathcal{J}$,
$\mathcal{I}\not\leq_{\operatorname{P}}\mathcal{J}$, resp.) is absolute (see
[11, p. 210]), so $\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{J}$
($\mathcal{I}\not\leq_{\operatorname{KB}}\mathcal{J}$,
$\mathcal{I}\not\leq_{\operatorname{P}}\mathcal{J}$, resp.) holds in ZFC.
Then, using item (1) ((2) or (3), resp.) we obtain, under CH, that there
exists an $\mathcal{I}$-ultrafilter (weak $\mathcal{I}$-ultrafilter or
$\mathcal{I}$-point, resp.) which is not a $\mathcal{J}$-point.

(6) This can be shown in the same way as item (5). ∎

###  4.3. Maximal ideals

######  Theorem 4.4.

Let $\mathcal{I}$ be a maximal ideal on $\omega$. If
$\mathcal{F}\subseteq\omega^{\omega}$ contains the identity function, then

$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin},\mathcal{F})=\chi(\mathcal{I}^{*}),$

where
$\chi(\mathcal{I}^{*})=\min\\{|\mathcal{B}|:\mathcal{B}\subseteq\mathcal{I}^{*}\land\forall
A\in\mathcal{I}^{*}\,\exists B\in\mathcal{B}\,(B\subseteq^{*}A)\\}$.

###### Proof.

First we show
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin},\mathcal{F})\geq\chi(\mathcal{I}^{*})$.
Take a family $\mathcal{A}$ with SFIP and such that
$|\mathcal{A}|<\chi(\mathcal{I}^{*})$. Take any $f\in\mathcal{F}$. We have 2
cases.

Case 1. There is $F\in[\mathcal{A}]^{<\omega}$ with $f[\bigcap
F]\in\mathcal{I}$. Then we put $B=f[\bigcap F]$ and observe that $\bigcap
G\cap f^{-1}[B]\supseteq\bigcap G\cap\bigcap F\notin\mathrm{Fin}$ for each
$G\in[\mathcal{A}]^{<\omega}$.

Case 2. For each $F\in[\mathcal{A}]^{<\omega}$, $f[\bigcap
F]\notin\mathcal{I}$. Since $\mathcal{I}$ is maximal, $f[\bigcap
F]\in\mathcal{I}^{*}$ for each $F\in[\mathcal{A}]^{<\omega}$. Since
$|[\mathcal{A}]^{<\omega}|=|\mathcal{A}|<\chi(\mathcal{I}^{*})$, there is
$C\in\mathcal{I}^{*}$ such that $f[\bigcap F]\not\subseteq^{*}C$ for each
$F\in[\mathcal{A}]^{<\omega}$. Then we put $B=\omega\setminus C$. Since
$\bigcap F\cap f^{-1}[B]=\bigcap F\setminus f^{-1}[C]$ for each
$F\in[\mathcal{A}]^{<\omega}$, $|\bigcap F\cap f^{-1}[B]|\geq|f[\bigcap
F]\setminus C|=\omega$.

We conclude that
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin},\mathcal{F})>|\mathcal{A}|$
and consequently
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin},\mathcal{F})\geq\chi(\mathcal{I}^{*})$.

Now we show
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin},\mathcal{F})\leq\chi(\mathcal{I}^{*})$.
Fix $\mathcal{G}\subseteq\mathcal{I}^{*}$ with
$|\mathcal{G}|<\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin},\mathcal{F})$.
Since $\mathcal{G}$ has SFIP and
$|\mathcal{G}|<\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin},\mathcal{F})$,
for the identity function $f\in\mathcal{F}$, we can find $B\in\mathcal{I}$
such that $\mathcal{G}\cup\\{B\\}=\mathcal{G}\cup\\{f^{-1}[B]\\}$ has SFIP.
Let $C=\omega\setminus B$. Then $C\in\mathcal{I}^{*}$ and for each
$G\in\mathcal{G}$ we have $|G\setminus C|=|G\cap B|=\omega$, so
$G\not\subseteq^{*}C$. Thus $\chi(\mathcal{I}^{*})>|\mathcal{G}|$, and
consequently,
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin},\mathcal{F})\leq\chi(\mathcal{I}^{*})$.
∎

Again as a simple corollary, we obtain another result of Flašková from [19].

######  Corollary 4.5 ([19, Proposition 2.1]).

There exists an $\mathcal{I}$-ultrafilter for each maximal ideal $\mathcal{I}$
such that $\chi(\mathcal{I}^{*})=\mathfrak{c}$.

###### Proof.

Apply Theorems 3.2 and 4.4. ∎

##  5\. Let’s play a game

In this section we develop an infinite game which will enable us to replace CH
or $\mathfrak{p}=\mathfrak{c}$ with
$\operatorname{cov}(\mathcal{M})=\mathfrak{c}$ in some of our results.

######  Definition 5.1.

Consider the following game $G(\mathcal{J},A,f)$ associated to an ideal
$\mathcal{J}$ on $\omega$, $A\notin\mathcal{J}$ and a function $f:A\to\omega$
which is $\mathcal{J}$-to-one (i.e. $f^{-1}[\\{n\\}]\in\mathcal{J}$ for every
$n$). In the first move Player I plays $k_{0}\in\omega$ and then in the $n$th
move Player II plays $G_{n}\in\mathrm{Fin}$ and Player I responses in the
$(n+1)$st move with a pair $(F_{n},k_{n+1})\in\mathrm{Fin}\times\omega$ such
that $F_{n}\cap G_{n}=\emptyset$ and $|F_{n}|\leq k_{n}$. At the end Player I
is declared the winner if
$f^{-1}[\bigcup_{n\in\omega}F_{n}]\notin\mathcal{J}\restriction A$. Otherwise
Player II is the winner.

Next result characterizes winning strategies in $G(\mathcal{J},A,f)$ and
connects the considered game with the property of containing a tall summable
ideal.

######  Lemma 5.2.

  1. (1)

Player I has a winning strategy in $G(\mathcal{J},A,f)$ if and only if there
are families $\mathcal{F}_{0},\mathcal{F}_{1},\dots\subseteq\mathrm{Fin}$ such
that:

    1. (a)

$\sup\\{|F|:F\in\mathcal{F}_{n}\\}<\infty$ for each $n$,

    2. (b)

$\sup\\{\min(F):F\in\mathcal{F}_{n}\\}=\infty$ for each $n$,

    3. (c)

if $F_{n}\in\mathcal{F}_{n}$ for each $n$, then
$f^{-1}[\bigcup_{n\in\omega}F_{n}]\notin\mathcal{J}\restriction A$.

  2. (2)

Player II has a winning strategy in $G(\mathcal{J},A,f)$ if and only if there
is a tall summable ideal $\mathcal{I}_{g}$ such that $f$ witnesses
$\mathcal{I}_{g}\leq_{\operatorname{K}}\mathcal{J}\restriction A$.

  3. (3)

If $\mathcal{J}\restriction A$ is a Borel ideal, then the game
$G(\mathcal{J},A,f)$ is determined for all $f$.

  4. (4)

If $\mathcal{J}$ is Borel such that for every $A\notin\mathcal{J}$ the ideal
$\mathcal{J}\restriction A$ is not $\leq_{\operatorname{K}}$-above any tall
summable ideal, then Player I has a winning strategy in $G(\mathcal{J},A,f)$,
for all $A$ and $f$.

###### Proof.

(1) Assume that there are families $\mathcal{F}_{n}$ satisfying (a)-(c). Then
Player I has a winning strategy by playing in his initial move
$k_{0}=\sup\\{|F|:F\in\mathcal{F}_{0}\\}$ and then in each next move (in
response to some $G_{n}\in\mathrm{Fin}$) any $F_{n}\in\mathcal{F}_{n}$ such
that $F_{n}\cap G_{n}=\emptyset$ (such $F_{n}$ exists from the property (b))
and $k_{n+1}=\sup\\{|F|:F\in\mathcal{F}_{n+1}\\}$.

Now, assume that Player I has a winning strategy
$\$:\mathrm{Fin}^{<\omega}\to\mathrm{Fin}\times\omega$. Let
$\pi_{\mathrm{Fin}}:\mathrm{Fin}\times\omega\to\mathrm{Fin}$ and
$\pi_{\omega}:\mathrm{Fin}\times\omega\to\omega$ be the projections onto the
first and the second coordinate, respectively (so $\pi_{\mathrm{Fin}}(F,n)=F$
and $\pi_{\omega}(F,n)=n$ for all $(F,n)\in\mathrm{Fin}\times\omega$). Let
$\mathcal{F}(\emptyset)=\\{\pi_{\mathrm{Fin}}(\$(G)):G\in\mathrm{Fin}\\}$ and

$\mathcal{F}(G_{0},\ldots,G_{n})=\\{\pi_{\mathrm{Fin}}(\$(G_{0},\ldots,G_{n},G)):G\in\mathrm{Fin}\\}$

for all $(G_{0},\ldots,G_{n})\in\mathrm{Fin}^{<\omega}$. Observe that for each
$F\in\mathcal{F}(G_{0},\ldots,G_{n})$ we have
$|F|\leq\pi_{\omega}(\$(G_{0},\ldots,G_{n}))$, and
$\sup\\{\min(F):F\in\mathcal{F}(G_{0},\ldots,G_{n})\\}=\infty$ as for each
$m\in\omega$ minimum of the set
$\pi_{\mathrm{Fin}}(\$(G_{0},\ldots,G_{n},[0,m]))\in\mathcal{F}(G_{0},\ldots,G_{n})$
is greater than $m$. Thus, the families $\mathcal{F}(G_{0},\ldots,G_{n})$
satisfy (a) and (b). As $\mathrm{Fin}^{<\omega}$ is countable, it suffices to
check (c). For each $(G_{0},\ldots,G_{n})\in\mathrm{Fin}^{<\omega}$, take
$F_{(G_{0},\ldots,G_{n})}\in\mathcal{F}(G_{0},\ldots,G_{n})$. Let
$B=\bigcup\\{F_{(G_{0},\ldots,G_{n})}:(G_{0},\ldots,G_{n})\in\mathrm{Fin}^{<\omega}\\}$.
To finish the proof, we have to show that
$f^{-1}[B]\notin\mathcal{J}\restriction A$.

Since $F_{\emptyset}\in\mathcal{F}(\emptyset)$, there is
$H_{0}\in\mathrm{Fin}$ such that
$F_{\emptyset}=\pi_{\mathrm{Fin}}(\$(H_{0}))$. Since
$F_{(H_{0})}\in\mathcal{F}(H_{0})$, there is $H_{1}\in\mathrm{Fin}$ such that
$F_{(H_{0})}=\pi_{\mathrm{Fin}}(\$(H_{0},H_{1}))$. Continuing in this way we
can inductively pick an infinite sequence $H_{0},H_{1},\ldots\in\mathrm{Fin}$
such that
$F_{(H_{0},\dots,H_{n})}=\pi_{\mathrm{Fin}}(\$(H_{0},\ldots,H_{n},H_{n+1}))\in\mathcal{F}(H_{0},\dots,H_{n})$
for each $n\in\omega$. Then

$f^{-1}[B]\supseteq
f^{-1}\left[\bigcup_{n\in\omega}F_{(H_{0},\dots,H_{n})}\right]\notin\mathcal{J}\restriction
A$

as $\bigcup_{n\in\omega}F_{(H_{0},\dots,H_{n})}$ constitutes an outcome of a
play of the game along the winning strategy for Player I.

(2) Suppose that $f:A\to\omega$ witnesses
$\mathcal{I}_{g}\leq_{\operatorname{K}}\mathcal{J}\restriction A$. Then Player
II has a winning strategy by playing in each move a set
$[0,m_{n}]\in\mathrm{Fin}$ such that $k_{n}g(i)<1/2^{n}$ for all $i>m_{n}$ as
in this case

$\sum_{i\in\bigcup_{n\in\omega}F_{n}}g(i)\leq\sum_{n\in\omega}\frac{1}{2^{n}}<+\infty,$

hence $\bigcup_{n\in\omega}F_{n}\in\mathcal{I}_{g}$ and consequently
$f^{-1}[\bigcup_{n\in\omega}F_{n}]\in\mathcal{J}\restriction A$.

Now, assume that Player II has a winning strategy
$\$:\omega\times(\mathrm{Fin}\times\omega)^{<\omega}\to\mathrm{Fin}$.

Put $M_{0}=\$(1)$, $M_{1}=\$(2)$ and define inductively:

$\begin{split}M_{n+1}=&\bigcup\\{\$(k_{0},F_{0},k_{1},F_{1}\ldots,k_{l},F_{l},k_{l+1}):l\leq
n,\forall i\leq l\left(|F_{i}|\leq k_{i}\right),\\\ &\forall i\leq
l\left(F_{i}\cap\$(k_{0},F_{0},\ldots,k_{i})=\emptyset\right),\forall i\leq
l+1\left(k_{i}\leq n+2\right),\bigcup_{i\leq n}F_{i}\subseteq
M_{n}\\}\end{split}$

for all $n>0$. Without loss of generality we may assume that
$(M_{n})_{n\in\omega}$ is a strictly increasing sequence of finite sets and
$\bigcup_{n\in\omega}M_{n}=\omega$ (as playing larger sets by Player II only
increases chances of winning).

Define $g:\omega\to[0,+\infty)$ by $g(i)=\frac{1}{n+1}$ whenever $i\in
M_{n+1}\setminus M_{n}$ and $g(i)=2$ for all $i\in M_{0}$. Observe that
$\mathcal{I}_{g}$ is tall as $\lim_{i\to\infty}g(i)=0$. We claim that $f$
witnesses $\mathcal{I}_{g}\leq_{\operatorname{K}}\mathcal{J}\restriction A$.

For every $B\in\mathcal{I}_{g}$ there is $k\in\omega$ such that

$\frac{|B\cap M_{n}|}{n}\leq\sum_{i\in B\cap M_{n}}g(i)\leq k,$

for all $n>0$. Hence, $B$ can be partitioned into $M_{0}\cap B$ (for which we
have $f^{-1}[M_{0}\cap B]\subseteq f^{-1}[M_{0}]\in\mathcal{J}\restriction A$
as $M_{0}\in\mathrm{Fin}$ and $f^{-1}[\\{i\\}]\in\mathcal{J}\restriction A$
for all $i\in\omega$) and $k$ many sets $C$ satisfying the property $|C\cap
M_{n}|\leq n$ for all $n>0$. Thus, to finish the proof we need to show that
for each set $C\subseteq\omega$ such that $|C\cap M_{n}|\leq n$ for all $n$ we
have $f^{-1}[C]\in\mathcal{J}\restriction A$.

Fix $C\subseteq\omega$ as above and let $C_{n}=C\cap(M_{n+1}\setminus M_{n})$
for all $n\in\omega$. Then for each $n\in\omega$ we have $|C_{2n+2}|\leq 2n+3$
and

$C_{2n+2}\cap\$(1,C_{0},3,C_{2},\ldots,2n+1,C_{2n},2n+3)\subseteq C_{2n+2}\cap
M_{2n+2}=\emptyset.$

Thus, $C^{\prime}=\bigcup_{n\in\omega}C_{2n}$ is an outcome of a play of the
game along the winning strategy for Player II. Hence,
$f^{-1}[C^{\prime}]\in\mathcal{J}\restriction A$. Similarly one can show that
$f^{-1}[C^{\prime\prime}]\in\mathcal{J}\restriction A$ where
$C^{\prime\prime}=\bigcup_{n\in\omega}C_{2n+1}=C\setminus C^{\prime}$.

(3) We will use Martin’s theorem on Borel determinacy. Indeed, using the
notation from [27, Section 20A], $G(\mathcal{J},A,f)$ is equivalent to the
game $G(T,\phi^{-1}[\mathcal{J}\restriction A])$ where

$T=\left\\{(F_{0},\ldots,F_{n})\in\mathrm{Fin}^{<\omega}:\forall{i\in\omega}\left(|F_{3i}|=1\text{
and }F_{3i+1}\cap F_{3i+2}=\emptyset\text{ and }\right.\right.$
$\left.\left.\left(|F_{3i+2}|\leq x\text{, where
}\\{x\\}=F_{3i}\right)\right)\right\\}$

and $\phi:[T]\to\mathcal{P}(A)$ given by
$\phi((F_{0},F_{1},\ldots))=f^{-1}[\bigcup_{i\in\omega}F_{3i+2}]$ is a
function of Baire class $1$ ($[T]\subseteq\mathrm{Fin}^{\omega}$ denotes the
set of all infinite branches of the tree $T$).

(4) Let $A\notin\mathcal{J}$ and $f:A\to\omega$ be such that
$f^{-1}[\\{n\\}]\in\mathcal{J}$ for all $n\in\omega$. Since
$\mathcal{J}\restriction A$ is not $\leq_{\operatorname{K}}$-above any tall
summable ideal, Player II cannot have a winning strategy in
$G(\mathcal{J},A,f)$ by item (2). However, the game $G(\mathcal{J},A,f)$ is
determined by item (3) as $\mathcal{J}$ being Borel implies that
$\mathcal{J}\restriction A$ is Borel as well. Thus, Player I has a winning
strategy in $G(\mathcal{J},A,f)$. ∎

The following immediate corollary seems to be interesting in itself as the
property of containing a tall summable ideal is quite natural in the studies
of ideals on $\omega$.

######  Corollary 5.3.

Let $\mathcal{J}$ be a Borel ideal. Then $\mathcal{J}$ does not contain a tall
summable ideal if and only if there are families
$\mathcal{F}_{0},\mathcal{F}_{1},\ldots\subseteq\mathrm{Fin}$ such that

  1. (1)

$\sup\\{|F|:F\in\mathcal{F}_{n}\\}<\infty$ for each $n$,

  2. (2)

$\sup\\{\min(F):F\in\mathcal{F}_{n}\\}=\infty$ for each $n$,

  3. (3)

if $F_{n}\in\mathcal{F}_{n}$ for each $n$, then
$\bigcup_{n\in\omega}F_{n}\notin\mathcal{J}$.

###### Proof.

It follows from Lemma 5.2 applied to the game
$G(\mathcal{J},\omega,\text{id})$. ∎

At this point we turn our attention to providing examples of ideals for which
Player I has a winning strategy in $G(\mathcal{J},A,f)$ for all $A$ and $f$.

######  Proposition 5.4.

If
$\mathcal{J}\in\\{\mathrm{Fin}^{2},\mathcal{ED}_{\mathrm{Fin}},\mathcal{ED},\mathrm{conv}\\}$,
then the ideal $\mathcal{J}\restriction A$ is not
$\leq_{\operatorname{K}}$-above any tall summable ideal for every
$A\notin\mathcal{J}$ (in particular, Player I has a winning strategy in
$G(\mathcal{J},A,f)$ for all $A$ and $f$).

###### Proof.

For $\mathcal{J}\in\\{\mathrm{Fin}^{2},\mathcal{ED}_{\mathrm{Fin}}\\}$, it
follows from Propositions 2.3(5, 6) and 2.5.

For $\mathcal{J}=\mathrm{conv}$, it follows from the facts that
$\mathrm{Fin}^{2}$ has the required property and that
$\mathrm{conv}\restriction A\leq_{\operatorname{K}}\mathrm{Fin}^{2}$ for every
$A\notin\mathrm{conv}$ (by [24] for each $A\notin\mathrm{conv}$ one can find
$B\notin\mathrm{conv}$, $B\subseteq A$ such that $\mathrm{conv}\restriction B$
is isomorphic to $\mathrm{Fin}^{2}$ and this particular isomorphism is a
witness for $\mathrm{conv}\restriction
A\leq_{\operatorname{K}}\mathrm{Fin}^{2}$).

For $\mathcal{J}=\mathcal{ED}$, it follows from the facts that
$\mathcal{ED}_{\mathrm{Fin}}$ has the required property and that
$\mathcal{ED}\restriction A\leq_{\operatorname{K}}\mathcal{ED}_{\mathrm{Fin}}$
for every $A\notin\mathcal{ED}$ (as $A\notin\mathcal{ED}$ allows to construct
inductively an increasing sequence $(m_{n})$ such that $|A_{(m_{n})}|\geq n$
and using this sequence one can define an injective function $f:\Delta\to A$
satisfying $f(n,k)\in\\{m_{n}\\}\times A_{(m_{n})}$, for each
$(n,k)\in\Delta$, which witnesses $\mathcal{ED}\restriction
A\leq_{\operatorname{K}}\mathcal{ED}_{\mathrm{Fin}}$).

The “in particular” part follows from Lemma 5.2(4). ∎

######  Proposition 5.5.

Suppose that $\mathcal{J}=\mathrm{Fin}(\phi)$ is a $\bf{\Sigma^{0}_{2}}$ ideal
such that:

$(\star)\ \forall_{n\in\omega}\ \exists_{k\in\omega}\
\forall_{A\notin\mathrm{Fin}(\phi)}\ \exists_{F\in\mathrm{Fin}}\ (|F|\leq
k\land\phi(F\cap A)\geq n).$

Then for each $A\notin\mathcal{J}$ the ideal $\mathcal{J}\restriction A$ is
not $\leq_{\operatorname{K}}$-above any tall summable ideal (in particular,
Player I has a winning strategy in $G(\mathcal{J},A,f)$ for all $A$ and $f$).

###### Proof.

The ”in particular” part follows from Lemma 5.2(4).

Let $A\notin\mathcal{J}$, $f:A\to\omega$ and $\mathcal{I}_{g}$ be a tall
summable ideal. If $f^{-1}[\\{n\\}]\notin\mathcal{J}$ for some $n\in\omega$
then we are done, so suppose that $f^{-1}[\\{n\\}]\in\mathcal{J}$ for all
$n\in\omega$.

We will inductively pick finite sets $B_{n}$, for $n\in\omega$, such that:

  * •

$\phi(f^{-1}[B_{n}])\geq n$;

  * •

$\sum_{i\in B_{n}}g(i)\leq\frac{1}{2^{n}}$.

In the $n$th step, using condition $(\star)$ find $k\in\omega$ such that:

$\forall_{C\notin\mathrm{Fin}(\phi)}\ \exists_{F\in\mathrm{Fin}}\ (|F|\leq
k\land\phi(F\cap C)\geq n).$

Since $\mathcal{I}_{g}$ is tall, $\lim_{i\to\infty}g(i)=0$. Hence, there is
$G\in\mathrm{Fin}$ such that $g(i)\leq\frac{1}{k\cdot 2^{n}}$ for all
$i\in\omega\setminus G$. Then $A\setminus f^{-1}[G]\notin\mathcal{J}$ (as
$A\notin\mathcal{J}$ and $f^{-1}[G]\in\mathcal{J}$), so there is
$F_{n}\in\mathrm{Fin}$ such that $|F_{n}|\leq k$ and
$\phi(F_{n}\cap(A\setminus f^{-1}[G]))\geq n$. Put $B_{n}=f[F_{n}]$ and
observe that $\sum_{i\in B_{n}}g(i)\leq k\cdot\frac{1}{k\cdot
2^{n}}=\frac{1}{2^{n}}$.

Define $B=\bigcup_{n\in\omega}B_{n}$. Then
$\phi(f^{-1}[B])\geq\phi(f^{-1}[B_{n}])\geq n$ for all $n\in\omega$, so
$f^{-1}[B]\notin\mathcal{J}\restriction A$. On the other hand, $\sum_{i\in
B}g(i)\leq\sum_{n\in\omega}\sum_{i\in
B_{n}}g(i)\leq\sum_{n\in\omega}\frac{1}{2^{n}}<\infty$, so
$B\in\mathcal{I}_{g}$. ∎

######  Proposition 5.6.

The ideals $\mathrm{Fin}$, $\mathcal{ED}$, $\mathcal{ED}_{\mathrm{Fin}}$ and
$\mathcal{W}$ all are examples of $\bf{\Sigma^{0}_{2}}$ ideals satisfying
condition $(\star)$ from Proposition 5.5.

###### Proof.

We will show that $\mathcal{W}$ and $\mathcal{ED}$ satisfy condition $(\star)$
from Proposition 5.5. The cases of $\mathrm{Fin}$ and
$\mathcal{ED}_{\mathrm{Fin}}$ are similar.

Observe that $\mathcal{W}=\mathrm{Fin}(\phi_{\mathcal{W}})$ where

$\phi_{\mathcal{W}}(A)=\sup\\{n\in\omega:A\text{ contains an arithmetic
progression of length }n\\}$

for all $A\subseteq\omega$. We claim that $k=n$ works for all $n$. Fix
$n\in\omega$ and $A\notin\mathcal{W}$. Then $A$ contains arbitrary long finite
arithmetic progressions. In particular, there is an arithmetic progression
$F\subseteq A$ of length $n$. Clearly, $|F|=n$ and $\phi_{\mathcal{W}}(F\cap
A)=\phi_{\mathcal{W}}(F)=n$.

Similarly, $\mathcal{ED}=\mathrm{Fin}(\phi_{\mathcal{ED}})$, where
$\phi_{\mathcal{ED}}(A)=\min\\{n\in\omega:\forall m\geq n(|A_{(m)}|\leq n)\\}$
for all $A\subseteq\omega^{2}$ (see [33]). Again, we claim that $k=n+1$ works
for all $n$. Fix $n\in\omega$ and $A\notin\mathcal{ED}$. Then
$\phi_{\mathcal{ED}}(A)>n$, so $|A_{(m)}|>n$ for some $m\geq n$. Let
$F\subseteq A_{(m)}$ be any set of cardinality $n+1$. Then $|\\{m\\}\times
F|=|F|=n+1$ and $\phi_{\mathcal{ED}}((\\{m\\}\times F)\cap
A)=\phi_{\mathcal{ED}}(\\{m\\}\times F)\geq n+1$, so $\\{m\\}\times F$ is the
required set. ∎

We are ready for the main result of this section.

######  Theorem 5.7.

If $\mathcal{I}$ contains a tall summable ideal and Player I has a winning
strategy in $G(\mathcal{J},A,f)$ for all $A$ and $f$ (in particular, if
$\mathcal{J}$ is a Borel ideal such that $\mathcal{J}\restriction A$ is not
$\leq_{\operatorname{K}}$-above any tall summable ideal for each
$A\notin\mathcal{J}$), then

$\operatorname{cov}(\mathcal{M})\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\omega^{\omega}).$

###### Proof.

The “in particular” part follows from Lemma 5.2(4).

Since $\mathcal{I}$ contains a tall summable ideal, there is
$g:\omega\to[0,+\infty)$ such that $\lim_{i\to\infty}g(i)=0$ and
$\mathcal{I}_{g}=\\{A\subseteq\omega:\sum_{a\in
A}g(a)<\infty\\}\subseteq\mathcal{I}$

Fix a family $\mathcal{A}$ having $\mathcal{J}^{+}$-FIP and such that
$|\mathcal{A}|<\operatorname{cov}(\mathcal{M})$. Let $f\in\omega^{\omega}$. We
need to show that $\\{f^{-1}[B]\\}\cup\mathcal{A}$ has $\mathcal{J}^{+}$-FIP
for some $B\in\mathcal{I}$.

If there is $n\in\omega$ such that $\\{f^{-1}[\\{n\\}]\\}\cup\mathcal{A}$ has
$\mathcal{J}^{+}$-FIP, then the proof is finished in this case.

Now, suppose that $\\{f^{-1}[\\{n\\}]\\}\cup\mathcal{A}$ does not have
$\mathcal{J}^{+}$-FIP for any $n\in\omega$.

We define a poset $(\mathbb{P},\leq)$ as follows:

$\mathbb{P}=\left\\{p\in\mathrm{Fin}\setminus\\{\emptyset\\}:\sum_{i\in
p}g(i)\leq\left(2-\frac{1}{2^{|p|}}\right)\cdot\frac{1}{1+\min p}\right\\}$

and $q\leq p$ $\iff$

  1. (1)

$q\supseteq p$,

  2. (2)

$\min(q\setminus p)>\max p$ [here $\min\emptyset=\omega$ by convention].

Let

$T=\left\\{H\in[\mathcal{A}]^{<\omega}:f^{-1}[\\{n\\}]\cap\bigcap
H\notin\mathcal{J}\text{ for infinitely many $n$}\right\\}.$

For each $H\in[\mathcal{A}]^{<\omega}\setminus T$, there is $k_{H}\in\omega$
such that $f^{-1}[\\{n\\}]\cap\bigcap H\in\mathcal{J}$ for each $n\geq k_{H}$.
Then, $\bigcap H\setminus f^{-1}[\\{i:i<k_{H}\\}]\notin\mathcal{J}$
(otherwise, $\\{f^{-1}[\\{n\\}]\\}\cup\mathcal{A}$ would have
$\mathcal{J}^{+}$-FIP for some $n<k_{H}$) and $f\restriction(\bigcap
H\setminus f^{-1}[\\{i:i<k_{H}\\}])$ is $\mathcal{J}$-to-1. Since Player I has
a winning strategy in $G(\mathcal{J},\bigcap H\setminus
f^{-1}[\\{i:i<k_{H}\\}]),f\restriction(\bigcap H\setminus
f^{-1}[\\{i:i<k_{H}\\}]))$, we can use Lemma 5.2(1) to obtain families
$\mathcal{F}_{0}^{H},\mathcal{F}_{1}^{H},\dots\subseteq\mathrm{Fin}$ such
that:

  1. (1)

$\sup\\{|F|:F\in\mathcal{F}_{n}^{H}\\}<\infty$ for each $n$,

  2. (2)

$\sup\\{\min(F):F\in\mathcal{F}_{n}^{H}\\}=\infty$ for each $n$,

  3. (3)

if $F_{n}\in\mathcal{F}_{n}^{H}$ for each $n$, then
$f^{-1}[\bigcup_{n\in\omega}F_{n}]\notin\mathcal{J}\restriction(\bigcap
H\setminus f^{-1}[\\{i:i<k_{H}\\}]))$.

Let us define the following sets:

  1. (1)

$D_{H}=\\{p\in\mathbb{P}:f^{-1}[p]\cap\bigcap H\notin\mathcal{J}\\}$ for each
$H\in T$,

  2. (2)

$E^{n}_{H}=\\{p\in\mathbb{P}:F\subseteq p\text{ for some
}F\in\mathcal{F}_{n}^{H}\\}$ for each $n\in\omega$ and
$H\in[\mathcal{A}]^{<\omega}\setminus T$.

Below, we show that these sets are dense in $(\mathbb{P},\leq)$.

(1) Let $H\in T$ and $p\in\mathbb{P}$. Since $\lim_{i\to\infty}g(i)=0$, there
is $t\in\omega$ such that

$g(i)<\left(\frac{1}{2^{|p|}}-\frac{1}{2^{|p|+1}}\right)\cdot\frac{1}{1+\min
p}$

for all $i>t$. Since $H\in T$, there is $n>\max(t,\max(p))$ with
$f^{-1}[\\{n\\}]\cap\bigcap H\notin\mathcal{J}$. Let $q=p\cup\\{n\\}$. Then
$q\in\mathbb{P}$ as

$\begin{split}\sum_{i\in q}g(i)&<\sum_{i\in
p}g(i)+\left(\frac{1}{2^{|p|}}-\frac{1}{2^{|p|+1}}\right)\cdot\frac{1}{1+\min
p}\leq\\\ &\leq\left(2-\frac{1}{2^{|p|}}\right)\cdot\frac{1}{1+\min
p}+\left(\frac{1}{2^{|p|}}-\frac{1}{2^{|p|+1}}\right)\cdot\frac{1}{1+\min
p}\\\ &=\left(2-\frac{1}{2^{|p|+1}}\right)\cdot\frac{1}{1+\min
p}=\left(2-\frac{1}{2^{|q|}}\right)\cdot\frac{1}{1+\min q}.\end{split}$

Since $q\in D_{H}$ and $q\leq p$, $D_{H}$ is dense in $(\mathbb{P},\leq)$.

(2) Let $n\in\omega$, $H\in[\mathcal{A}]^{<\omega}\setminus T$ and
$p\in\mathbb{P}$. Put $k_{n}=\sup\\{|F|:F\in\mathcal{F}_{n}^{H}\\}$. Since
$\lim_{i\to\infty}g(i)=0$, there is $t\in\omega$ such that

$g(i)<\frac{1}{k_{n}}\cdot\left(\frac{1}{2^{|p|}}-\frac{1}{2^{|p|+k_{n}}}\right)\cdot\frac{1}{1+\min
p}$

for all $i>t$. Since $\sup\\{\min(F):F\in\mathcal{F}_{n}^{H}\\}=\infty$, there
is $F\in\mathcal{F}_{n}^{H}$ with $\min(F)>\max(t,\max(p))$. Let $q=p\cup F$.
Then $q\in\mathbb{P}$ as

$\begin{split}\sum_{i\in q}g(i)&<\sum_{i\in p}g(i)+\sum_{i\in
F}\frac{1}{k_{n}}\cdot\left(\frac{1}{2^{|p|}}-\frac{1}{2^{|p|+k_{n}}}\right)\cdot\frac{1}{1+\min
p}\leq\\\ &\leq\left(2-\frac{1}{2^{|p|}}\right)\cdot\frac{1}{1+\min
p}+\left(\frac{1}{2^{|p|}}-\frac{1}{2^{|p|+k_{n}}}\right)\cdot\frac{1}{1+\min
p}\\\ &=\left(2-\frac{1}{2^{|p|+k_{n}}}\right)\cdot\frac{1}{1+\min
p}\leq\left(2-\frac{1}{2^{|q|}}\right)\cdot\frac{1}{1+\min q}.\end{split}$

Since $q\in E^{n}_{H}$ and $q\leq p$, $E_{H}^{n}$ is dense in
$(\mathbb{P},\leq)$.

Let $\mathcal{D}$ be the family of all the sets of the form $D_{H}$ and
$E_{H}^{n}$.

Since $\mathbb{P}$ is countable and
$|\mathcal{D}|<\operatorname{cov}(\mathcal{M})$, there is a filter
$\mathcal{G}\subseteq\mathbb{P}$ such that $\mathcal{G}\cap D\neq\emptyset$
for each $D\in\mathcal{D}$ (see e.g. [6, Theorem 7.13])).

We define $B=\bigcup\mathcal{G}$. If we show that $B\in\mathcal{I}$ and
$\\{f^{-1}[B]\\}\cup\mathcal{A}$ has $\mathcal{J}^{+}$-FIP, then the proof
will be finished in this case.

Let $B=\\{b_{i}:\ i\in\omega\\}$ be the increasing enumeration of $B$. For
each $i\in\omega$, there is $p_{i}\in\mathcal{G}$ with $b_{i}\in p_{i}$. Since
$\mathcal{G}$ is a filter, without loss of generality we may assume that
$p_{0}\subseteq p_{1}\subseteq\dots$. Then

$\sum_{i\in\omega}g(b_{i})=\lim_{i\to\infty}\sum_{j\in
p_{i}}g(j)\leq\lim_{i\to\infty}\left(2-\frac{1}{2^{|p_{i}|}}\right)\cdot\frac{1}{1+\min
p_{i}}=\frac{2}{1+\min B}<\infty,$

so $B\in\mathcal{I}_{g}\subseteq\mathcal{I}$.

Finally, we show that $\\{f^{-1}[B]\\}\cup\mathcal{A}$ has
$\mathcal{J}^{+}$-FIP. Let $H\in[\mathcal{A}]^{<\omega}$.

If $H\in T$, take $p\in\mathcal{G}\cap D_{H}$. Then $f^{-1}[B]\cap\bigcap
H\supseteq f^{-1}[p]\cap\bigcap H\notin\mathcal{J}$.

If $H\in[\mathcal{A}]^{<\omega}\setminus T$, take $p_{n}\in\mathcal{G}\cap
E_{H}^{n}$ for each $n\in\omega$, and pick $F_{n}\in\mathcal{F}_{n}^{H}$ such
that $F_{n}\subseteq p_{n}$. Then

$f^{-1}[B]\cap\bigcap H\supseteq
f^{-1}\left[\bigcup_{n\in\omega}p_{n}\right]\cap\bigcap H\supseteq
f^{-1}\left[\bigcup_{n\in\omega}F_{n}\right]\cap\bigcap H\notin\mathcal{J}.$

∎

######  Corollary 5.8.

Assume $\operatorname{cov}(\mathcal{M})=\mathfrak{c}$.

  1. (1)

If $\mathcal{I}$ contains a tall summable ideal and Player I has a winning
strategy in $G(\mathcal{J},A,f)$ for all $A$ and $f$ (in particular, if
$\mathcal{J}$ is a Borel ideal such that $\mathcal{J}\restriction A$ is not
$\leq_{\operatorname{K}}$-above any tall summable ideal for each
$A\notin\mathcal{J}$), then there exists an $\mathcal{I}$-ultrafilter which is
not a $\mathcal{J}$-point.

  2. (2)

There exists an $\mathcal{I}$-ultrafilter for each tall analytic P-ideal ideal
$\mathcal{I}$. (It is an extension of [18, Proposition 2.5.3], where the
author proved under $\operatorname{cov}(\mathcal{M})=\mathfrak{c}$ that there
exists an $\mathcal{I}$-ultrafilter for each summable ideal.)

  3. (3)

For each tall analytic P-ideal $\mathcal{I}$, there exists an
$\mathcal{I}$-ultrafilter which is not a $\mathcal{W}$-point. (It is an
extension of [18, Proposition 2.5.2], where the author proved under
$\operatorname{cov}(\mathcal{M})=\mathfrak{c}$ that there exists an
$\mathcal{I}_{1/n}$-ultrafilter which is not a $\mathcal{W}$-ultrafilter.)

###### Proof.

(1) follows from Theorems 5.7 and 3.2. (2) follows from (1) and Propositions
2.2. (3) follows from (1), Propositions 2.2, 5.5 and 5.6. ∎

##  6\. P-points which are not $\mathcal{J}$-ultrafilters

Now we turn our attention to ideals $\mathcal{J}$ such that it is consistent
that there is a P-point which is not a $\mathcal{J}$-ultrafilter. We start
with an immediate consequence of our previous results.

######  Proposition 6.1.

Let $\mathcal{J}$ be a $\text{P}^{+}(\mathcal{J})$-ideal which is
$\leq_{K}$-homogeneous.

  1. (1)

Assume CH. The following conditions are equivalent.

    1. (a)

$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathcal{J}$.

    2. (b)

There is a P-point which is not a $\mathcal{J}$-point.

    3. (c)

There is a P-point which is not a weak $\mathcal{J}$-ultrafilter.

    4. (d)

There is a P-point which is not a $\mathcal{J}$-ultrafilter.

  2. (2)

If $\mathcal{J}$ is $F_{\sigma}$, then CH can be relaxed to the assumption
$\mathfrak{p}=\mathfrak{c}$.

###### Proof.

It follows from Theorems 4.3(1) and 2.4(1). In the case of $F_{\sigma}$-ideals
additionally we need to use Theorems 4.3(4). ∎

Although the assumptions on $\mathcal{J}$ in the above proposition seem to be
strong, it can be applied for instance for
$\mathcal{J}=\mathcal{ED}_{\mathrm{Fin}}$ and $\mathcal{J}=\mathcal{W}$ (by
Propositions 2.6(1), 2.5 and 2.3(3)). However, we cannot drop the assumption
on the ideal $\mathcal{J}$ in general. Indeed, it is known that
$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathrm{conv}$ ([24]), but all
P-points (i.e. $\mathrm{Fin}^{2}$-ultrafilters) are
$\mathrm{conv}$-ultrafilters by Theorem 2.4(1). Nevertheless, in Theorem 6.3
we show that the assumption on the ideal $\mathcal{J}$ can be removed if we
replace the Katětov order $\leq_{\operatorname{K}}$ by the property that
$\mathcal{J}$ can be extended to a $P^{+}$-ideal. In order to show it, we will
need the following rather easy result about $P^{+}$-ideals.

######  Lemma 6.2.

Let $\mathcal{I}$ and $\mathcal{J}$ be ideals on $\omega$.

  1. (1)

If $\mathcal{I}\leq_{\operatorname{K}}\mathcal{J}$ and $\mathcal{J}$ is a
$P^{+}$-ideal, then $\mathcal{I}$ can be extended to a $P^{+}$-ideal.

  2. (2)

If $\mathcal{J}$ is a $P^{+}$-ideal and
$\mathcal{A}\subseteq\mathcal{P}(\omega)$ is a countable family such that
$\omega\notin\mathcal{J}(\mathcal{A})$, then $\mathcal{J}(\mathcal{A})$ is a
$P^{+}$-ideal.

  3. (3)

If $\mathcal{J}$ is not a $P^{+}$-ideal, then there is a countable family
$\mathcal{A}\subseteq\mathcal{P}(\omega)$ such that
$\omega\notin\mathcal{J}(\mathcal{A})$ and
$\mathrm{Fin}^{2}\leq_{\operatorname{K}}\mathcal{J}(\mathcal{A})$.

###### Proof.

(1) Let $f:\omega\to\omega$ be such that $f^{-1}[B]\in\mathcal{J}$ for each
$B\in\mathcal{I}$. Let $\mathcal{K}=\\{B:f^{-1}[B]\in\mathcal{J}\\}$. It is
easy to see that $\mathcal{K}$ is an ideal and
$\mathcal{I}\subseteq\mathcal{K}$. We show that $\mathcal{K}$ is a
$P^{+}$-ideal. Let $(B_{n})\subseteq\mathcal{K}^{+}$ and $B_{n}\supseteq
B_{n+1}$ for each $n$. Then $f^{-1}[B_{n}]\in\mathcal{J}^{+}$ and
$f^{-1}[B_{n}]\supseteq f^{-1}[B_{n+1}]$ for each $n$. Since $\mathcal{J}$ is
a $P^{+}$-ideal, there is $A\notin\mathcal{J}$ such that $A\setminus
f^{-1}[B_{n}]$ is finite for each $n$. Then $B=f[A]\notin\mathcal{K}$ and
$B\setminus B_{n}$ is finite for each $n$.

(2) Let $\mathcal{A}=\\{A_{n}:n\in\omega\\}$ be such that
$\omega\notin\mathcal{J}(\mathcal{A})$. Take any decreasing sequence
$\\{B_{n}:n\in\omega\\}\subseteq\mathcal{J}(\mathcal{A})^{+}$ and define
$C_{0}=\omega$ and $C_{n+1}=B_{n}\setminus\bigcup_{i<n}A_{i}$. Then
$C_{n}\notin\mathcal{J}(\mathcal{A})$ and $C_{n}\supseteq C_{n+1}$ for each
$n$. Since $\mathcal{J}$ is a $P^{+}$-ideal, there is
$C^{\prime}\notin\mathcal{J}$ such that $C^{\prime}\setminus C_{n}$ is finite
for each $n$. Define $C=C^{\prime}\cap C_{1}$ and note that
$C\notin\mathcal{J}$ as $C^{\prime}\setminus C=C^{\prime}\setminus C_{1}$ is
finite. Once we show that $C\notin\mathcal{J}(\mathcal{A})$, the proof will be
finished. Suppose that $C\in\mathcal{J}(\mathcal{A})$. Then there is $n$ such
that $C\setminus\bigcup_{i<n}A_{i}\in\mathcal{J}$. Moreover,
$C\cap\bigcup_{i<n}A_{i}$ is finite since $C\cap\bigcup_{i<n}A_{i}\subseteq
C\setminus C_{n+1}$. Thus, $C\in\mathcal{J}$, a contradiction.

(3) Since $\mathcal{J}$ is not a $P^{+}$-ideal, there is a decreasing sequence
$(B_{n})$ of sets not belonging to $\mathcal{J}$ such that if $X\setminus
B_{n}\in\mathrm{Fin}$ for all $n$, then $X\in\mathcal{J}$. Let
$A_{0}=\omega\setminus B_{0}$ and $A_{n+1}=B_{n}\setminus B_{n+1}$ for all
$n$. Observe that $\bigcup_{i\leq n}A_{i}=\omega\setminus
B_{n}\notin\mathcal{J}^{*}$, so
$\omega\notin\mathcal{J}(\\{A_{n}:n\in\omega\\})$. What is more, any one-to-
one function $f:\omega\to\omega^{2}$ such that
$f[A_{n}]\subseteq\\{n\\}\times\omega$ witnesses
$\mathrm{Fin}^{2}\leq_{\operatorname{K}}\mathcal{J}(\\{A_{n}:n\in\omega\\})$.
∎

######  Theorem 6.3.

Let $\mathcal{J}$ be an ideal on $\omega$.

  1. (1)

The following conditions are equivalent.

    1. (a)

$\mathcal{J}$ is extendable to a $P^{+}$-ideal.

    2. (b)

$\mathrm{Fin}^{2}\not\preceq_{\omega^{\omega},1-1}^{\omega_{1}}\mathcal{J}$.

    3. (c)

$\mathrm{Fin}^{2}\not\preceq_{\omega^{\omega},\mathrm{Fin}-1}^{\omega_{1}}\mathcal{J}$.

    4. (d)

$\mathrm{Fin}^{2}\not\preceq_{\omega^{\omega},\omega^{\omega}}^{\omega_{1}}\mathcal{J}$.

  2. (2)

Under CH, the above conditions are equivalent to the following conditions.

    1. (e)

There is a P-point which is not a $\mathcal{J}$-point.

    2. (f)

There is a P-point which is not a weak $\mathcal{J}$-ultrafilter.

    3. (g)

There is a P-point which is not a $\mathcal{J}$-ultrafilter.

  3. (3)

If $\mathcal{J}$ is Borel, then CH can be relaxed to the assumption
$\mathfrak{p}=\mathfrak{c}$ in (2).

  4. (4)

If $\mathcal{J}$ is Borel, then the following conditions are equivalent.

    1. (a)

It is consistent that there exists a P-point which is not a
$\mathcal{J}$-point (weak $\mathcal{J}$-ultrafilter or
$\mathcal{J}$-ultrafilter, resp.).

    2. (b)

Under CH, there exists a P-point which is not a $\mathcal{J}$-point (weak
$\mathcal{J}$-ultrafilter or $\mathcal{J}$-ultrafilter, resp.).

In particular, if one can find a P-point which is not a $\mathcal{J}$-point
(weak $\mathcal{J}$-ultrafilter or $\mathcal{J}$-ultrafilter, resp.) in some
sophisticated model of ZFC, then one could do it already under CH.

###### Proof.

(1a)$\implies$(1b) Let $\mathcal{J}^{\prime}\supseteq\mathcal{J}$ and
$\mathcal{J}^{\prime}$ be a $P^{+}$-ideal. By Lemma 6.2(2),
$\mathcal{J}^{\prime}(\mathcal{A})$ is a $P^{+}$-ideal for each countable
family $\mathcal{A}\subseteq\mathcal{P}(\omega)$ such that
$\omega\notin\mathcal{J}^{\prime}(\mathcal{A})$. So
$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathcal{J}^{\prime}(\mathcal{A})$,
by Proposition 2.3(3). Thus,
$\mathrm{Fin}^{2}\not\preceq_{\omega^{\omega},1-1}^{\omega_{1}}\mathcal{J}$.

The implications (1b)$\implies$(1c) and (1c)$\implies$(1d) are obvious.

(1d)$\implies$(1a) Since
$\mathrm{Fin}^{2}\not\preceq_{\omega^{\omega},\omega^{\omega}}^{\omega_{1}}\mathcal{J}$,
there is an ideal $\mathcal{J}^{\prime}\geq_{\operatorname{K}}\mathcal{J}$
such that
$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathcal{J}^{\prime}(\mathcal{A})$
for each countable $\mathcal{A}\subseteq\mathcal{P}(\omega)$ such that
$\omega\notin\mathcal{J}^{\prime}(\mathcal{A})$. By Lemma 6.2(3),
$\mathcal{J}^{\prime}$ is a $P^{+}$-ideal, hence by Lemma 6.2(1),
$\mathcal{J}$ can be extended to a $P^{+}$-ideal.

(1b)$\implies$ (2e) Under CH, it follows from Corollary 3.5(1) and Theorem
2.4.

The implications (2e)$\implies$ (2f) and (2f)$\implies$ (2g) are obvious (and
hold in ZFC).

(2g)$\implies$ (1a) Take a P-point $\mathcal{U}$ which is not a
$\mathcal{J}$-ultrafilter (i.e.
$\mathcal{J}\leq_{\operatorname{K}}\mathcal{U}^{*}$). Then $\mathcal{U}^{*}$
is a $P^{+}$-ideal, so $\mathcal{J}$ can be extended to a $P^{+}$-ideal by
Lemma 6.2(1). (Note that this argument works in ZFC.)

(3) Taking into account that only the proof of the implication “(1b)$\implies$
(2e)” was not done in ZFC, and (1a)$\iff$(1b) holds in ZFC, we only need to
show the implication “(1a)$\implies$(2e)” under the assumption
$\mathfrak{p}=\mathfrak{c}$ with $\mathcal{J}$ being Borel.

Assume that $\mathfrak{p}=\mathfrak{c}$ and $\mathcal{J}$ is a Borel ideal
that can be extended to a $P^{+}$-ideal. It is known that if a Borel ideal can
be extended to a $P^{+}$-ideal, then it can be extended to an
$F_{\sigma}$-ideal (see e.g. [33, Theorem 3.2.7] or [22, Corollary 3.7]). Let
$\widehat{\mathcal{J}}$ be an $F_{\sigma}$ ideal such that
$\mathcal{J}\subseteq\widehat{\mathcal{J}}$. Then
$\widehat{\mathcal{J}}\restriction A$ is also an $F_{\sigma}$ ideal for each
$A\notin\widehat{\mathcal{J}}$. Thus,
$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\widehat{\mathcal{J}}\restriction
A$ for each $A\notin\widehat{\mathcal{J}}$ by Proposition 2.3(3). By Theorems
4.1, 3.6 and 2.9,
$\mathfrak{u}(\mathrm{Fin}^{2},\omega^{\omega},\widehat{\mathcal{J}},1-1)>\mathfrak{c}=|\omega^{\omega}|$.
By Theorem 3.4, we obtain a $\mathrm{Fin}^{2}$-ultrafilter (hence a P-point by
Theorem 2.4(1)) which is not a $\widehat{\mathcal{J}}$-point (so it is not a
$\mathcal{J}$-point as well).

(4) The implication “(4b)$\implies$(4a)” is obvious, so below we show only the
implication “(4a)$\implies$(4b)”. Assume that it is consistent that there
exists a P-point that is not a $\mathcal{J}$-point (weak
$\mathcal{J}$-ultrafilter or $\mathcal{J}$-ultrafilter, resp.). Then
consistently
$\mathfrak{u}(\mathrm{Fin}^{2},\omega^{\omega},\mathcal{J},1-1)=\mathfrak{c}^{+}$
($\mathfrak{u}(\mathrm{Fin}^{2},\omega^{\omega},\mathcal{J},Fin-1)=\mathfrak{c}^{+}$
or
$\mathfrak{u}(\mathrm{Fin}^{2},\omega^{\omega},\mathcal{J},\omega^{\omega})=\mathfrak{c}^{+}$,
resp.) by Theorem 3.4, so in particular it is consistent that
$\mathrm{Fin}^{2}\not\preceq_{\omega^{\omega},1-1}^{\omega_{1}}\mathcal{J}$
($\mathrm{Fin}^{2}\not\preceq_{\omega^{\omega},Fin-1}^{\omega_{1}}\mathcal{J}$
or
$\mathrm{Fin}^{2}\not\preceq_{\omega^{\omega},\omega^{\omega}}^{\omega_{1}}\mathcal{J}$,
resp.). Then, by item (1), it is consistent that $\mathcal{J}$ can be extended
to a $P^{+}$-ideal, and since $\mathcal{J}$ is Borel, $\mathcal{J}$ can be
extended to an $F_{\sigma}$-ideal (by [33, Theorem 3.2.7] or [22, Corollary
3.7]). However, if $\mathcal{J}$ is a Borel ideal, then the sentence
“$\mathcal{J}$ can be extended to an $F_{\sigma}$-ideal” is absolute, so
$\mathcal{J}$ can be extended to an $F_{\sigma}$ ideal in ZFC. Then, using (2)
we obtain, under CH, that there exists a P-point which is not a
$\mathcal{J}$-point (weak $\mathcal{J}$-ultrafilter or
$\mathcal{J}$-ultrafilter, resp.). ∎

An ideal $\mathcal{I}$ has the _property $\operatorname{FinBW}$_ (in short:
$\mathcal{I}\in\operatorname{FinBW}$) if for each bounded sequence
$(x_{n})_{n\in\omega}$ of reals there is $A\notin\mathcal{I}$ such that the
subsequence $(x_{n})_{n\in A}$ is convergent ([14]).

The ideal $\mathcal{I}_{d}=\\{A\subseteq\omega:\lim_{n\to\infty}|A\cap
n|/n=0\\}$ of asymptotic density zero sets, the ideal
$\mathcal{I}_{u}=\\{A\subseteq\omega:\lim_{n\to\infty}\max\\{|A\cap\\{m,m+1,\ldots,m+n-1\\}|:m\in\omega\\}/n=0\\}$
of Banach (a.k.a. uniform) density zero sets, the ideal $\mathrm{nwd}$, the
ideal $\mathrm{conv}$ and the ideal $\mathcal{L}$ generated by lacunary sets
(recall: a set $A\subseteq\omega$ is _lacunary_ ([12]) if
$\lim_{n\to\infty}(a_{n+1}-a_{n})=\infty$, where $(a_{n})_{n\in\omega}$ is the
increasing enumeration of all elements of $A$) do not have the
$\operatorname{FinBW}$ property (see [21, Example 3], [1, Corollary 1], [14,
Example 3.1], [2, Proposition 6.4] and [28, Lemma 3.2], respectively).

######  Proposition 6.4.

If $\mathcal{J}\notin\operatorname{FinBW}$, then every P-point is a
$\mathcal{J}$-ultrafilter. In particular, every P-point is an
$\mathcal{I}_{d}$-ultrafilter, $\mathcal{I}_{u}$-ultrafilter,
$\mathrm{nwd}$-ultrafilter, $\mathrm{conv}$-ultrafilter and
$\mathcal{L}$-ultrafilter (note that the cases of the ideals $\mathcal{I}_{d}$
and $\mathcal{L}$ were earlier done in a different manner in [18, Proposition
2.3.3]).

###### Proof.

Let $\mathcal{U}$ be a P-point and suppose that $\mathcal{U}$ is not a
$\mathcal{J}$-ultrafilter i.e.
$\mathcal{J}\leq_{\operatorname{K}}\mathcal{U}^{*}$. Since
$\mathcal{J}\notin\operatorname{FinBW}$, then
$\mathrm{conv}\leq_{\operatorname{K}}\mathcal{J}$ ([33, Section 2.7], see also
[2, Proposition 6.4]). Consequently,
$\mathrm{conv}\leq_{\operatorname{K}}\mathcal{U}^{*}$, so $\mathcal{U}$ is not
a P-point by Theorem 2.4(1), a contradiction. ∎

It is known ([14, Proposition 4.1 and 3.4]) that
$\mathcal{J}\in\operatorname{FinBW}$ for every ideal $\mathcal{J}$ which can
be extended to an $F_{\sigma}$ ideal. In the realm of analytic P-ideals,
$\mathcal{J}\in\operatorname{FinBW}$ is even equivalent to extendability of
$\mathcal{J}$ to an $F_{\sigma}$ ideal ([2, Proposition 6.5]). It was a
question of Hrušak ([23, Question 5.16]) whether the same holds in the realm
of all Borel ideals, but the answer to his question is negative as was
recently showed by Kwela ([30]).

######  Theorem 6.5.

If $\mathcal{J}$ is an $F_{\sigma}$ ideal and $\mathcal{I}$ is not a weak
P-ideal, then

$\operatorname{cov}(\mathcal{M})\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$

for every $\mathcal{F}\subseteq\omega^{\omega}$.

###### Proof.

Let $\phi$ be a lsc submeasure such that $\mathcal{J}=\mathrm{Fin}(\phi)$. Let
$\mathcal{A}$ be a family having $\mathcal{J}^{+}$-FIP and such that
$|\mathcal{A}|<\operatorname{cov}(\mathcal{M})$. Fix $f\in\mathcal{F}$.

Since $\mathcal{I}$ is not a weak P-ideal, there is a partition
$\\{B_{n}:n\in\omega\\}\subseteq\mathcal{I}$ of $\omega$ such that for each
$B\notin\mathcal{I}$ there is $n$ with $|B_{n}\cap B|=\omega$. For each
$n\in\omega$, we define $C_{n}=f^{-1}[B_{n}]$. We have 2 cases.

Case 1. There is $n\in\omega$ such that $\bigcap F\cap C_{n}\notin\mathcal{J}$
for each finite $F\subseteq\mathcal{A}$. Then, we put $C=B_{n}$. Since
$C\in\mathcal{I}$ and $\mathcal{A}\cup\\{f^{-1}[C]\\}$ has
$\mathcal{J}^{+}$-FIP, the proof is finished in this case.

Case 2. For each $n\in\omega$ there exists a finite $F\subseteq\mathcal{A}$
with $\bigcap F\cap C_{n}\in\mathcal{J}$.

We define a poset $(\mathbb{P},\leq)$ as follows:

$\mathbb{P}=\left\\{(H,n)\in[\omega]^{<\omega}\times\omega:H\subseteq\bigcup_{i<n}C_{i}\right\\}$

and $(H_{1},n_{1})\leq(H_{2},n_{2})\iff$

  1. (1)

$H_{1}\supseteq H_{2}$,

  2. (2)

$n_{1}\geq n_{2}$,

  3. (3)

$\max H_{2}<\min(H_{1}\setminus H_{2})$ [here $\max\emptyset=-1$ and
$\min\emptyset=\omega$ by convention],

  4. (4)

$(H_{1}\setminus H_{2})\cap\bigcup_{i<n_{2}}C_{i}=\emptyset$.

Now we define sets

  1. (1)

$D_{F,N}=\\{(H,n)\in\mathbb{P}:\phi(\bigcap F\cap H)>N\\}$ for each finite
$F\subseteq\mathcal{A}$ and $N\in\omega$,

  2. (2)

$E_{k}=\\{(H,n)\in\mathbb{P}:n>k\\}$ for each $k\in\omega$

and below show that these sets are dense in $(\mathbb{P},\leq)$.

First we show that sets $D_{F,N}$ are dense. Take any $(H,n)\in\mathbb{P}$. In
the present case, we know that for each $i<n$ there is a finite
$F_{i}\subseteq\mathcal{A}$ with $\bigcap F_{i}\cap C_{i}\in\mathcal{J}$.
Since $\mathcal{A}$ has $\mathcal{J}^{+}$-FIP, $X=\bigcap_{i<n}\bigcap
F_{i}\notin\mathcal{J}$. Consequently, there is a finite set
$H^{\prime}\subseteq X\setminus(\bigcup_{i<n}C_{i}\cup\max H)$ with
$\phi(H^{\prime})>N$. We put $G=H\cup H^{\prime}$ and
$m=\max(\\{i\in\omega:G\cap C_{i}\neq\emptyset\\}\cup\\{n\\})$. Since
$(G,m)\in D_{F,N}$ and $(G,m)\leq(H,n)$, we obtain denseness of $D_{F,N}$.

Now we show that $E_{k}$ are dense. Take any $(H,n)\in\mathbb{P}$. Let $G=H$
and $m=\max(n,k+1)$. Since $(G,m)\in E_{k}$ and $(G,m)\leq(H,n)$, we obtain
denseness of $E_{k}$.

Let $\mathcal{D}$ be the family of all the sets of the form $D_{F,N}$ and
$E_{k}$.

Since $\mathbb{P}$ is countable and
$|\mathcal{D}|<\operatorname{cov}(\mathcal{M})$, there is a filter
$\mathcal{G}\subseteq\mathbb{P}$ such that $\mathcal{G}\cap D\neq\emptyset$
for each $D\in\mathcal{D}$ (see e.g. [6, Theorem 7.13])).

We define

$B=f[A],\text{\ \ where\ \ }A=\bigcup\\{H:(H,n)\in\mathcal{G}\text{ for some
$n\in\omega$}\\}.$

If we show that $B\in\mathcal{I}$ and $\mathcal{A}\cup\\{f^{-1}[B]\\}$ has
$\mathcal{J}^{+}$-FIP, the proof will be finished.

First we show that $\mathcal{A}\cup\\{f^{-1}[B]\\}$ has $\mathcal{J}^{+}$-FIP.
Take any finite set $F\subseteq\mathcal{A}$. Let $N\in\omega$. Since
$\mathcal{G}$ is $\mathcal{D}$-generic, there is $(H,n)\in\mathcal{G}\cap
D_{F,N}$. In particular, $H\subseteq A$ and $\phi(\bigcap F\cap H)>N$. Since
$A\subseteq f^{-1}[B]$, we obtain $\phi(\bigcap F\cap f^{-1}[B])>N$.
Consequently, $\phi(\bigcap F\cap f^{-1}[B])=\infty$, so $\bigcap F\cap
f^{-1}[B]\notin\mathcal{J}$.

Finally, we show that $B\in\mathcal{I}$. If we show that $A\cap C_{k}$ is
finite for each $k$, then $B\cap B_{k}$ will be finite for each $k$, and
consequently $B\in\mathcal{I}$.

Take any $k\in\omega$. Since $\mathcal{G}$ is $\mathcal{D}$-generic, there is
$(H,n)\in\mathcal{G}\cap E_{k}$. We claim that $A\cap C_{k}\subseteq H$ (in
particular, it means that $A\cap C_{k}$ is finite).

Take any $a\in A\cap C_{k}$. Then there is $(G,m)\in\mathcal{G}$ such that
$a\in G$. Since $\mathcal{G}$ is a filter, there is $(K,l)\in\mathcal{G}$ such
that $(K,l)\leq(H,n)$ and $(K,l)\leq(G,m)$. We have 2 cases.

Case 1. Assume that $m\leq k$. Since $(G,m)\in\mathbb{P}$,
$G\subseteq\bigcup_{i<m}C_{i}$, and consequently, $a\notin C_{k}$, a
contradiction, so this case is not possible.

Case 2. Assume that $m>k$. Since $(K,l)\leq(H,n)$, $(K\setminus
H)\cap\bigcup_{i<n}C_{i}=\emptyset$. Since $(H,n)\in E_{k}$, $n>k$.
Consequently, $(K\setminus H)\cap C_{k}=\emptyset$. Suppose, in the sake of
contradiction, that $a\notin H$. Since $a\in G\subseteq K$, $a\in K\setminus
H$. Since $a\in C_{k}$, $a\in(K\setminus H)\cap C_{k}$, a contradiction. ∎

The following corollary provides an extension of [19, Theorem 3.1], where the
author proved that under $\operatorname{cov}(\mathcal{M})=\mathfrak{c}$, there
exists a P-point which is not a $\mathcal{J}$-ultrafilter for each
$F_{\sigma}$-ideal $\mathcal{J}$.

######  Corollary 6.6.

Assume $\operatorname{cov}(\mathcal{M})=\mathfrak{c}$. If an ideal
$\mathcal{J}$ can be extended to an $F_{\sigma}$ ideal (in particular, if
$\mathcal{J}$ is an analytic P-ideal with the property $\operatorname{FinBW}$)
and $\mathcal{I}$ is not a weak P-ideal, then there exists an
$\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-point. In particular,
then there exists a P-point which is not a $\mathcal{J}$-point.

###### Proof.

Let $\mathcal{K}$ be an $F_{\sigma}$ ideal with
$\mathcal{J}\subseteq\mathcal{K}$. By Theorems 3.2 and 6.5, there exists an
ultrafilter $\mathcal{U}$ such that
$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{U}^{*}$ and
$\mathcal{K}\subseteq\mathcal{U}^{*}$. Then
$\mathcal{J}\subseteq\mathcal{U}^{*}$, so $\mathcal{U}$ is not a
$\mathcal{J}$-point. The first “in particular” part follows from the fact that
any analytic P-ideal with the property $\operatorname{FinBW}$ can be extended
to an $F_{\sigma}$-ideal (see [14, Theorem 4.2]). The second “in particular”
part follows from an easy fact that the ideal $\mathrm{Fin}^{2}$ is not a weak
P-ideal, and not-so-easy fact that $\mathrm{Fin}^{2}$-ultrafilters coincide
with P-points (see Theorem 2.4(1)). ∎

A set $A\subseteq\omega$ is _thin_ ([7]) if
$\lim_{n\to\infty}a_{n}/a_{n+1}=0$, where $(a_{n})_{n\in\omega}$ is the
increasing enumeration of elements of $A$. Using the ratio test for the
convergence of a series, one can easily see that the ideal $\mathcal{T}$
generated by thin sets is contained in the summable ideal
$\mathcal{I}_{1/n}=\\{A\subseteq\omega:\sum_{n\in A}1/n<\infty\\}$. Thus,
using the above corollary, we see that there exists a P-point which is not a
$\mathcal{T}$-point (note that this particular instance of the above corollary
in the case of the ideal $\mathcal{T}$ was earlier proved in [19, Theorem
3.4]).

##  7\. $\mathcal{I}$-ultrafilters which are not P-points

In this section we are interested in ideals $\mathcal{I}$ such that it is
consistent that there is an $\mathcal{I}$-ultrafilter which is not a P-point.
It will turn out that such ideals can be characterized using the Katětov
order. However, we will start this section with some sufficient conditions for
existence of $\mathcal{I}$-ultrafilter which is not a P-point. Results of this
section combined with the ones from the previous section will give us a
characterization of Borel ideals $\mathcal{I}$ for which the notions of
P-point and $\mathcal{I}$-ultrafilter coincide.

######  Theorem 7.1.

Let $\mathcal{F}\subseteq\omega^{\omega}$ and $\mathcal{I}$ be an ideal on
$\omega$.

  1. (1)

If $\mathcal{I}\not\leq_{\mathcal{F}}\mathrm{Fin}^{2}$ and $\mathcal{I}$ is a
tall ideal (in particular, if $\mathcal{I}$ is a tall P-ideal), then

$\max\\{\operatorname{\operatorname{add}^{*}}(\mathcal{I}),\omega_{1}\\}\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin}^{2},\mathcal{F}).$

  2. (2)

If $\mathcal{I}$ contains a tall summable ideal, then

$\operatorname{cov}(\mathcal{M})\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin}^{2},\mathcal{F}).$

  3. (3)

If $\mathcal{I}$ is a tall analytic P-ideal, then

$\max(\operatorname{\operatorname{add}^{*}}(\mathcal{I}),\operatorname{cov}(\mathcal{M}))\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin}^{2},\mathcal{F}).$

###### Proof.

(1) follows from Theorems 2.8, 4.1 and Proposition 2.5. For the “in
particular” part we additionally need Proposition 2.3(5). (2) follows from
Theorem 5.7 and Proposition 5.4. (3) follows from (1), (2) and Proposition
2.2. ∎

######  Corollary 7.2.

Let $\mathcal{I}$ be an ideal on $\omega$.

  1. (1)

Assume
$\max(\operatorname{\operatorname{add}^{*}}(\mathcal{I}),\omega_{1})=\mathfrak{c}$.

    1. (a)

If $\mathcal{I}\not\leq_{\operatorname{K}}\mathrm{Fin}^{2}$
($\mathcal{I}\not\leq_{\operatorname{KB}}\mathrm{Fin}^{2}$ or
$\mathcal{I}\not\leq_{\operatorname{P}}\mathrm{Fin}^{2}$, resp.), then there
exists an $\mathcal{I}$-ultrafilter (weak $\mathcal{I}$-ultrafilter or
$\mathcal{I}$-point, resp.) which is not a P-point.

    2. (b)

If $\mathcal{I}$ is a tall P-ideal, then there exists an
$\mathcal{I}$-ultrafilter which is not a P-point.

  2. (2)

Assume $\operatorname{cov}(\mathcal{M})=\mathfrak{c}$. If $\mathcal{I}$
contains a tall summable ideal, then there exists an $\mathcal{I}$-ultrafilter
which is not a P-point.

  3. (3)

Assume
$\max(\operatorname{\operatorname{add}^{*}}(\mathcal{I}),\operatorname{cov}(\mathcal{M}))=\mathfrak{c}$.
If $\mathcal{I}$ is a tall analytic P-ideal, then there exists an
$\mathcal{I}$-ultrafilter which is not a P-point.

###### Proof.

It follows from Theorems 2.8, 7.1, 3.2 and 2.4(1). ∎

Next result characterizes ideals $\mathcal{I}$ such that it is consistent that
there is an $\mathcal{I}$-ultrafilter which is not a P-point.

######  Theorem 7.3.

Let $\mathcal{I}$ be an ideal on $\omega$.

  1. (1)

Under CH, the following conditions are equivalent.

    1. (a)

$\mathcal{I}\not\leq_{\operatorname{K}}\mathrm{Fin}^{2}$
($\mathcal{I}\not\leq_{\operatorname{KB}}\mathrm{Fin}^{2}$ or
$\mathcal{I}\not\leq_{\operatorname{P}}\mathrm{Fin}^{2}$, resp.).

    2. (b)

There exists an $\mathcal{I}$-ultrafilter (weak $\mathcal{I}$-ultrafilter or
$\mathcal{I}$-point, resp.) which is not a P-point.

  2. (2)

If $\mathcal{I}$ is a tall Borel ideal, then the following conditions are
equivalent.

    1. (a)

It is consistent that there exists an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not a
P-point.

    2. (b)

Under CH, there exists an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not a
P-point.

In particular, if one can find an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not a
P-point in some sophisticated model of ZFC, then one could do it already under
CH.

###### Proof.

(1a) $\implies$ (1b) By Theorem 2.8,
$\mathfrak{p}(\mathrm{Fin}^{2})=\mathfrak{c}$ under CH, so Corollary 7.2
finishes the proof.

(1b) $\implies$ (1a) It follows from Theorem 2.4(1).

(2) Follows from Theorems 4.3(5), 2.4(1) and Proposition 2.5. ∎

At this point we are already able to characterize Borel ideals $\mathcal{I}$
for which the notions of P-point and $\mathcal{I}$-ultrafilter coincide.

######  Theorem 7.4.

Let $\mathcal{I}$ be a Borel ideal. The following conditions are equivalent:

  1. (1)

the notions of P-point and $\mathcal{I}$-ultrafilter coincide,

  2. (2)

$\mathcal{I}\leq_{K}\mathrm{Fin}^{2}$ and $\mathcal{I}$ is not extendable to
an $F_{\sigma}$-ideal.

###### Proof.

By [33, Theorem 3.2.7], a Borel ideal $\mathcal{I}$ can be extended to a
$P^{+}$-ideal iff it can be extended to an $F_{\sigma}$-ideal. Thus it
suffices to apply Theorems 6.3 and 7.3. ∎

The ideals $\mathcal{BI}$ and
$\mathrm{conv}(\mathcal{I}_{d},(\frac{1}{2^{n+1}}))$ introduced in [31] and
[30], respectively, are examples of Borel ideals below the ideal
$\mathrm{Fin}^{2}$ in the Katětov order (by the definition of
$\mathrm{conv}(\mathcal{I}_{d},(\frac{1}{2^{n+1}}))$ and results of [31], we
have
$\mathrm{conv}(\mathcal{I}_{d},(\frac{1}{2^{n+1}}))\subseteq\mathrm{conv}\leq_{K}\mathcal{BI}\leq_{K}\mathrm{Fin}^{2}$)
and not extendable to an $F_{\sigma}$-ideal (this is shown in [30]). Thus, an
ultrafilter $\mathcal{U}$ is a P-point iff $\mathcal{U}$ is a
$\mathcal{BI}$-ultrafilter iff $\mathcal{U}$ is a
$\mathrm{conv}(\mathcal{I}_{d},(\frac{1}{2^{n+1}}))$-ultrafilter.

Next two corollaries are not new, however we present them in order to show how
they can be derived from our results.

######  Corollary 7.5 (Folklore).

Assume CH. There exists a weak $\mathcal{ED}_{\mathrm{Fin}}$-ultrafilter which
is not a P-point (i.e. there exists a Q-point which is not a P-point).

###### Proof.

It follows from Theorem 7.3(1) and Proposition 2.3(1). ∎

######  Corollary 7.6 ([18, Proposition 2.1.14]).

Assume CH. If $\mathcal{I}$ is a tall P-ideal, then there exists an
$\mathcal{I}$-ultrafilter which is not a P-point.

###### Proof.

It follows from Theorem 7.3 and Proposition 2.3(5). ∎

In [19, Theorem 3.2] Flašková strengthened Corollary 7.6 by replacing CH with
$\mathfrak{p}=\mathfrak{c}$. However, the method of the proof of her stronger
result is different from the proofs we were trying to capture in the cardinal
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$. The
following question can act as a test whether the cardinal
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{J},\mathcal{F})$ can
encompass more cases than it was cooked up for.

######  Question 7.7.

Does $\mathfrak{p}=\mathfrak{c}$ imply
$\operatorname{\mathfrak{fip}}(\mathcal{I},\mathrm{Fin}^{2},\omega^{\omega})=\mathfrak{c}$
for each tall P-ideal?

On the other hand, our results provides another corollary which does not
follow from the above mentioned result of Flaškova.

######  Corollary 7.8.

If $\mathcal{I}$ is a tall P-ideal and
$\operatorname{\operatorname{add}^{*}}(\mathcal{I})=\mathfrak{c}$, then there
exists an $\mathcal{I}$-ultrafilter which is not a P-point.

###### Proof.

It follows from Corollary 7.2 and Propositon 2.3(5). ∎

##  8\. Q-points and $\mathcal{J}$-ultrafilters

This section is a counterpart of the previous two sections – we study Q-points
instead of P-points.

###  8.1. $\mathcal{I}$-ultrafilters which are not Q-points

In the case of characterizing ideals $\mathcal{I}$ such that it is consistent
that there is an $\mathcal{I}$-ultrafilter which is not a Q-point, again it
will turn out that the Katětov order is a proper tool. We start with
sufficient conditions for existence of $\mathcal{I}$-ultrafilter which is not
a Q-point.

######  Theorem 8.1.

Let $\mathcal{F}\subseteq\omega^{\omega}$ and $\mathcal{I}$ be an ideal on
$\omega$.

  1. (1)

If $\mathcal{I}\not\leq_{\mathcal{F}}\mathcal{ED}_{\mathrm{Fin}}$ and
$\mathcal{I}$ is a tall ideal (in particular, if $\mathcal{I}$ is a tall
P-ideal), then

$\max\\{\operatorname{\operatorname{add}^{*}}(\mathcal{I}),\mathfrak{p}(\mathcal{ED}_{\mathrm{Fin}})\\}\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{ED}_{\mathrm{Fin}},\mathcal{F}).$

  2. (2)

If $\mathcal{I}$ contains a tall summable ideal, then

$\operatorname{cov}(\mathcal{M})\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{ED}_{\mathrm{Fin}},\mathcal{F}).$

  3. (3)

If $\mathcal{I}$ is a tall analytic P-ideal, then

$\max(\operatorname{\operatorname{add}^{*}}(\mathcal{I}),\mathfrak{p}(\mathcal{ED}_{\mathrm{Fin}}),\operatorname{cov}(\mathcal{M}))\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{ED}_{\mathrm{Fin}},\mathcal{F}).$

###### Proof.

(1) follows from Theorem 4.1(1) and Proposition 2.5. For the “in particular”
part additionally one has to use Proposition 2.3(6). (2) follows from Theorem
5.7 and Proposition 5.4. (3) follows from (1), (2) and Proposition 2.2. ∎

######  Corollary 8.2.

Let $\mathcal{I}$ be an ideal on $\omega$.

  1. (1)

Assume
$\max(\operatorname{\operatorname{add}^{*}}(\mathcal{I}),\mathfrak{p}(\mathcal{ED}_{\mathrm{Fin}}))=\mathfrak{c}$.

    1. (a)

If $\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{ED}_{\mathrm{Fin}}$
($\mathcal{I}\not\leq_{\operatorname{KB}}\mathcal{ED}_{\mathrm{Fin}}$ or
$\mathcal{I}\not\leq_{\operatorname{P}}\mathcal{ED}_{\mathrm{Fin}}$, resp.),
then there exists an $\mathcal{I}$-ultrafilter (weak $\mathcal{I}$-ultrafilter
or $\mathcal{I}$-point, resp.) which is not a Q-point.

    2. (b)

If $\mathcal{I}$ is a tall P-ideal, then there exists an
$\mathcal{I}$-ultrafilter which is not a Q-point.

  2. (2)

Assume $\operatorname{cov}(\mathcal{M})=\mathfrak{c}$. If $\mathcal{I}$
contains a tall summable ideal, then there exists an $\mathcal{I}$-ultrafilter
which is not a Q-point.

  3. (3)

Assume
$\max(\operatorname{\operatorname{add}^{*}}(\mathcal{I}),\mathfrak{p}(\mathcal{ED}_{\mathrm{Fin}}),\operatorname{cov}(\mathcal{M}))=\mathfrak{c}$.
If $\mathcal{I}$ is a tall analytic P-ideal, then there exists an
$\mathcal{I}$-ultrafilter which is not a Q-point.

###### Proof.

It follows from Theorems 8.1, 3.2 and 2.4(2). ∎

Next result characterizes, with the use of Katětov order, ideals $\mathcal{I}$
such that it is consistent that there is an $\mathcal{I}$-ultrafilter which is
not a Q-point.

######  Theorem 8.3.

Let $\mathcal{I}$ be an ideal on $\omega$.

  1. (1)

Under $\mathfrak{p}=\mathfrak{c}$, the following conditions are equivalent.

    1. (a)

$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{ED}_{\mathrm{Fin}}$
($\mathcal{I}\not\leq_{\operatorname{KB}}\mathcal{ED}_{\mathrm{Fin}}$ or
$\mathcal{I}\not\leq_{\operatorname{P}}\mathcal{ED}_{\mathrm{Fin}}$, resp.).

    2. (b)

There exists an $\mathcal{I}$-ultrafilter (weak $\mathcal{I}$-ultrafilter or
$\mathcal{I}$-point, resp.) which is not a Q-point.

  2. (2)

If $\mathcal{I}$ is a tall Borel ideal, then the following conditions are
equivalent.

    1. (a)

It is consistent that there exists an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not a
Q-point.

    2. (b)

Under CH, there exists an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not a
Q-point.

In particular, if one can find an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not a
Q-point in some sophisticated model of ZFC, then one could do it already under
CH.

###### Proof.

(1a) $\implies$ (1b) Since the ideal $\mathcal{ED}_{\mathrm{Fin}}$ is
$F_{\sigma}$, $\mathfrak{p}(\mathcal{ED}_{\mathrm{Fin}})\geq\mathfrak{p}$ by
Theorem 2.9. Then Corollary 8.2 finishes the proof.

(1b) $\implies$ (1a) It follows from Theorem 2.4(2).

(2) Follows from Theorems 4.3(5) and 2.4(2) and Proposition 2.5. ∎

######  Corollary 8.4 (Folklore).

Assume $\mathfrak{p}=\mathfrak{c}$. There exists a
$\mathrm{Fin}^{2}$-ultrafilter which is not a Q-point (i.e. there exists a
P-point which is not a Q-point).

###### Proof.

It follows from Theorems 8.3 and 2.4(1) and Proposition 2.3(3), because
$\mathcal{ED}_{\mathrm{Fin}}$ is an $F_{\sigma}$ ideal, so it is a
$P^{+}$-ideal by Proposition 2.6(1). ∎

###  8.2. Q-points which are not $\mathcal{J}$-ultrafilters

######  Theorem 8.5.

Let $\mathcal{J}$ be a $\text{P}^{+}(\mathcal{J})$-ideal which is
$\leq_{\operatorname{KB}}$-homogeneous.

  1. (1)

Assume CH. The following conditions are equivalent.

    1. (a)

$\mathcal{ED}_{\mathrm{Fin}}\not\leq_{\operatorname{KB}}\mathcal{J}$.

    2. (b)

There is a Q-point which is not a $\mathcal{J}$-point.

    3. (c)

There is a Q-point which is not a weak $\mathcal{J}$-ultrafilter.

  2. (2)

If $\mathcal{J}$ is $F_{\sigma}$, then CH can be relaxed to the assumption
$\mathfrak{p}=\mathfrak{c}$.

###### Proof.

It follows from Theorems 4.3(2) and 2.4(2). In the case of $F_{\sigma}$-ideals
additionally we need to use Theorems 4.3(4). ∎

Q-points cannot be characterized as $\mathcal{J}$-ultrafilters for any ideal
$\mathcal{J}$ – to obtain a Q-point which is not a $\mathcal{J}$-ultrafilter
one needs to assume nothing about an ideal $\mathcal{J}$ as the following
result of Flaškova shows.

######  Proposition 8.6 ([18, Proposition 2.4.7]).

Assume $\operatorname{cov}(\mathcal{M})=\mathfrak{c}$. For each ideal
$\mathcal{J}$, there exists a Q-point which is not a
$\mathcal{J}$-ultrafilter.

##  9\. Selective ultrafilters and $\mathcal{J}$-ultrafilters

###  9.1. Selective ultrafilters which are not $\mathcal{J}$-ultrafilters

######  Proposition 9.1.

Let $\mathcal{J}$ be a $\text{P}^{+}(\mathcal{J})$-ideal which is
$\leq_{K}$-homogeneous.

  1. (1)

Assume CH. The following conditions are equivalent.

    1. (a)

$\mathcal{ED}\not\leq_{\operatorname{K}}\mathcal{J}$.

    2. (b)

There is a selective ultrafilter which is not a $\mathcal{J}$-point.

    3. (c)

There is a selective ultrafilter which is not a weak
$\mathcal{J}$-ultrafilter.

    4. (d)

There is a selective ultrafilter which is not a $\mathcal{J}$-ultrafilter.

  2. (2)

If $\mathcal{J}$ is an $F_{\sigma}$-ideal, then CH can be relaxed to the
assumption $\mathfrak{p}=\mathfrak{c}$.

###### Proof.

(1) It follows from Theorems 4.3(1) and 2.4(3). For (2) one additionally needs
Theorems 4.3(4). ∎

In the above theorem, we cannot drop the assumptions on the ideal
$\mathcal{J}$ in general. Indeed, it is known that
$\mathcal{ED}\not\leq_{\operatorname{K}}\mathcal{R}$ (see [24, Section 2]),
but all selective ultrafilters (i.e. $\mathcal{ED}$-ultrafilters) are
$\mathcal{R}$-points (Theorem 2.4(3)). Although, in Theorem 9.3, we show that
the assumption on the ideal $\mathcal{J}$ can be removed if we replace the
Katětov order $\leq_{\operatorname{K}}$ by the property that $\mathcal{J}$ can
be extended to a selective ideal.

######  Lemma 9.2.

Let $\mathcal{I}$ and $\mathcal{J}$ be ideals on $\omega$.

  1. (1)

If $\mathcal{I}\leq_{\operatorname{K}}\mathcal{J}$ and $\mathcal{J}$ is a
selective ideal, then $\mathcal{I}$ can be extended to a selective ideal.

  2. (2)

If $\mathcal{J}$ is a selective ideal and
$\mathcal{A}\subseteq\mathcal{P}(\omega)$ is a countable family such that
$\omega\notin\mathcal{J}(\mathcal{A})$, then $\mathcal{J}(\mathcal{A})$ is a
selective ideal.

  3. (3)

If $\mathcal{J}$ is not a selective ideal, then there is a countable family
$\mathcal{A}\subseteq\mathcal{P}(\omega)$ such that
$\omega\notin\mathcal{J}(\mathcal{A})$ and
$\mathcal{ED}\leq_{\operatorname{K}}\mathcal{J}(\mathcal{A})$.

###### Proof.

(1) Let $f:\omega\to\omega$ be such that $f^{-1}[B]\in\mathcal{J}$ for each
$B\in\mathcal{I}$. Let $\mathcal{K}=\\{B:f^{-1}[B]\in\mathcal{J}\\}$. It is
easy to see that $\mathcal{K}$ is an ideal and
$\mathcal{I}\subseteq\mathcal{K}$. We show that $\mathcal{K}$ is a selective
ideal. Let $(B_{n})\subseteq\mathcal{K}^{+}$ be a decreasing sequence. Then
$f^{-1}[B_{n}]\in\mathcal{J}^{+}$ and $f^{-1}[B_{n}]\supseteq f^{-1}[B_{n+1}]$
for each $n$. Since $\mathcal{J}$ is a selective ideal, there is
$A\in\mathcal{J}^{+}$ such that $|A\cap(f^{-1}[B_{n}]\setminus
f^{-1}[B_{n+1}])|\leq 1$ for each $n$. Then $B=f[A]\in\mathcal{K}^{+}$ and
$|B\cap(B_{n}\setminus B_{n+1})|\leq 1$ for each $n$.

(2) Let $\mathcal{A}=\\{A_{n}:n\in\omega\\}$ be such that
$\omega\notin\mathcal{J}(\mathcal{A})$. Take any decreasing sequence
$B_{n}\in\mathcal{J}(\mathcal{A})^{+}$. Let $C_{0}=\omega$ and
$C_{n+1}=B_{n}\setminus\bigcup_{i<n}A_{i}$. Then
$C_{n}\in\mathcal{J}(\mathcal{A})^{+}$ and $C_{n}\supseteq C_{n+1}$ for each
$n$. Since $C_{n}\in\mathcal{J}^{+}$ and $\mathcal{J}$ is a selective ideal,
there is $C^{\prime}\in\mathcal{J}^{+}$ such that
$|C^{\prime}\cap(C_{n}\setminus C_{n+1})|\leq 1$ for each $n$. Define
$C=C^{\prime}\cap C_{1}$. Then $C\in\mathcal{J}^{+}$ as $|C^{\prime}\setminus
C|=|C^{\prime}\cap(C_{0}\setminus C_{1})|\leq 1$. Once we show that
$C\in\mathcal{J}(\mathcal{A})^{+}$, the proof will be finished. Suppose that
$C\in\mathcal{J}(\mathcal{A})$. Then there is $n$ such that
$C\setminus\bigcup_{i<n}A_{i}\in\mathcal{J}$. But $C\cap\bigcup_{i<n}A_{i}$ is
finite (it has at most $n$ elements), so $C\in\mathcal{J}$, a contradiction.

(3) Since $\mathcal{J}$ is not a selective ideal, there is a decreasing
sequence $B_{n}\in\mathcal{J}^{+}$ such that if $|X\cap(B_{n}\setminus
B_{n+1})|\leq 1$ for all $n$ then $X\in\mathcal{J}$. Let
$A_{0}=\omega\setminus B_{0}$ and $A_{n+1}=B_{n}\setminus B_{n+1}$ for all
$n$. Observe that $\bigcup_{i\leq n}A_{i}=\omega\setminus
B_{n}\notin\mathcal{J}^{*}$, so
$\omega\notin\mathcal{J}(\\{A_{n}:n\in\omega\\})$. What is more, any one-to-
one function $f:\omega\to\omega^{2}$ such that
$f[A_{n}]\subseteq\\{n\\}\times\omega$ witnesses
$\mathcal{ED}\leq_{\operatorname{K}}\mathcal{J}(\\{A_{n}:n\in\omega\\})$. ∎

######  Theorem 9.3.

Let $\mathcal{J}$ be an ideal on $\omega$.

  1. (1)

The following conditions are equivalent.

    1. (a)

$\mathcal{J}$ is extendable to a selective ideal.

    2. (b)

$\mathcal{ED}\not\preceq_{\omega^{\omega},1-1}^{\omega_{1}}\mathcal{J}$.

    3. (c)

$\mathcal{ED}\not\preceq_{\omega^{\omega},\mathrm{Fin}-1}^{\omega_{1}}\mathcal{J}$.

    4. (d)

$\mathcal{ED}\not\preceq_{\omega^{\omega},\omega^{\omega}}^{\omega_{1}}\mathcal{J}$.

  2. (2)

Under CH, the above conditions are equivalent to the following condition.

    1. (e)

There is a selective ultrafilter which is not a $\mathcal{J}$-point.

    2. (f)

There is a selective ultrafilter which is not a weak
$\mathcal{J}$-ultrafilter.

    3. (g)

There is a selective ultrafilter which is not a $\mathcal{J}$-ultrafilter.

###### Proof.

(1a)$\implies$(1b) Let $\mathcal{J}^{\prime}$ be a selective ideal such that
$\mathcal{J}^{\prime}\supseteq\mathcal{J}$. By Lemma 9.2(2),
$\mathcal{J}^{\prime}(\mathcal{A})$ is a selective ideal for each countable
family $\mathcal{A}\subseteq\mathcal{P}(\omega)$ such that
$\omega\notin\mathcal{J}^{\prime}(\mathcal{A})$. So
$\mathcal{ED}\not\leq_{\operatorname{K}}\mathcal{J}^{\prime}(\mathcal{A})$, by
Proposition 2.3(4). Thus,
$\mathcal{ED}\not\preceq_{\omega^{\omega},1-1}^{\omega_{1}}\mathcal{J}$.

The implications (1b)$\implies$(1c) and (1c)$\implies$(1d) are obvious.

(1d)$\implies$(1a) Since
$\mathcal{ED}\not\preceq_{\omega^{\omega},\omega^{\omega}}^{\omega_{1}}\mathcal{J}$,
there is an ideal $\mathcal{J}^{\prime}\geq_{\operatorname{K}}\mathcal{J}$
such that
$\mathcal{ED}\not\leq_{\operatorname{K}}\mathcal{J}^{\prime}(\mathcal{A})$ for
each countable $\mathcal{A}\subseteq\mathcal{P}(\omega)$ such that
$\omega\notin\mathcal{J}^{\prime}(\mathcal{A})$. By Lemma 9.2(3),
$\mathcal{J}^{\prime}$ is a selective ideal, hence by Lemma 9.2(1),
$\mathcal{J}$ can be extended to a selective ideal.

(1b)$\implies$ (2e) Under CH, it follows from Corollary 3.5(1) and Theorem
2.4(3).

The implications (2e)$\implies$(2f) and (2f)$\implies$(2g) are obvious (and
hold in ZFC).

(2g)$\implies$ (1a) Take a selective ultrafilter $\mathcal{U}$ which is not a
$\mathcal{J}$-ultrafilter (i.e.
$\mathcal{J}\leq_{\operatorname{K}}\mathcal{U}^{*}$). Then $\mathcal{U}^{*}$
is a selective ideal, so $\mathcal{J}$ can be extended to a selective ideal by
Lemma 9.2(1). (Note that this argument works in ZFC.) ∎

###  9.2. $\mathcal{I}$-ultrafilters which are not selective

######  Theorem 9.4.

If $\mathcal{I}$ contains a tall summable ideal, then

$\operatorname{cov}(\mathcal{M})\leq\operatorname{\mathfrak{fip}}(\mathcal{I},\mathcal{ED},\omega^{\omega}).$

###### Proof.

It follows from Theorem 5.7 and Proposition 5.4. ∎

######  Corollary 9.5.

Assume $\operatorname{cov}(\mathcal{M})=\mathfrak{c}$. If $\mathcal{I}$
contains a tall summable ideal, then there exists an $\mathcal{I}$-ultrafilter
which is not a selective ultrafilter.

###### Proof.

It follows from Theorems 9.4, 3.2 and 2.4(3). ∎

The characterization of ideals $\mathcal{I}$ such that it is consistent that
there exists an $\mathcal{I}$-ultrafilter which is not selective occurs to be
a simple consequence of our previous considerations.

######  Theorem 9.6.

Let $\mathcal{I}$ be an ideal on $\omega$.

  1. (1)

Under CH, the following conditions are equivalent.

    1. (a)

($\mathcal{I}\not\leq_{\operatorname{K}}\mathrm{Fin}^{2}$ or
$\mathcal{I}\not\leq_{\operatorname{K}}\mathcal{ED}_{\mathrm{Fin}}$)
(($\mathcal{I}\not\leq_{\operatorname{KB}}\mathrm{Fin}^{2}$ or
$\mathcal{I}\not\leq_{\operatorname{KB}}\mathcal{ED}_{\mathrm{Fin}}$) or
($\mathcal{I}\not\leq_{P}\mathrm{Fin}^{2}$ or
$\mathcal{I}\not\leq_{P}\mathcal{ED}_{\mathrm{Fin}}$), resp.).

    2. (b)

There exists an $\mathcal{I}$-ultrafilter (weak $\mathcal{I}$-ultrafilter or
$\mathcal{I}$-point, resp.) which is not selective.

  2. (2)

If $\mathcal{I}$ is a tall Borel ideal, then the following conditions are
equivalent.

    1. (a)

It is consistent that there exists an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not
selective.

    2. (b)

Under CH, there exists an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not
selective.

In particular, if one can find an $\mathcal{I}$-ultrafilter (weak
$\mathcal{I}$-ultrafilter or $\mathcal{I}$-point, resp.) which is not
selective in some sophisticated model of ZFC, then one could do it already
under CH.

###### Proof.

Follows from Theorems 7.3, 8.3 and the fact that an ultrafilter is selective
if and only if it is both a P-point and a Q-point. ∎

##  10\. By-products

In this section we collect some by-products of our studies.

######  Proposition 10.1.

Any $G_{\delta\sigma}$-ideal is extendable to a $F_{\sigma}$-ideal.

###### Proof.

Let $\mathcal{J}$ be a $G_{\delta\sigma}$-ideal. We will show that
$\mathrm{Fin}^{2}\not\preceq^{\omega_{1}}_{\omega^{\omega},1-1}\mathcal{J}$.
By Theorem 6.3, this will finish the proof, because for Borel ideals
extendability to $P^{+}$-ideals is equivalent to extendability to
$F_{\sigma}$-ideals (see [33, Theorem 3.2.7]). We claim that
$\mathcal{J}^{\prime}=\mathcal{J}$ witnesses
$\mathrm{Fin}^{2}\not\preceq^{\omega_{1}}_{\omega^{\omega},1-1}\mathcal{J}$.
Let $\mathcal{A}=\\{A_{i}:\ i\in\omega\\}\subseteq\mathcal{P}(\omega)$ be such
that $|\mathcal{A}|=\omega$ and $\omega\notin\mathcal{J}(\mathcal{A})$.
Observe that

$\mathcal{J}(\mathcal{A})=\left\\{B\subseteq\omega:\exists
n\in\omega(B\setminus(A_{0}\cup\ldots\cup
A_{n})\in\mathcal{J})\right\\}=\bigcup_{n\in\omega}\phi_{n}^{-1}[\mathcal{J}],$

where $\phi_{n}:\mathcal{P}(\omega)\to\mathcal{P}(\omega)$ is given by
$\phi_{n}(B)=B\setminus(A_{0}\cup\ldots\cup A_{n})$. It is easy to check that
each $\phi_{n}$ is continuous. Thus, $\mathcal{J}(\mathcal{A})$ is
$G_{\delta\sigma}$. Since the ideal $\mathcal{J}(\mathcal{A})$ is
$\Pi_{4}^{0}$, its rank is less than or equal to $1$ (by [13, Theorem 9.1];
see [13, Definition 3.1] for the definition of rank of an analytic ideal).
Hence, using [13, Theorem 7.5] we get that
$\mathrm{Fin}^{2}\not\sqsubseteq\mathcal{J}(\mathcal{A})$. Thus,
$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathcal{J}(\mathcal{A})$ by [2,
Example 4.1]. ∎

A set $A\subseteq\omega$ is _lacunary_ ([12]) if
$\lim_{n\to\infty}(a_{n+1}-a_{n})=\infty$, where $(a_{n})_{n\in\omega}$ is the
increasing enumeration of all elements of $A$. The ideal $\mathcal{L}$
generated by lacunary sets was examined for instance in [18] (where the author
used the name $SC$-set for a lacunary set) and [2]. It is not difficult to
show that $\mathcal{L}$ is not a $P^{+}$-ideal (see [18, Lemma 1.2.8])

######  Proposition 10.2.

The ideal $\mathcal{L}$ is a $P^{+}(\mathcal{L})$-ideal.

###### Proof.

We will use the following characterization ([18, Lemma 1.2.7]):

$A\notin\mathcal{L}\iff\forall_{n\in\omega}\exists_{d\in\omega}\forall_{K\in\omega}\exists_{k\geq
K}\,(|A\cap[k,k+d]|\geq n).$

Let $A_{n}\notin\mathcal{L}$ and $A_{n}\supseteq A_{n+1}$ for each
$n\in\omega$. Using the above characterization for each set $A_{n}$
separately, we find $d_{n}\in\omega$ such that for each $K\in\omega$ there is
$k\geqslant K$ with $|A_{n}\cap[k,k+d_{n}]|\geq n$. Then, for each $n$, we
choose an increasing sequence $k^{n}_{0}<k^{n}_{1}<\dots$ such that
$\\{k^{n}_{i}:i\in\omega\\}\in\mathcal{L}$ and
$|A_{n}\cap[k^{n}_{i},k^{n}_{i}+d_{n}]|\geq n$ for each $i$. Finally, we
define

$A=\bigcup_{n\in\omega}\left(\bigcup_{i\in\omega}A_{n}\cap[k^{n}_{i},k^{n}_{i}+d_{n}]\right).$

Using the above mentioned characterization, it is easy to see that
$A\notin\mathcal{L}$.

Below, we show that $A\setminus A_{n}\in\mathcal{L}$ for each $n$.

First, we observe that for each $n$ the set

$B_{n}=\bigcup_{i\in\omega}A_{n}\cap[k^{n}_{i},k^{n}_{i}+d_{n}]$

is a union of $d_{n}+1$ lacunary sets, so $B_{n}\in\mathcal{L}$. Then, taking
into account that $A_{0}\supseteq A_{1}\supseteq\dots$, we notice that
$A\setminus A_{n}\subseteq\bigcup_{i<n}B_{i}\in\mathcal{L}$. ∎

######  Proposition 10.3.

The ideal $\mathcal{L}$ is not $\leq_{\operatorname{K}}$-homogeneous (in
particular, it is not homogeneous).

###### Proof.

Since $\mathcal{L}\subseteq\mathcal{I}_{d}$ ([18, Lemma 1.4.4] and
$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathcal{I}_{d}$ (by Proposition
2.3(2) and the fact that $\mathcal{I}_{d}$ is a P-ideal), we obtain that
$\mathrm{Fin}^{2}\not\leq_{\operatorname{K}}\mathcal{L}$. Moreover, by
Proposition 10.2, $\mathcal{L}$ is a $P^{+}(\mathcal{L})$-ideal.

Now, assume towards contradiction that $\mathcal{L}$ is
$\leq_{\operatorname{K}}$-homogeneous. Then, using Theorem 6.1, under CH we
obtain a P-point which is not an $\mathcal{L}$-ultrafilter. However, this
contradicts Proposition 6.4. ∎

######  Proposition 10.4.

The ideal $\mathcal{R}$ generated by homogeneous sets in some fixed instance
of the random graph on $\omega$ is not $\leq_{\operatorname{K}}$-homogeneous
(in particular, it is not homogeneous).

###### Proof.

Note that $\mathcal{ED}\not\leq_{\operatorname{K}}\mathcal{R}$ (see [24,
Section 2]) and $\mathcal{R}$ is a $P^{+}(\mathcal{R})$-ideal (by Proposition
2.6(1) and the fact that $\mathcal{R}$ is an $F_{\sigma}$ ideal – see [33,
page 30]).

Now, assume towards contradiction that $\mathcal{R}$ is
$\leq_{\operatorname{K}}$-homogeneous. Then, using Theorem 9.1, under CH we
would obtain a selective ultrafilters which is not an
$\mathcal{R}$-ultrafilter. However, this contradicts Theorem 2.4(3). ∎

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