# A unified approach to Hindman, Ramsey and van der Waerden spaces

Rafał Filipów  Institute of Mathematics
Faculty of Mathematics, Physics and Informatics
University of Gdańsk
ul. Wita Stwosza 57
80-308 Gdańsk
Poland<EMAIL_ADDRESS>http://mat.ug.edu.pl/~rfilipow ,  Krzysztof
Kowitz  Institute of Mathematics
Faculty of Mathematics
Physics and Informatics
University of Gdańsk
ul. Wita Stwosza 57
80-308 Gdańsk
Poland<EMAIL_ADDRESS>and  Adam Kwela  Institute of
Mathematics
Faculty of Mathematics
Physics and Informatics
University of Gdańsk
ul. Wita Stwosza 57
80-308 Gdańsk
Poland<EMAIL_ADDRESS>https://mat.ug.edu.pl/~akwela

###### Abstract.

For many years, there have been conducting research (e.g. by Bergelson,
Furstenberg, Kojman, Kubiś, Shelah, Szeptycki, Weiss) into sequentially
compact spaces that are, in a sense, topological counterparts of some
combinatorial theorems, for instance Ramsey’s theorem for coloring graphs,
Hindman’s finite sums theorem and van der Waerden’s arithmetical progressions
theorem. These spaces are defined with the aid of different kinds of
convergences: IP-convergence, R-convergence and ordinary convergence.

The first aim of this paper is to present a unified approach to these various
types of convergences and spaces. Then, using this unified approach, we prove
some general theorems about existence of the considered spaces and show that
all results obtained so far in this subject can be derived from our theorems.

The second aim of this paper is to obtain new results about the specific types
of these spaces. For instance, we construct a Hausdorff Hindman space that is
not an $\mathcal{I}_{1/n}$-space and a Hausdorff differentially compact space
that is not Hindman. Moreover, we compare Ramsey spaces with other types of
spaces. For instance, we construct a Ramsey space that is not Hindman and a
Hindman space that is not Ramsey.

The last aim of this paper is to provide a characterization that shows when
there exists a space of one considered type that is not of the other kind.
This characterization is expressed in purely combinatorial manner with the aid
of the so-called Katětov order that has been extensively examined for many
years so far.

This paper may interest the general audience of mathematicians as the results
we obtain are on the intersection of topology, combinatorics, set theory and
number theory.

###### Key words and phrases:

Ramsey’s theorem, Hindman’s theorem, van der Waerden’s theorem, sequentially
compact space, compact space, Bolzano–Weierstrass theorem, Ramsey space,
Hindman space, differentially compact space, ideal, filter, almost disjoint
families, Katětov order

###### 2020 Mathematics Subject Classification:

Primary: 54A20 05D10, 03E75. Secondary: 03E02, 54D30, 54H05, 26A03, 40A35,
40A05, 03E17, 03E15, 03E05, 03E35.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. I Partition regular operations
    1. 3 Partition regular operations and ideals associated with them
    2. 4 Restrictions and small accretions
    3. 5 Topological complexity of partition regular operations
    4. 6 P-like properties
    5. 7 Katětov order
    6. 8 Tallness and homogeneity
  4. II $\operatorname{FinBW}$ spaces
    1. 9 A convergence with respect to partition regular functions
    2. 10 $\operatorname{FinBW}$ spaces
    3. 11 Inclusions between $\operatorname{FinBW}$ classes
  5. III Distinguishing between $\operatorname{FinBW}$ classes
    1. 12 Mrówka spaces and their compactifications
    2. 13 Mrówka for maximal almost disjoint families
    3. 14 Distinguishing between $\operatorname{FinBW}$ classes via Katětov order on ideals
    4. 15 Distinguishing between $\operatorname{FinBW}$ classes via Katětov order on partition regular functions
  6. IV Characterizations
    1. 16 Characterizations of distinguishness between $\operatorname{FinBW}$ classes via Katětov order
    2. 17 Non-Hausdorff world
    3. 18 Hindman (Ramsey, differentially compact) spaces that are not in $\operatorname{FinBW}(\mathcal{I})$ and vice versa

###  1\. Introduction

For more than twenty years, many mathematicians have been examining
sequentially compact spaces that are, in a sense, topological counterparts of
some combinatorial theorems, for instance Ramsey’s theorem for coloring
graphs, Hindman’s finite sums theorem and van der Waerden’s arithmetical
progressions theorem [5, 6, 18, 19, 20, 22, 24, 25, 26, 29, 33, 34, 50, 55,
56, 57, 58, 59, 60, 61, 71]. These spaces are defined with the aid of
different kinds of convergences: IP-convergence, R-convergence and ordinary
convergence.

We start our brief overview of these spaces with the ones defined using
ordinary convergence. A topological space $X$ is called:

  * •

_van der Waerden_ [56] if for every sequence $\langle
x_{n}\rangle_{n\in\mathbb{N}}$ in $X$ there exists a convergent subsequence
$\langle x_{n}\rangle_{n\in A}$ with $A$ being an AP-set (i.e. $A$ contains
arithmetic progressions of arbitrary finite length);

  * •

an _$\mathcal{I}_{1/n}$ -space_ [29] if for every sequence $\langle
x_{n}\rangle_{n\in\mathbb{N}}$ in $X$ there exists a convergent subsequence
$\langle x_{n}\rangle_{n\in A}$ with $A$ having the property that the series
of reciprocals of elements of $A$ diverges.

In fact both mentioned classes of spaces are special cases of a more general
notion. A nonempty family $\mathcal{I}\subseteq\mathcal{P}(\mathbb{N})$ of
subsets of $\mathbb{N}$ is an _ideal on $\mathbb{N}$_ if it is closed under
taking subsets and finite unions of its elements,
$\mathbb{N}\notin\mathcal{I}$ and $\mathcal{I}$ contains all finite subsets of
$\mathbb{N}$ (it is easy to see that the family
$\mathcal{I}_{1/n}=\\{A\subseteq\mathbb{N}:\sum_{n\in A}1/n<\infty\\}$ is an
ideal on $\mathbb{N}$, and it follows from van der Waerden’s theorem [73] that
the family $\mathcal{W}=\\{A\subseteq\mathbb{N}:\text{$A$ is not an AP-
set}\\}$ is an ideal on $\mathbb{N}$). If $\mathcal{I}$ is an ideal on
$\mathbb{N}$ then a topological space $X$ is called an _$\mathcal{I}$ -space_
[29] if for every sequence $\langle x_{n}\rangle_{n\in\mathbb{N}}$ in $X$
there exists a converging subsequence $\langle x_{n}\rangle_{n\in A}$ with
$A\notin\mathcal{I}$. In particular, van der Waerden spaces coincide with
$\mathcal{W}$-spaces.

Now we want to turn our attention to spaces defined with the aid of different
kinds of convergence. We start with Hindman spaces. A set
$A\subseteq\mathbb{N}$ is an _IP-set_ [34] if there exists an infinite set
$D\subseteq\mathbb{N}$ such that $\operatorname{FS}(D)\subseteq A$ where
$\operatorname{FS}(D)$ denotes the set of all finite sums of distinct elements
of $D$. The family $\mathcal{H}=\\{A\subseteq\mathbb{N}:\text{$A$ is not an
IP-set}\\}$ is an ideal on $\mathbb{N}$ (it follows from Hindman’s theorem
[42]).

An _IP-sequence_ in $X$ is a sequence indexed by $\operatorname{FS}(D)$ for
some infinite $D\subseteq\mathbb{N}$. An IP-sequence $\langle
x_{n}\rangle_{n\in\operatorname{FS}(D)}$ in a topological space $X$ is _IP-
convergent_ [34] to a point $x\in X$ if for every neighborhood $U$ of $x$
there exists $m\in\mathbb{N}$ so that $x_{n}\in U$ for every
$n\in\operatorname{FS}(D\setminus\\{0,1,\dots,m\\})$ (then $x$ is called the
_IP-limit_ of the sequence).

Since only finite spaces are $\mathcal{H}$-spaces [55], Kojman replaced the
ordinary convergence with IP-convergence (introduced by Furstenberg and Weiss
[34]) to define a meaningful topological counterpart of Hindman’s finite sums
theorem. Namely, a topological space $X$ is called _Hindman_ [55] if for every
sequence $\langle x_{n}\rangle_{n\in\mathbb{N}}$ in $X$ there exists an
infinite set $D\subseteq\mathbb{N}$ such that the subsequence $\langle
x_{n}\rangle_{n\in\operatorname{FS}(D)}$ IP-converges to some $x\in X$.

We finish our brief overview of classes of sequentially compact spaces with
Ramsey spaces. Let $[A]^{2}$ denote the set of all pairs of elements of $A$. A
sequence $\langle x_{n}\rangle_{n\in[D]^{2}}$ in $X$ (indexed by pairs of
natural numbers from some infinite set $D\subseteq\mathbb{N}$) _R-converges_
[6, 5] to a point $x\in X$ if for every neighborhood $U$ of $x$ there is a
finite set $F$ such that $x_{\\{a,b\\}}\in U$ for all distinct $a,b\in
D\setminus F$. A topological space $X$ is called _Ramsey_ [59] if for every
sequence $\langle x_{n}\rangle_{n\in[\mathbb{N}]^{2}}$ in $X$ there exists an
infinite set $D\subseteq\mathbb{N}$ such that the subsequence $\langle
x_{n}\rangle_{n\in[D]^{2}}$ R-converges to some $x\in X$.

We say that an ideal $\mathcal{I}$ (on $\mathbb{N}$) is below an ideal
$\mathcal{J}$ in the _Katětov order_ [53] if there is a function
$f:\mathbb{N}\to\mathbb{N}$ such that $f^{-1}[A]\in\mathcal{J}$ for every
$A\in\mathcal{I}$. Note that Katětov order has been extensively examined (even
in its own right) for many years so far [2, 3, 8, 12, 10, 39, 40, 43, 44, 46,
47, 48, 65, 66, 68, 70, 76].

There are three objectives of this paper. The first aim is to present a
unified approach to these various types of convergences and spaces. This is
achieved in sections in Part I with the aid of partition regular functions
(Definition 3.1), a convergence with respect to partition regular functions
(Definition 9.1) and a subclass of sequentially compact spaces defined using
this new kind of convergence (see Definition 10.1). Then using this approach,
we prove some general theorems about those classes of spaces (Theorem 10.5)
and show that all results obtained so far in this subject can be derived from
our theorems (see sections in Parts II and III).

The second aim of this paper is to obtain new results concerning specific
types of these spaces: Ramsey spaces, Hindman spaces, van der Waerden spaces
and $\mathcal{I}_{1/n}$-spaces. For instance, we construct a Hausdorff Hindman
space that is not an $\mathcal{I}_{1/n}$-space (Corollary 14.10(2)) – this
gives a positive answer to a question posed by Flašková [27] (so far only non-
Hausdorff answer to this question was known [22, Theorem 2.5]). We also
construct a Hausdorff so-called differentially compact space that is not
Hindman (Corollary 14.9(3)) which yields the negative answer to a question
posed by Shi [71, Question 4.2.2] and other authors [20, Problem 1],[58,
Question 3]. Moreover, we compare Ramsey spaces with other types of spaces (so
far Ramsey spaces were only examined in their own right without comparing them
with other kinds of spaces [6, 59, 11]). For instance, we construct a Ramsey
space that is not Hindman and a Hindman space that is not Ramsey (Corollary
14.9).

The final aim of this paper is to provide a characterization that shows when
there exists a space of one considered type that is not of the other type
(Theorem 16.1 and other results in Part IV). This characterization is
expressed in purely combinatorial manner with the aid of the Katětov order or
its counterpart in the realm of partition regular functions (Definition 7.3).

###  2\. Preliminaries

In the paper we are exclusively interested in Hausdorff topological spaces
with one exception (Sections 17 and 18) where we were unable to obtain results
for Hausdorff spaces but succeeded in constructing a topological space with
unique limits of sequences.

Following von Neumann, we identify an ordinal number $\alpha$ with the set of
all ordinal numbers less than $\alpha$. In particular, the smallest infinite
ordinal number $\omega=\\{0,1,\dots\\}$ is equal to the set $\mathbb{N}$ of
all natural numbers, and each natural number $n=\\{0,\dots,n-1\\}$ is equal to
the set of all natural numbers less than $n$. Using this identification, we
can for instance write $n\in k$ instead of $n<k$ and $n<\omega$ instead of
$n\in\omega$ or $A\cap n$ instead of $A\cap\\{0,1,\dots,n-1\\}$.

If $A\subseteq\omega$ and $n\in\omega$, we write $A+n=\\{a+n:a\in A\\}$ and
$A-n=\\{a-n:a\in A,a>n\\}$.

We write $[A]^{2}$ to denote the set of all unordered pairs of elements of
$A$, $[A]^{<\omega}$ to denote the family of all finite subsets of $A$,
$[A]^{\omega}$ to denote the family of all infinite countable subsets of $A$
and $\mathcal{P}(A)$ to denote the family of all subsets of $A$.

We say that a family $\mathcal{A}$ of subsets of a set $\Lambda$ is an _almost
disjoint family on $\Lambda$_ if

  1. (1)

$|A|=|\Lambda|$ for every $A\in\mathcal{A}$ and

  2. (2)

$|A\cap B|<|\Lambda|$ for all distinct elements $A,B\in\mathcal{A}$.

By $A\sqcup B$ we denote the _disjoint union_ of sets $A$ and $B$:

$A\sqcup B=(A\times\\{0\\})\cup(B\times\\{1\\})=\\{(x,0):x\in
A\\}\cup\\{(y,1):y\in B\\}.$

For families of sets $\mathcal{A}\subseteq\mathcal{P}(\Lambda)$ and
$\mathcal{B}\subseteq\mathcal{P}(\Sigma)$, we write
$\mathcal{A}\oplus\mathcal{B}=\\{A\sqcup
B:A\in\mathcal{A},B\in\mathcal{B}\\}.$

A nonempty family $\mathcal{I}\subseteq\mathcal{P}(\Lambda)$ of subsets of
$\Lambda$ is an _ideal on $\Lambda$_ if it is closed under taking subsets and
finite unions of its elements, $\Lambda\notin\mathcal{I}$ and $\mathcal{I}$
contains all finite subsets of $\Lambda$. By $\mathrm{Fin}(\Lambda)$ we denote
the family of all finite subsets of $\Lambda$. For $\Lambda=\omega$, we write
$\mathrm{Fin}$ instead of $\mathrm{Fin}(\omega)$. For an ideal $\mathcal{I}$
on $\Lambda$, we write
$\mathcal{I}^{+}=\\{A\subseteq\Lambda:A\notin\mathcal{I}\\}$ and call it the
_coideal of $\mathcal{I}$_, and we write $\mathcal{I}^{*}=\\{\Lambda\setminus
A:A\in\mathcal{I}\\}$ and call it the _filter dual to $\mathcal{I}$_. For an
ideal $\mathcal{I}$ on $\Lambda$ and $A\in\mathcal{I}^{+}$, it is easy to see
that $\mathcal{I}\restriction A=\\{A\cap B:B\in\mathcal{I}\\}$ is an ideal on
$A$.

In our research the following ideal on $\omega^{2}$ plays an important role:

$\mathrm{Fin}^{2}=\left\\{C\subseteq\omega^{2}:\\{n\in\omega:\\{k\in\omega:(n,k)\in
C\\}\notin\mathrm{Fin}\\}\in\mathrm{Fin}\right\\}.$

We say that a function $f:\Lambda\to\Sigma$ is _$\mathcal{I}$ -to-one_ if
$f^{-1}(\sigma)\in\mathcal{I}$ for every $\sigma\in\Sigma$.

A set $A\subseteq X$ is $F_{\sigma}$ ($G_{\delta}$, $F_{\sigma\delta}$, etc.,
resp.) in a topological space $X$ if $A$ is a union of a countable family of
closed sets ($A$ is an intersection of a countable family of open sets, $A$ is
an intersection of a countable family of $F_{\sigma}$ sets, etc., resp.).

For a function $f:X\to Y$ and a set $A\subseteq X$, we write $f\restriction A$
to denote the restriction of $f$ to the set $A$.

##  Part I Partition regular operations

###  3\. Partition regular operations and ideals associated with them

Below we introduce a notion that proved to be a convenient tool allowing to
grasp the common feature of different kinds of convergences related to
Hindman, Ramsey and van der Waerden spaces.

######  Definition 3.1.

Let $\Lambda$ and $\Omega$ be countable infinite sets. Let $\mathcal{F}$ be a
nonempty family of infinite subsets of $\Omega$ such that $F\setminus
K\in\mathcal{F}$ for every $F\in\mathcal{F}$ and a finite set
$K\subseteq\Omega$. We say that a function
$\rho:\mathcal{F}\to[\Lambda]^{\omega}$ is _partition regular_ if

(M):

$\forall E,F\in\mathcal{F}\,\left(E\subseteq
F\implies\rho(E)\subseteq\rho(F)\right)$,

(R):

$\forall F\in\mathcal{F}\,\forall A,B\subseteq\Lambda\,\left(\rho(F)=A\cup
B\implies\exists E\in\mathcal{F}\,(\rho(E)\subseteq A\lor\rho(E)\subseteq
B)\right).$

(S):

$\forall F\in\mathcal{F}\,\exists E\in\mathcal{F}\,(E\subseteq F\land\forall
a\in\rho(E)\,\exists K\in[\Omega]^{<\omega}(a\notin\rho(E\setminus K)))$.

In our considerations, we use the following easy observation concerning
condition . ‣ Partition regular operations and ideals associated with them of
Definition 3.1.

######  Proposition 3.2.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ (with
$\mathcal{F}\subseteq[\Omega]^{\omega}$) be a partition regular function. Then
for every $F\in\mathcal{F}$ there is $E\in\mathcal{F}$ such that $E\subseteq
F$ and for every finite set $L\subseteq\Lambda$ there exists a finite set
$K\subseteq\Omega$ such that $\rho(E\setminus K)\subseteq\rho(E)\setminus L$.

###### Proof.

For $F\in\mathcal{F}$, let $E\in\mathcal{F}$ be as in condition . ‣ Partition
regular operations and ideals associated with them of Definition 3.1. Let
$L\subseteq\Lambda$ be a finite set. For every $a\in\rho(E)$, we take a finite
set $K_{a}$ such that $a\notin\rho(E\setminus K_{a})$. Then
$K=\bigcup\\{K_{a}:a\in\rho(E)\cap L\\}$ is finite and $\rho(E\setminus
K)\subseteq\rho(E)\setminus L$. ∎

The following easy proposition reveals basic relationships between partition
regular functions and ideals.

######  Proposition 3.3.

  1. (1)

If $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ is partition regular, then

$\mathcal{I}_{\rho}=\\{A\subseteq\Lambda:\forall
F\in\mathcal{F}\,(\rho(F)\not\subseteq A)\\}.$

is an ideal on $\Lambda$.

  2. (2)

If $\mathcal{I}$ is an ideal on $\Lambda$, then the function

$\rho_{\mathcal{I}}:\mathcal{I}^{+}\to[\Lambda]^{\omega}\text{\ \ \ given by \
\ }\rho_{\mathcal{I}}(A)=A$

is partition regular and $\mathcal{I}=\mathcal{I}_{\rho_{\mathcal{I}}}.$

######  Remark.

If $\rho$ is partition regular and $\tau=\rho_{\mathcal{I}_{\rho}}$, then
$\mathcal{I}_{\tau}=\mathcal{I}_{\rho}$ but, as we will see, in general,
$\rho\neq\tau$. More important, $\tau$ may miss some crucial properties which
$\rho$ possesses (e.g. $P$-like properties – see Proposition 6.5(3)(4)).

Below we present the most important examples of partition regular functions
that were our prototypes while we were thinking on a unified approach to
Hindman, Ramsey and van der Waerden spaces.

####  3.1. Hindman’s finite sums theorem

Let the function $\operatorname{FS}:[\omega]^{\omega}\to[\omega]^{\omega}$ be
given by

$\operatorname{FS}(D)=\left\\{\sum_{n\in\alpha}n:\alpha\in[D]^{<\omega}\setminus\\{\emptyset\\}\right\\}$

i.e. $\operatorname{FS}(D)$ is the set of all finite non-empty sums of
distinct elements of $D$.

A set $D\subseteq\omega$ is _sparse_ [55, p. 1598] if for each
$n\in\operatorname{FS}(D)$ there exists the unique set $\alpha\subseteq D$
such that $n=\sum_{i\in\alpha}i$. This unique set will be denoted by
$\alpha_{D}(n)$. For instance, the set $E=\\{2^{i}:i\in\omega\\}$ is sparse,
and in the sequel, we write $\alpha(n)$ instead of $\alpha_{E}(n)$.

A sparse set $D\subseteq\omega$ is _very sparse_ [22, p. 894] if
$\alpha_{D}(x)\cap\alpha_{D}(y)\neq\emptyset$ implies
$x+y\notin\operatorname{FS}(D)$ for every $x,y\in\operatorname{FS}(D)$.

######  Theorem 3.4 (Hindman).

The function $\operatorname{FS}$ is partition regular and the family

$\mathcal{H}=\mathcal{I}_{\operatorname{FS}}=\\{A\subseteq\omega:\forall
D\in[\omega]^{\omega}\,(\operatorname{FS}(D)\not\subseteq A)\\}$

is an ideal on $\omega$. The ideal $\mathcal{H}$ is called _the Hindman ideal_
[29, p. 109]. It is known that sets from $\mathcal{H}^{+}$ (that are called
_IP-sets_) are examples of so-called Poincaré sequences111A set
$W\subseteq\mathbb{Z}$ is called a _Poincaré sequence_ [33, Definition 3.6 at
p. 72] if for any measure preserving system $(X,\mathcal{B},\mu,T)$ and
$A\in\mathcal{B}$ with $\mu(A)>0$ we have $\mu(T^{-n}[A]\cap A)>0$ for some
$n\in W$, $n\neq 0$. that play an important role in the study of recurrences
in topological dynamics [33, p. 74].

###### Proof.

It is easy to see that condition . ‣ Partition regular operations and ideals
associated with them of Definition 3.1 is satisfied for $\operatorname{FS}$.
Condition . ‣ Partition regular operations and ideals associated with them of
Definition 3.1 holds for $\operatorname{FS}$ as in this case it is the well
known Hindman’s finite sums theorem [42, Theorem 3.1],[4, Theorem 3.5]. To see
that condition . ‣ Partition regular operations and ideals associated with
them of Definition 3.1 holds for $\operatorname{FS}$, it is enough to notice
[55, p. 1598] that every infinite set $F\subseteq\omega$ has an infinite
sparse subset $G\subseteq F$ which obviously satisfies condition . ‣ Partition
regular operations and ideals associated with them. Finally, Proposition
3.3(1) shows that $\mathcal{H}$ is an ideal on $\omega$. ∎

The following lemma will be used in some proofs regarding properties of the
function $\operatorname{FS}$.

######  Lemma 3.5 ([55, Lemma 7]).

If $D$ is an infinite sparse set, then there exists a set
$S=\\{s_{i}:i\in\omega\\}\subseteq\operatorname{FS}(D)$ such that for every
$i\in\omega$ we have $s_{i}<s_{i+1}$ and

$\max\alpha_{D}(s_{i})<\min\alpha_{D}(s_{i+1})\text{ and
}\max\alpha(s_{i})<\min\alpha(s_{i+1}).$

####  3.2. Ramsey’s theorem for coloring graphs

######  Theorem 3.6 (Ramsey).

Let $r:[\omega]^{\omega}\to\left[[\omega]^{2}\right]^{\omega}$ be given by

$r(H)=[H]^{2}=\\{\\{x,y\\}\subseteq[\omega]^{2}:x,y\in H,x\neq y\\}$

i.e. $r(H)$ is the set of all unordered pairs of elements of $H$. Then $r$ is
partition regular and the family

$\mathcal{R}=\mathcal{I}_{r}=\\{A\subseteq[\omega]^{2}:\forall
H\in[\omega]^{\omega}\,([H]^{2}\not\subseteq A)\\}$

is an ideal on $[\omega]^{2}$. The ideal $\mathcal{R}$ is called the _Ramsey
ideal_ [65, 48]. (If we identify a set $A\subseteq[\omega]^{2}$ with a graph
$G_{A}=(\omega,A)$, the ideal $\mathcal{R}$ can be seen as an ideal consisting
of graphs without infinite complete subgraphs).

###### Proof.

It is easy to see that condition . ‣ Partition regular operations and ideals
associated with them of Definition 3.1 is satisfied for $r$. Condition . ‣
Partition regular operations and ideals associated with them of Definition 3.1
holds for $r$ as in this case it is the well known Ramsey’s theorem for
coloring graphs [69, Theorem A],[38, Theorem 1.5]. To see that condition . ‣
Partition regular operations and ideals associated with them of Definition 3.1
holds for $r$, it is enough to notice that for every $\\{a,b\\}\in[F]^{2}$ we
have $\\{a,b\\}\notin[F\setminus\\{a,b\\}]^{2}$. Finally, Proposition 3.3(1)
shows that $\mathcal{R}$ is an ideal on $[\omega]^{2}$. ∎

####  3.3. The positive differences and the associated ideal

Let the function $\Delta:[\omega]^{\omega}\to[\omega]^{\omega}$ be given by

$\Delta(E)=\left\\{a-b:a,b\in E,a>b\right\\}$

i.e. $\Delta(E)$ is the set of all positive differences of distinct elements
of $E$.

We say that a set $E\subseteq\omega$ is _$\mathcal{D}$ -sparse_ [20, p. 2009]
if for every $a\in\Delta(E)$ there are unique elements $b,c\in E$ such that
$a=b-c$.

######  Proposition 3.7.

The function $\Delta$ is partition regular and the family

$\mathcal{D}=\mathcal{I}_{\Delta}=\\{A\subseteq\omega:\forall
E\in[\omega]^{\omega}\,(\Delta(E)\not\subseteq A)\\}$

is an ideal on $\omega$ such that $\mathcal{D}\subsetneq\mathcal{H}$. It is
known that sets from $\mathcal{D}^{+}$ are examples of so-called Poincaré
sequences [33, p. 74].

###### Proof.

It is easy to see that condition . ‣ Partition regular operations and ideals
associated with them of Definition 3.1 is satisfied for $\Delta$. It is known
[20, Proposition 4.1] that condition . ‣ Partition regular operations and
ideals associated with them of Definition 3.1 holds for $\Delta$. To see that
condition . ‣ Partition regular operations and ideals associated with them of
Definition 3.1 holds for $\Delta$, it is enough to notice [20, Proposition
4.3(2)] that every infinite set $F\subseteq\omega$ has an infinite
$\mathcal{D}$-sparse subset $G\subseteq F$ which obviously satisfies condition
. ‣ Partition regular operations and ideals associated with them. Finally,
Proposition 3.3(1) shows that $\mathcal{D}$ is an ideal on $\omega$ and it is
known [71, Proposition 4.2.1],[20, Proposition 4.1] that
$\mathcal{D}\subsetneq\mathcal{H}$. ∎

####  3.4. The summable ideal

######  Proposition 3.8.

The family

$\mathcal{I}_{1/n}=\left\\{A\subseteq\omega:\sum_{n\in
A}\frac{1}{n+1}<\infty\right\\}$

is an ideal on $\omega$. The ideal $\mathcal{I}_{1/n}$ is called the _summable
ideal_ [64, Definition 1.6],[62, Example 3],[75, p. 238],[51, p. 411]. The
function $\rho_{\mathcal{I}_{1/n}}:\mathcal{I}_{1/n}^{+}\to[\omega]^{\omega}$
given by $\rho_{\mathcal{I}_{1/n}}(A)=A$ is partition regular and
$\mathcal{I}_{1/n}=\mathcal{I}_{\rho_{\mathcal{I}_{1/n}}}$.

###### Proof.

It is easy to show that $\mathcal{I}_{1/n}$ is an ideal on $\omega$, whereas
Proposition 3.3(2) gives the required properties of
$\rho_{\mathcal{I}_{1/n}}$. ∎

####  3.5. van der Waerden’s arithmetical progressions theorem

######  Theorem 3.9 (van der Waerden).

A set $A\subseteq\omega$ is called an _AP-set_ if it contains an arithmetic
progressions of arbitrary finite length. The family

$\mathcal{W}=\\{A\subseteq\omega:\text{$A$ is not an AP-set}\\}$

is an ideal on $\omega$. The ideal $\mathcal{W}$ is called the _van der
Waerden ideal_ [29, p. 107]. The function
$\rho_{\mathcal{W}}:\mathcal{W}^{+}\to[\omega]^{\omega}$ given by
$\rho_{\mathcal{W}}(A)=A$ is partition regular and
$\mathcal{W}=\mathcal{I}_{\rho_{\mathcal{W}}}$.

###### Proof.

It is easy to see that all conditions from the definition of an ideal but
additivity are satisfied, whereas additivity is the well known van der
Waerden’s arithmetical progressions theorem [73],[38, Theorem 2.1]. Finally,
Proposition 3.3(2) gives the required properties of $\rho_{\mathcal{W}}$. ∎

####  3.6. Ideals on directed sets

Finally, we introduce a class of partition regular functions which are
connected with ideals on directed sets [17, 18]. Recall, that $(\Lambda,<)$ is
a _directed set_ if the relation $<$ is an upward directed strict partial
order on $\Lambda$.

Let $(\Lambda,<)$ be a _directed set_ such that $\Lambda$ is infinite
countable. A set $B\subseteq\Lambda$ is _cofinal_ in $(\Lambda,<)$ if for
every $\lambda\in\Lambda$ there is $b\in B$ with $\lambda<b$. A family
$\mathcal{I}$ of subsets of $\Lambda$ is an _ideal on $(\Lambda,<)$_ [18,
Definition 2.2] if $\mathcal{I}$ is an ideal on $\Lambda$ and $\mathcal{I}$
contains all sets which are _not_ cofinal. A family $\mathcal{B}$ of subsets
of $\Lambda$ is a _coideal basis on $(\Lambda,<)$_ [18, Definition 2.4] if
$\mathcal{B}\neq\emptyset$, all sets in $\mathcal{B}$ are cofinal and if
$C\cup D\in\mathcal{B}$, then there exists $B\in\mathcal{B}$ such that
$B\subseteq C$ or $B\subseteq D$. In particular, for every ideal $\mathcal{I}$
on $(\Lambda,<)$ the family $\mathcal{I}^{+}$ is a coideal basis on
$(\Lambda,<)$. It is known [17, Proposition 2.7] that $\mathcal{I}$ is an
ideal on $(\Lambda,<)$ if and only if there exists a coideal basis
$\mathcal{B}$ on $(\Lambda,<)$ such that
$\mathcal{I}=\\{A\subseteq\Lambda:\forall B\in\mathcal{B}\,(B\not\subseteq
A)\\}$.

The following easy proposition reveals basic relationships between partition
regular functions and ideals on directed sets.

######  Proposition 3.10.

Let $(\Lambda,<)$ be a directed set.

  1. (1)

If $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ is a partition regular function
such that $\rho(F)$ is cofinal for every $F\in\mathcal{F}$, then

$\mathcal{I}_{\rho}=\\{A\subseteq\Lambda:\forall
F\in\mathcal{F}\,(\rho(F)\not\subseteq A)\\}.$

is an ideal on $(\Lambda,<)$.

  2. (2)

For a coideal basis $\mathcal{B}$ on $(\Lambda,<)$ (in particular for
$\mathcal{B}=\mathcal{I}^{+}$, where $\mathcal{I}$ is an ideal on
$(\Lambda,<)$), we define

$\widehat{\mathcal{B}}=\\{B\setminus
K:B\in\mathcal{B},K\in[\Lambda]^{<\omega}\\}.$

Then the function
$\rho_{\mathcal{B}}:\widehat{\mathcal{B}}\oplus\mathrm{Fin}(\Lambda)^{*}\to[\Lambda]^{\omega}$
given by

$\rho_{\mathcal{B}}((B\setminus K)\sqcup C)=(B\setminus
K)\cap\\{\lambda\in\Lambda:\forall\lambda^{\prime}\in(\Lambda\setminus
C)\,(\lambda^{\prime}<\lambda)\\}$

is a partition regular function such that $\rho_{\mathcal{B}}((B\setminus
K)\sqcup C)$ is cofinal for every $(B\setminus L)\sqcup
C\in\widehat{\mathcal{B}}\oplus\mathrm{Fin}(\Lambda)^{*}$ and
$\mathcal{I}_{\rho_{\mathcal{B}}}=\\{A\subseteq\Lambda:\forall
B\in\mathcal{B}\,(B\not\subseteq A)\\}.$

###  4\. Restrictions and small accretions

####  4.1. Restrictions of partition regular operations

For $B\notin\mathcal{I}_{\rho}$, we define a family $\mathcal{F}\restriction
B=\\{E\in\mathcal{F}:\rho(E)\subseteq B\\}$ and a function $\rho\restriction
B:\mathcal{F}\restriction B\to[B]^{\omega}$ by $(\rho\restriction
B)(E)=\rho(E)$ (i.e. $\rho\restriction
B=\rho\restriction(\mathcal{F}\restriction B)$). The following easy
proposition reveals relationships between restriction of a function $\rho$ and
restriction of an ideal $\mathcal{I}_{\rho}$.

######  Proposition 4.1.

If $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ is partition regular and
$B\notin\mathcal{I}_{\rho}$, then $\rho\restriction B$ is partition regular
and $\mathcal{I}_{\rho\restriction B}=\mathcal{I}_{\rho}\restriction B$.

####  4.2. Small accretions of partition regular operations

We will need the following notion in the last part of the paper for
characterization that shows when there exists a space of one considered type
that is not of the other type (Theorem 16.1).

######  Definition 4.2.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ (with
$\mathcal{F}\subseteq[\Omega]^{\omega}$) be a partition regular function.

  1. (1)

A set $F\in\mathcal{F}$ has _small accretions_ if
$\rho(F)\setminus\rho(F\setminus K)\in\mathcal{I}_{\rho}$ for every finite set
$K$.

  2. (2)

$\rho$ has _small accretions_ if for every $E\in\mathcal{F}$ there is
$F\in\mathcal{F}$ such that $F\subseteq E$ and $F$ has small accretions.

######  Proposition 4.3.

If
$\rho\in\\{\operatorname{FS},r,\Delta\\}\cup\\{\rho_{\mathcal{I}}:\text{$\mathcal{I}$
is an ideal}\\}$, then $\rho$ has small accretions.

###### Proof for $\rho=\rho_{\mathcal{I}}$ where $\mathcal{I}$ is an ideal.

The function $\rho$ has small accretions, since for every
$A\in\mathcal{I}^{+}$ and finite $K\subseteq\Lambda$ we have
$\rho_{\mathcal{I}}(A)\setminus\rho_{\mathcal{I}}(A\setminus
K)=A\setminus(A\setminus K)\subseteq K\in\mathcal{I}$. ∎

###### Proof for $\rho=\operatorname{FS}$.

It is known [22, Lemma 2.2] that every infinite set $E\subseteq\omega$ has an
infinite very sparse subset $F\subseteq E$, so if we show that every very
sparse set has small accretions, the proof will be finished.

Let $F\subseteq\omega$ be an infinite very sparse set and $K\subseteq\omega$
be a finite set. Assume towards contradiction that
$\operatorname{FS}(D)\subseteq\operatorname{FS}(F)\setminus\operatorname{FS}(F\setminus
K)=\\{x\in\operatorname{FS}(F):\ \alpha_{F}(x)\cap K\neq\emptyset\\}$ for some
$D\in[\omega]^{\omega}$. Since $K$ is finite, we can find $x,y\in D$, $x\neq
y$, such that $\alpha_{F}(x)\cap\alpha_{F}(y)\neq\emptyset$. But then
$x+y\in\operatorname{FS}(D)\setminus\operatorname{FS}(F)$, a contradiction. ∎

###### Proof for $\rho=r$.

The function $r$ has small accretions, since for every $A\in[\omega]^{\omega}$
and finite $K\subseteq\omega$ we have $r(A)\setminus r(A\setminus
K)=[A]^{2}\setminus[A\setminus K]^{2}=\\{\\{i,j\\}:\ i\in A\cap K,j\in
A\\}\in\mathcal{R}$. ∎

###### Proof for $\rho=\Delta$.

It is known [20, Proposition 4.3(2)] that every infinite set
$E\subseteq\omega$ has an infinite $\mathcal{D}$-sparse subset $F\subseteq E$,
so if we show that every $\mathcal{D}$-sparse set has small accretions, the
proof will be finished.

Let $F\subseteq\omega$ be an infinite $\mathcal{D}$-sparse set and
$K\subseteq\omega$ be a finite set. It is known [20, Proposition 4.3(1)] that
then $F-n\in\mathcal{D}$ for every $n<\min F$, and consequently, $\\{a-b:a\in
F\setminus K,b\in F\cap K\\}\cap\omega\in\mathcal{D}$. Thus,
$\Delta(F)\setminus\Delta(F\setminus K)=\\{a-b:a\in F\cap K,b\in
F,a>b\\}\cup(\\{a-b:a\in F\setminus K,b\in F\cap
K\\}\cap\omega)\in\mathcal{D}$ as a finite union of sets from $\mathcal{D}$. ∎

###  5\. Topological complexity of partition regular operations

If $\Lambda$ is a countable infinite set, then we consider
$2^{\Lambda}=\\{0,1\\}^{\Lambda}$ as a product (with the product topology) of
countably many copies of a discrete topological space $\\{0,1\\}$. Since
$2^{\Lambda}$ is a Polish space [54, p. 13] and $[\Lambda]^{\omega}$ is a
$G_{\delta}$ subset of $2^{\Lambda}$, we obtain that $[\Lambda]^{\omega}$ is a
Polish space as well [54, Theorem 3.11]. In particular, if $\Lambda$ and
$\Omega$ are countable infinite and $\mathcal{F}\subseteq[\Omega]^{\omega}$,
we say that a partition regular function
$\rho:\mathcal{F}\to[\Lambda]^{\omega}$ is _continuous_ if $\rho$ is a
continuous function from a topological subspace $\mathcal{F}$ into a
topological space $[\Lambda]^{\omega}$.

By identifying subsets of $\Lambda$ with their characteristic functions, we
equip $\mathcal{P}(\Lambda)$ with the topology of the space $2^{\Lambda}$ and
therefore we can assign topological notions to ideals on $\Lambda$. In
particular, an ideal $\mathcal{I}$ is _Borel_ (_analytic_ , _coanalytic_ ,
resp.) if $\mathcal{I}$ is a Borel (analytic, coanalytic, resp.) subset of
$2^{\Lambda}$. Recall, a set $A\subseteq X$ is _analytic_ if there is a Polish
space $Y$ and a Borel set $B\subseteq X\times Y$ such that $A$ is a projection
of $B$ onto the first coordinate [54, Exercise 14.3], and a set $C\subseteq X$
is coanalytic if $X\setminus C$ is an analytic set.

######  Proposition 5.1.

If a partition regular function $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ (with
$\mathcal{F}\subseteq[\Omega]^{\omega}$) is continuous and $\mathcal{F}$ is a
closed subset of $[\Omega]^{\omega}$, then the ideal $\mathcal{I}_{\rho}$ is
coanalytic.

###### Proof.

We will show that
$\mathcal{I}_{\rho}^{+}=\mathcal{P}(\Lambda)\setminus\mathcal{I}_{\rho}$ is an
analytic set. Let
$B=\\{(A,F)\in\mathcal{P}(\Lambda)\times\mathcal{F}:\rho(F)\subseteq A\\}.$
Since $B\subseteq\mathcal{P}(\Lambda)\times[\Omega]^{\omega}$ and
$\mathcal{I}_{\rho}^{+}$ is a projection of $B$ onto the first coordinate, we
only need to show that $B$ is a Borel set. It suffices to show that
$C=(\mathcal{P}(\Lambda)\times[\Omega]^{\omega})\setminus B$ is an open set,
since

$B=((\mathcal{P}(\Lambda)\times[\Omega]^{\omega})\setminus
C)\cap(\mathcal{P}(\Lambda)\times\mathcal{F}).$

Let $(A,F)\in C$. We have two cases: (1) $F\notin\mathcal{F}$ or (2)
$F\in\mathcal{F}$.

_Case (1)_. Since $\mathcal{F}$ is closed, there is an open set
$U\subseteq[\Omega]^{\omega}$ with $F\in U$ and $U\cap\mathcal{F}=\emptyset$.
Then $W=\mathcal{P}(\Lambda)\times U$ is open and $(A,F)\in W\subseteq C$.

_Case (2)_. Since $\rho(F)\not\subseteq A$, there is $a\in\rho(F)\setminus A$.
Let $V=\\{D\in\mathcal{P}(\Lambda):a\in D\\}$. Then $V$ is an open and closed
set, $A\notin V$ and $\rho(F)\in V$. Since $\rho$ is continuous at the point
$F$, there is an open set $U\subseteq[\Omega]^{\omega}$ such that $F\in U$ and
$\rho[U]\subseteq V$. Then $W=(\mathcal{P}(\Lambda)\setminus V)\times U$ is
open and $(A,F)\in W\subseteq C$. ∎

######  Proposition 5.2.

  1. (1)

The ideals $\mathcal{I}_{1/n}$ and $\mathcal{W}$ are $F_{\sigma}$.

  2. (2)

The functions $\operatorname{FS}$ and $r$ are continuous.

  3. (3)

The function $\Delta$ is not continuous. In fact, the function $\Delta$ is
discontinuous at every point $A$ such that $\Delta(A)\neq\omega$.

  4. (4)

If $\mathcal{L}=\\{A\in[\omega]^{\omega}:\forall
n\in\omega\,(e_{A}(n+1)-e_{A}(n)>e_{A}(n))\\}$ where $e_{A}:\omega\to A$ is
the increasing enumeration of a set $A\subseteq\omega$, then
$\mathcal{I}_{\Delta}=\mathcal{I}_{\Delta\restriction\mathcal{L}}$,
$\mathcal{L}$ is closed and $\Delta\restriction\mathcal{L}$ is continuous.

  5. (5)

The ideals $\mathcal{H}$, $\mathcal{R}$ and $\mathcal{D}$ are coanalytic.

###### Proof.

(1) It is known that $\mathcal{I}_{1/n}$ and $\mathcal{W}$ are $F_{\sigma}$
[64, Example 1.5], [23, Example 4.12].

(2) _Case of $\operatorname{FS}$_. Let $D\in[\omega]^{\omega}$ and let $U$ be
an open basic neighborhood of $\operatorname{FS}(D)$. Then there exists a
finite set $G\subseteq\omega$ such that
$U=\\{B\in[\omega]^{\omega}:B\cap\\{0,1,\dots,\max G\\}=G\\}$. Let
$F=D\cap\\{0,1,\dots,\max G\\}$. Then
$V=\\{A\in[\omega]^{\omega}:A\cap\\{0,1,\dots,\max G\\}=F\\}$ is an open
neighborhood of $D$ and $\operatorname{FS}[V]\subseteq U$.

_Case of $r$_. Let $D\in[\omega]^{\omega}$ and let $U$ be an open basic
neighborhood of $[D]^{2}$. There exists a finite set $G\subseteq[\omega]^{2}$
such that $U=\\{B\in\left[[\omega]^{2}\right]^{\omega}:B\cap[N]^{2}=G\\}$,
where $N=\max\\{\max\\{p,q\\}:\\{p,q\\}\in G\\}$. Then
$V=\\{A\in[\omega]^{\omega}:A\cap N=D\\}$ is an open neighborhood of $D$ and
$r[V]\subseteq U$.

(3) Let $A\subseteq\omega$ be such that $b\notin\Delta(A)$ for some
$b\in\omega$. Then $U=\\{B\subseteq\omega:b\notin B\\}$ is an open
neighborhood of $\Delta(A)$. Let $V$ be an open basic neighborhood of $A$.
There is $N\in\omega$ such that $V=\\{C\subseteq\omega:C\cap N=A\cap N\\}$.
Then $C=(A\cap N)\cup(\omega\setminus N)\in V$ and $\Delta(C)=\omega\notin U$.
Hence the function $\Delta$ is discontinuous at the point $A$.

(4) It is obvious that
$\mathcal{I}_{\Delta}=\mathcal{I}_{\Delta\restriction\mathcal{L}}$. To show
that $\mathcal{L}$ is closed, notice that
$[\omega]^{\omega}\setminus\mathcal{L}$ is open as for each
$A\in[\omega]^{\omega}\setminus\mathcal{L}$ there is $n\in\omega$ such that
$e_{A}(n+1)-e_{A}(n)\leq e_{A}(n)$ and
$U=\\{C\in[\omega]^{\omega}:C\cap(e_{A}(n+1)+1)=A\cap(e_{A}(n+1)+1)\\}$ is an
open neighborhood of $A$ disjoint with $\mathcal{L}$.

Below we show that $\Delta\restriction\mathcal{L}$ is continuous. Let
$A\in\mathcal{L}$. We are going to show that the function
$\Delta\restriction\mathcal{L}$ is continuous at the point $A$. Let $U$ be a
neighborhood of $\Delta(A)$. Without loss of generality, we can assume that
there is $N\in\omega$ such that $U=\\{B\in[\omega]^{\omega}:B\cap
N=\Delta(A)\cap N\\}$. There exists $M\in\omega$ such that $e_{A}(M)>N$. Then
$V=\\{C\in[\omega]^{\omega}:C\cap(e_{A}(M)+1)=A\cap(e_{A}(M)+1)\\}$ is an open
neighborhood of $A$. Once we show that $\Delta[V\cap\mathcal{L}]\subseteq U$,
the proof will be finished. Let $C\in V\cap\mathcal{L}$. Since
$A,C\in\mathcal{L}$, we obtain
$\Delta(C)\cap(e_{A}(M)+1)=\Delta(C\cap(e_{A}(M)+1))=\Delta(A\cap(e_{A}(M)+1))=\Delta(A)\cap(e_{A}(M)+1)$.
But $N<e_{A}(M)$, hence $\Delta(C)\cap N=\Delta(A)\cap N$ and consequently
$\Delta(C)\in U$.

(5) It is known that $\mathcal{H}$ and $\mathcal{R}$ are coanalytic [23,
Example 4.11],[65, Lemma 1.6.24] (but it also follows from item (2) and
Proposition 5.1). It follows from item (4) and Proposition 5.1 that
$\mathcal{D}$ is coanalytic. ∎

###  6\. P-like properties

####  6.1. P-like properties of ideals

For $A,B\subseteq\Lambda$, we write $A\subseteq^{*}B$ if there is a finite set
$K\subseteq\Lambda$ with $A\setminus K\subseteq B$.

Let us recall definitions of P-like properties of ideals that are considered
in the literature [43, p. 2030]. An ideal $\mathcal{I}$ on $\Lambda$ is

  * •

_$P^{-}(\Lambda)$_ if for every $\subseteq$-decreasing sequence
$A_{n}\in\mathcal{I}^{+}$ with $A_{0}=\Lambda$ and $A_{n}\setminus
A_{n+1}\in\mathcal{I}$ for each $n\in\omega$ there exists
$B\in\mathcal{I}^{+}$ such that $B\subseteq^{*}A_{n}$ for each $n\in\omega$;

  * •

_$P^{-}$_ if for every $\subseteq$-decreasing sequence
$A_{n}\in\mathcal{I}^{+}$ with $A_{n}\setminus A_{n+1}\in\mathcal{I}$ for each
$n\in\omega$ there exists $B\in\mathcal{I}^{+}$ such that
$B\subseteq^{*}A_{n}$ for each $n\in\omega$;

  * •

_$P^{+}$_ if for every $\subseteq$-decreasing sequence
$A_{n}\in\mathcal{I}^{+}$ there exists $B\in\mathcal{I}^{+}$ such that
$B\subseteq^{*}A_{n}$ for each $n\in\omega$.

The following proposition reveals some implications between P-like properties
and provides equivalent forms of the properties $P^{-}(\Lambda)$ and $P^{-}$
that were considered in the literature [61] under the names _weak P-ideals_
and _hereditary weak P-ideals_ , where the author used them for in-depth
research on $\mathcal{I}$-spaces.

######  Proposition 6.1.

Let $\mathcal{I}$ be an ideal on an infinite countable set $\Lambda$.

  1. (1)

$\mathcal{I}\text{ is }P^{+}\implies\mathcal{I}\text{ is
}P^{-}\implies\mathcal{I}\text{ is }P^{-}(\Lambda)$.

  2. (2)

The implications from item (1) cannot be reversed.

  3. (3)

The following conditions are equivalent.

    1. (a)

$\mathcal{I}$ is $P^{-}(\Lambda)$ ($\mathcal{I}$ is $P^{-}$, resp.).

    2. (b)

For every partition $\mathcal{A}$ of $\Lambda$ (of any set
$C\in\mathcal{I}^{+}$, resp.) into sets from $\mathcal{I}$ there exists
$B\in\mathcal{I}^{+}$ such that $B\subseteq\Lambda$ ($B\subseteq C$, resp.)
and $B\cap A$ is finite for each $A\in\mathcal{A}$.

    3. (c)

$\mathcal{I}$ is a _weak P-ideal_ (_hereditary weak P-ideal_ , resp.) i.e. for
every countable family $\mathcal{A}\subseteq\mathcal{I}$ of subsets of
$\Lambda$ (subsets of any $C\in\mathcal{I}^{+}$, resp.) there exists
$B\in\mathcal{I}^{+}$ such that $B\subseteq\Lambda$ ($B\subseteq C$, resp.)
and $B\cap A$ is finite for each $A\in\mathcal{A}$.

###### Proof.

(1) Straightforward.

(2) The ideal $\mathrm{Fin}\oplus\mathrm{Fin}^{2}$ is
$P^{-}(\omega\sqcup\omega^{2})$ (the set $B=\omega\sqcup\emptyset$ works for
every sequence) but not $P^{-}$ (as witnessed by the sets
$A_{n}=\emptyset\sqcup((\omega\setminus n)\times\omega)$).

Below we show an example of a $P^{-}$ ideal that is not $P^{+}$. For a set
$A\subseteq\omega$, we define the asymptotic density of $A$ by
$\overline{d}(A)=\limsup_{n\to\infty}|A\cap n|/n$. Then the ideal
$\mathcal{I}_{d}=\\{A\subseteq\omega:\overline{d}(A)=0\\}$ is $P^{-}$ (see
e.g. [9, Corollary 1.1]). Now we show that $\mathcal{I}_{d}$ is not $P^{+}$.
Take a decreasing sequence $B_{n}\subseteq\omega$ such that
$0<\overline{d}(B_{n})<1/n$ for each $n\in\omega$. If $C\subseteq\omega$ is
such that $C\subseteq^{*}B_{n}$ for all $n\in\omega$, then
$\overline{d}(C)\leq\overline{d}(B_{n})\to 0$ as $n\to\infty$. Hence
$C\in\mathcal{I}_{d}$. This shows that $\mathcal{I}_{d}$ is not $P^{+}$.

(3) Straightforward. ∎

There are known relationships between topological complexity and P-like
properties.

######  Theorem 6.2 ([61, Proposition 4.9],[51, Lemma 1.2],[43, Theorem 3.7]).

  1. (1)

Each $G_{\delta\sigma\delta}$ (in particular, $F_{\sigma\delta}$) ideal is
$P^{-}$ (hence $P^{-}(\Lambda)$).

  2. (2)

Each $F_{\sigma}$ ideal is $P^{+}$ (hence $P^{-}$ and $P^{-}(\Lambda)$).

  3. (3)

If $\mathcal{I}$ is an analytic ideal, then the following conditions are
equivalent.

    1. (a)

There exists a $P^{+}$ ideal $\mathcal{J}$ with
$\mathcal{I}\subseteq\mathcal{J}$.

    2. (b)

There exists an $F_{\sigma}$ ideal $\mathcal{K}$ with
$\mathcal{I}\subseteq\mathcal{K}$.

####  6.2. P-like properties of partition regular operations

######  Definition 6.3.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular. For sets
$F\in\mathcal{F}$ and $B\subseteq\Lambda$, we write $\rho(F)\subseteq^{\rho}B$
if there is a finite set $K\subseteq\Omega$ with $\rho(F\setminus K)\subseteq
B$.

######  Remark.

We want to stress here that the relation “$\rho(F)\subseteq^{\rho}B$” is in
fact a relation between $F$ and $B$ and not between $\rho(F)$ and $B$ because
it can happen that $\rho(F)=\rho(G)$ and $\rho(F)\subseteq^{\rho}B$ but
$\rho(G)\not\subseteq^{\rho}B$. We decided that we write
$\rho(F)\subseteq^{\rho}B$ instead of $F\subseteq^{\rho}B$ as the former seems
more natural for us. The same remark applies to other notions involving
“$\rho(F)$” we defined earlier or we define later (e.g. Definitions 6.4 and
9.1).

The following properties will prove useful in the studies of classes of
sequentially compact spaces defined with the aid of partition regular
functions.

######  Definition 6.4.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular. We say that
$\rho$ is

  1. (1)

_$P^{-}(\Lambda)$_ if for every $\subseteq$-decreasing sequence
$A_{n}\in\mathcal{I}_{\rho}^{+}$ with $A_{0}=\Lambda$ and $A_{n}\setminus
A_{n+1}\in\mathcal{I}_{\rho}$ for each $n\in\omega$ there exists
$F\in\mathcal{F}$ such that $\rho(F)\subseteq^{\rho}A_{n}$ for each
$n\in\omega$;

  2. (2)

_$P^{-}$_ if for every $\subseteq$-decreasing sequence
$A_{n}\in\mathcal{I}_{\rho}^{+}$ with $A_{n}\setminus
A_{n+1}\in\mathcal{I}_{\rho}$ for each $n\in\omega$ there exists
$F\in\mathcal{F}$ such that $\rho(F)\subseteq^{\rho}A_{n}$ for each
$n\in\omega$;

  3. (3)

_$P^{+}$_ if for every $\subseteq$-decreasing sequence
$A_{n}\in\mathcal{I}_{\rho}^{+}$ there exists $F\in\mathcal{F}$ such that
$\rho(F)\subseteq^{\rho}A_{n}$ for each $n\in\omega$;

  4. (4)

_weak $P^{+}$_ if for every $E\in\mathcal{F}$ there exists $F\in\mathcal{F}$
such that $\rho(F)\subseteq\rho(E)$ and for every sequence
$\\{F_{n}:n\in\omega\\}\subseteq\mathcal{F}$ such that
$\rho(F)\supseteq\rho(F_{n})\supseteq\rho(F_{n+1})$ for each $n\in\omega$
there exists $G\in\mathcal{F}$ such that $\rho(G)\subseteq^{\rho}\rho(F_{n})$
for each $n\in\omega$.

The following result reveals basic properties of the above defined notions and
their connections with P-like properties of ideals.

######  Proposition 6.5.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular with
$\mathcal{F}\subseteq[\Omega]^{\omega}$. Let $\mathcal{I}$ be an ideal on
$\Lambda$.

  1. (1)

$\rho\text{ is }P^{+}\implies\rho\text{ is weak }P^{+}\implies\rho\text{ is
}P^{-}\implies\rho\text{ is }P^{-}(\Lambda)$.

  2. (2)

$\mathcal{I}$ is $P^{+}$ $\iff$ $\rho_{\mathcal{I}}$ is $P^{+}$, for every
ideal $\mathcal{I}$. Similar equivalences hold for $P^{-}$ and
$P^{-}(\Lambda)$, resp.

  3. (3)

The implications from item (1) cannot be reversed.

  4. (4)

If $\mathcal{I}_{\rho}$ is $P^{-}(\Lambda)$ ($P^{-}$, $P^{+}$, resp.), then
$\rho$ is $P^{-}(\Lambda)$ ($P^{-}$, $P^{+}$, resp.).

  5. (5)

The implications from item (4) cannot be reversed in case of $P^{-}(\Lambda)$
and $P^{-}$ properties.

###### Proof.

(1) Below we only show that if $\rho$ is weak $P^{+}$ then it is $P^{-}$ since
other implications are straightforward.

Let $A_{n}\in\mathcal{I}_{\rho}^{+}$ be a $\subseteq$-decreasing sequence with
$A_{n}\setminus A_{n+1}\in\mathcal{I}_{\rho}$ for each $n\in\omega$. Since
$A_{0}\in\mathcal{I}_{\rho}^{+}$, there is $E\in\mathcal{F}$ such that
$\rho(E)\subseteq A_{0}$. Using the fact that $\rho$ is weak $P^{+}$ we can
find $F\in\mathcal{F}$ with $\rho(F)\subseteq\rho(E)$ and such as in the
definition of weak $P^{+}$ property.

We will show that there is a sequence
$\\{F_{n}:n\in\omega\\}\subseteq\mathcal{F}$ such that
$\rho(F_{0})\subseteq\rho(F)$ and $\rho(F_{n+1})\subseteq\rho(F_{n})\cap
A_{n+1}$ for each $n\in\omega$. Indeed, since $\rho(F)\subseteq A_{0}$, it
suffices to put $F_{0}=F$. Suppose now that $F_{i}$ have been constructed for
$i\leq n$. Since $\rho(F_{n})\cap A_{n+1}=\rho(F_{n})\setminus(A_{n}\setminus
A_{n+1})\in\mathcal{I}^{+}_{\rho}$, there is $F_{n+1}\in\mathcal{F}$ with
$\rho(F_{n+1})\subseteq\rho(F_{n})\cap A_{n+1}$.

Since $F$ is as in the definition of weak $P^{+}$ property, there exists
$G\in\mathcal{F}$ such that $\rho(G)\subseteq^{\rho}\rho(F_{n})$ for each
$n\in\omega$. Thus, $\rho(G)\subseteq^{\rho}A_{n}$ for each $n\in\omega$.

(2) Straightforward.

(3) The cases of the second and third implications follow from Proposition
6.1(2) and item (2), where the proof of the fact that $\rho_{\mathcal{I}_{d}}$
is not weak $P^{+}$ is just a slight modification of the proof that
$\mathcal{I}_{d}$ is not $P^{+}$.

Now we show that the first implication cannot be reversed. Consider the ideal
$\mathcal{I}=\\{A\subseteq\omega\times\omega:\text{$A\cap(\\{n\\}\times\omega)$
is finite for every $n\in\omega$}\\}$. Then $\mathcal{I}$ is not $P^{+}$ as
witnessed by $A_{n}=(\omega\setminus n)\times\omega$, so $\rho_{\mathcal{I}}$
is not $P^{+}$ (by item (2)). However, we will show that $\rho_{\mathcal{I}}$
is weak $P^{+}$. Let $E\in\mathcal{I}^{+}$. Then there is $n\in\omega$ such
that $F=E\cap(\\{n\\}\times\omega)$ is infinite. Then $F\in\mathcal{I}^{+}$
and it is easy to see that if $F_{n}\in\mathcal{I}^{+}$ are such that
$F\supseteq F_{n}\supseteq F_{n+1}$ then one can pick $x_{n}\in F_{n}$ for
each $n\in\omega$ and $G=\\{x_{n}:n\in\omega\\}\in\mathcal{I}^{+}$ is such
that $G\setminus\\{x_{i}:i<n\\}\subseteq F_{n}$ for all $n\in\omega$.

(4) Proofs in all cases are very similar, so we only present a proof for the
property $P^{-}(\Lambda)$. Let $A_{n}\in\mathcal{I}_{\rho}^{+}$ be a
$\subseteq$-decreasing sequence with $A_{0}=\Lambda$ and $A_{n}\setminus
A_{n+1}\in\mathcal{I}_{\rho}$ for each $n\in\omega$. Since
$\mathcal{I}_{\rho}$ is $P^{-}(\Lambda)$, there is
$B\not\in\mathcal{I}_{\rho}$ such that for every $n\in\omega$ one can find a
finite set $K_{n}\subseteq\Omega$ such that $B\setminus K_{n}\subseteq A_{n}$.
From the fact that $B\not\in\mathcal{I}_{\rho}$, there is $F\in\mathcal{F}$
such that $\rho(F)\subseteq B$. Using Proposition 3.2, we can find
$E\in\mathcal{F}$ such that $E\subseteq F$ and for every $K_{n}$ there exists
a finite set $L_{n}\subseteq\Omega$ such that $\rho(E\setminus
L_{n})\subseteq\rho(E)\setminus K_{n}$. Then $\rho(E\setminus
L_{n})\subseteq\rho(E)\setminus K_{n}\subseteq B\setminus K_{n}\subseteq
A_{n}$, so $\rho$ is $P^{-}(\Lambda)$.

(5) In Proposition 6.7(3)(4) we will show that $\rho=\operatorname{FS}$ is
weak $P^{+}$, but $\mathcal{I}_{\rho}=\mathcal{H}$ is not $P^{-}(\omega)$. ∎

We will need the following lemma to show that $FS$, $r$ and $\Delta$ are not
$P^{+}$.

######  Lemma 6.6.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ (with
$\mathcal{F}\subseteq[\Omega]^{\omega}$) be a partition regular function such
that there exists a function $\tau:[\Omega]^{<\omega}\to\Lambda$ such that

  1. (1)

$\forall
F\in\mathcal{F}\,\forall\\{a,b\\}\in[F]^{2}\,(\tau\left(\\{a,b\\}\right)\in\rho(F))$,

  2. (2)

$\forall F\in\mathcal{F}\,\forall c\in\rho(F)\,\exists
S\in[F]^{<\omega}\,(\tau(S)=c)$,

  3. (3)

there exists a pairwise disjoint family
$\\{P_{n}:n\in\omega\\}\subseteq\mathcal{F}$ such that the family
$\\{\rho(P_{n}):n\in\omega\\}$ is also pairwise disjoint and the restriction
$\tau\restriction\left[\bigcup\\{P_{n}:n\in\omega\\}\right]^{<\omega}$ is one-
to-one.

Then $\rho$ is not $P^{+}$.

###### Proof.

Let $\\{P_{n}:n\in\omega\\}$ be as in item (3) of the lemma. For each
$n\in\omega$, we define $B_{n}=\bigcup\\{\rho(P_{i}):i\geq n\\}.$ Then
$B_{n}\in\mathcal{I}_{\rho}^{+}$ and $B\supseteq B_{n}\supseteq B_{n+1}$ for
each $n\in\omega$. If we show that there is no $G\in\mathcal{F}$ such that
$\rho(G)\subseteq^{\rho}B_{n}$ for every $n\in\omega$, the proof will be
finished. Suppose for sake of contradiction that there exists
$G\in\mathcal{F}$ such that for every $n\in\omega$ there exists a finite set
$K_{n}\subseteq\Omega$ with $\rho(G\setminus K_{n})\subseteq B_{n}$. We have
two cases:

  1. (1)

$|G\cap P_{n_{0}}|=\omega$ for some $n_{0}\in\omega$,

  2. (2)

$|G\cap P_{n}|<\omega$ for all $n\in\omega$.

_Case (1)._ We take distinct $a,b\in(G\cap P_{n_{0}})\setminus K_{n_{0}+1}$.
Since $a,b\in P_{n_{0}}\in\mathcal{F}$, we have
$\tau\left(\\{a,b\\}\right)\in\rho(P_{n_{0}})$. On the other hand, $a,b\in
G\setminus K_{n_{0}+1}\in\mathcal{F}$, so
$\tau\left(\\{a,b\\}\right)\in\rho(G\setminus K_{n_{0}+1})\subseteq
B_{n_{0}+1}$. Hence, there exists $i\geq n_{0}+1$ such that
$\tau\left(\\{a,b\\}\right)\in\rho(P_{i})$. A contradiction with
$\rho(P_{i})\cap\rho(P_{n_{0}})=\emptyset$.

_Case (2)._ In this case, there exists a strictly increasing sequence
$\\{k_{n}:n\in\omega\\}$ such that we can choose an element $x_{k_{n}}\in
G\cap P_{k_{n}}$ for each $n\in\omega$. Since $x_{k_{n}}$ are pairwise
distinct, there is $N\in\omega$ such that $x_{k_{n}}\in G\setminus K_{0}$ for
every $n\geq N$. In particular,
$\tau\left(\\{x_{k_{N}},x_{k_{N+1}}\\}\right)\in\rho(G\setminus
K_{0})\subseteq B_{0}$, and consequently there exists $i\in\omega$ such that
$\tau\left(\\{x_{k_{N}},x_{k_{N+1}}\\}\right)\in\rho(P_{i})$. Therefore there
is a finite set $S\subseteq P_{i}$ such that
$\tau(S)=\tau\left(\\{x_{k_{N}},x_{k_{N+1}}\\}\right)$. Since $P_{n}$ are
pairwise disjoint and $x_{k_{n}}\in P_{n}$, we obtain that $x_{k_{N}}\notin
P_{i}$ or $h_{k_{N}+1}\notin P_{i}$. Consequently,
$\\{x_{k_{N}},x_{k_{N+1}}\\}\neq S$, so
$\tau\restriction\left[\bigcup\\{P_{n}:n\in\omega\\}\right]^{<\omega}$ is not
one-to-one, a contradiction. ∎

######  Proposition 6.7.

  1. (1)

The ideals $\mathcal{W}$ and $\mathcal{I}_{1/n}$ are $P^{+}$ (hence, $P^{-}$
and $P^{-}(\omega)$) while $\rho_{\mathcal{W}}$ and $\rho_{\mathcal{I}_{1/n}}$
are $P^{+}$, weak $P^{+}$, $P^{-}$ and $P^{-}(\omega)$.

  2. (2)

If $\rho\in\\{\operatorname{FS},r,\Delta\\}$, then $\rho$ is not $P^{+}$,

  3. (3)

If $\rho\in\\{\operatorname{FS},r,\Delta\\}$, then $\rho$ is weak $P^{+}$
(hence $P^{-}$ and $P^{-}(\Lambda)$).

  4. (4)

If $\mathcal{I}\in\\{\mathcal{H},\mathcal{R},\mathcal{D}\\}$, then
$\mathcal{I}$ is not $P^{-}(\Lambda)$ (hence not $P^{+}$ and not $P^{-}$).

###### Proof.

(1) It follows from Theorem 6.2(2) and the fact that $\mathcal{W}$ and
$\mathcal{I}_{1/n}$ are $F_{\sigma}$ ideals (see Proposition 5.2(1)). The
“hence” part follows from Proposition 6.5.

(2) Below we show that $\rho$ is not $P^{+}$ separately for each $\rho$.

_Case of $\rho=FS$._ We define a function $\tau:[\omega]^{<\omega}\to\omega$
by $\tau(S)=\sum_{i\in S}i$. Then we take an infinite sparse set $P$ and a
partition $\\{P_{n}:n\in\omega\\}$ of $P$ into infinite sets. Lemma 6.6 shows
that $\operatorname{FS}$ is not $P^{+}$.

_Case of $\rho=r$._ We define a function
$\tau:[\omega]^{<\omega}\to[\omega]^{2}$ by $\tau(\\{a,b\\})=\\{a,b\\}$ for
distinct $a>b$ and $\tau(S)=\\{0,1\\}$ otherwise. Then we take a partition
$\\{P_{n}:n\in\omega\\}$ of $\omega$ into infinite sets. Lemma 6.6 shows that
$r$ is not $P^{+}$.

_Case of $\rho=\Delta$._ We define a function
$\tau:[\omega]^{<\omega}\to\omega$ by $\tau(\\{a,b\\})=a-b$ for distinct $a>b$
and $\tau(S)=0$ otherwise. Then we take an infinite $\mathcal{D}$-sparse set
$P$ and a partition $\\{P_{n}:n\in\omega\\}$ of $P$ into infinite sets. Lemma
6.6 shows that $\Delta$ is not $P^{+}$.

(3) The “hence” part follows from Proposition 6.5(1). Below we show that
$\rho$ is weak $P^{+}$ separately for each $\rho$.

_Case of $\rho=\operatorname{FS}$._ It is proved in [22, Lemma 2.3] (see also
[18, Example 2.9(2)]).

_Case of $\rho=r$._ For any $E\in[\omega]^{\omega}$ we take $F=E$. Let
$F_{n}\in[\omega]^{\omega}$ be such that
$[F]^{2}\supseteq[F_{n}]^{2}\supseteq[F_{n+1}]^{2}$ for each $n\in\omega$. We
pick $x_{n}\in F_{n}\setminus\\{x_{i}:i<n\\}$ for each $n\in\omega$. Then
$G=\\{x_{n}:n\in\omega\\}\in[\omega]^{\omega}$ and
$[G]^{2}\subseteq^{r}[F_{n}]^{2}$ for each $n\in\omega$.

_Case of $\rho=\Delta$._ Fix any $F\in[\omega]^{\omega}$. Inductively pick a
sequence $(x_{i})_{i\in\omega}\subseteq\omega$ such that $x_{i}\in F$,
$x_{i}<x_{i+1}$ and $x_{i+1}-x_{i}>x_{i}-x_{0}$ for all $i\in\omega$. Let
$E=\\{x_{i}:\ i\in\omega\\}\in[F]^{\omega}$.

Define $a_{i}=x_{i+1}-x_{i}$ for all $i\in\omega$ and observe that
$a_{i}=x_{i+1}-x_{i}>x_{i}-x_{0}=\sum_{j<i}a_{j}$. Put $A=\\{a_{i}:\
i\in\omega\\}$. By [22, proof of Lemma 2.2] the set $A$ is very sparse, i.e.
$A$ is sparse and if $\alpha_{A}(x)\cap\alpha_{A}(y)\neq\emptyset$ then
$x+y\notin\operatorname{FS}(A)$. Note that $\Delta(E)=\\{\sum_{i\in I}a_{i}:\
I\text{ is a finite interval in }\omega\\}\subseteq\operatorname{FS}(A)$.

Observe that if $\Delta(\\{y_{n}:\ n\in\omega\\})\subseteq\Delta(E)$, where
$y_{n}<y_{n+1}$ for all $n\in\omega$, then there is a partition of $\omega$
into finite intervals $(I_{n})_{n\in\omega}$ such that $\max I_{n}<\min
I_{n+1}$ and $y_{n+1}-y_{n}=\sum_{i\in I_{n}}a_{i}$. Indeed, as
$y_{n+1}-y_{n}\in\Delta(\\{y_{n}:\
n\in\omega\\})\subseteq\Delta(E)\subseteq\operatorname{FS}(A)$, for each
$n\in\omega$ there is a finite interval $I_{n}$ such that
$y_{n+1}-y_{n}=\sum_{i\in I_{n}}a_{i}$ (because $A$ is sparse, we get
$I_{n}=\alpha_{A}(y_{n+1}-y_{n})$). We need to show that the intervals $I_{n}$
are pairwise disjoint and cover $\omega$. Suppose first that $\sup
I_{n}+1<\inf I_{n+1}$ for some $n\in\omega$. Then
$y_{n+2}-y_{n}=(y_{n+2}-y_{n+1})+(y_{n+1}-y_{n})=\sum_{i\in I_{n}\cup
I_{n+1}}a_{i}$. On the other hand, $y_{n+2}-y_{n}\in\Delta(\\{y_{n}:\
n\in\omega\\})\subseteq\Delta(E)$, so $y_{n+2}-y_{n}=\sum_{i\in I}a_{i}$ for
some interval $I$. This contradicts uniqueness of $\alpha_{A}(y_{n+2}-y_{n})$
(because $A$ is sparse). Suppose now that $I_{n}\cap I_{n+1}\neq\emptyset$.
Then
$y_{n+2}-y_{n}=(y_{n+2}-y_{n+1})+(y_{n+1}-y_{n})\notin\operatorname{FS}(A)$
(because $A$ is very sparse), which contradicts $\Delta(\\{y_{n}:\
n\in\omega\\})\subseteq\Delta(E)\subseteq FS(A)$.

Fix any sequence $(F_{k})_{k\in\omega}\subseteq[\omega]^{\omega}$ such that
$\Delta(F_{k+1})\subseteq\Delta(F_{k})\subseteq\Delta(E)$ for all
$k\in\omega$. By the previous paragraph, with each $k\in\omega$ we can
associate a partition of $\omega$ into finite intervals $I^{k}_{n}$, i.e.,
$\Delta(F_{k})=\\{\sum_{i\in I}a_{i}:\ I=I^{k}_{j}\cup
I^{k}_{j+1}\cup\ldots\cup I^{k}_{j^{\prime}}\text{ for some }j<j^{\prime}\\}$.

Observe that actually for each $n,k\in\omega$ we have that
$I^{k+1}_{n}=\bigcup_{i\in I}I^{k}_{i}$ for some interval $I$. Indeed,
otherwise for some $n,k\in\omega$ we would have $x=\sum_{i\in
I^{k+1}_{n}}a_{i}\in\Delta(F_{k+1})\subseteq\Delta(F_{k})$, so $x=\sum_{i\in
I}a_{i}$ for some $I=I^{k}_{j}\cup I^{k}_{j+1}\cup\ldots\cup
I^{k}_{j^{\prime}}$, which contradicts that $A$ is sparse.

Inductively pick a sequence $(n_{k})_{k\in\omega}\subseteq\omega$ such that
for each $k\in\omega$ we have $n_{k+1}>n_{k}$ (so also
$a_{n_{k+1}}>a_{n_{k}}$) and $n_{k}=\min I^{k}_{j}$ for some $j\in\omega$.
Define $E^{\prime}=\\{x_{n_{k}}:\ k\in\omega\\}$. Notice that
$x_{n_{k+1}}-x_{n_{k}}=\sum_{i\in[n_{k},n_{k+1})}a_{i}$. Then for each
$k\in\omega$ we have
$\Delta(E^{\prime}\setminus[0,x_{n_{k}}))=\Delta(\\{x_{n_{i}}:\ i\geq
k\\})\subseteq\\{\sum_{i\in I}a_{i}:\ I=I^{k}_{j}\cup
I^{k}_{j+1}\cup\ldots\cup I^{k}_{j^{\prime}}\text{ for some
}j<j^{\prime}\\}=\Delta(F_{k})$.

(4) The “hence” part follows from Proposition 6.1. Below we show that
$\mathcal{I}$ is not $P^{-}(\Lambda)$ separately for each $\mathcal{I}$.

_Case of $\mathcal{I}=\mathcal{H}$._ Let $A_{k}=\\{2^{k}(2n+1):n\in\omega\\}$
for each $k\in\omega$. In [22, item (2) in the proof of Proposition 1.1], the
authors showed that $A_{k}\in\mathcal{H}$ for every $k\in\omega$, whereas in
[22, item (1) in the proof of Proposition 1.1] it is shown that for every
$B\notin\mathcal{H}$ there is $k\in\omega$ such that $B\cap A_{k}$ is
infinite. Thus, the family $\\{A_{k}:k\in\omega\\}$ witnesses the fact that
$\mathcal{H}$ is not $P^{-}(\omega)$.

_Case of $\mathcal{I}=\mathcal{R}$._ Let $A_{n}=\\{\\{k,i\\}:i>k\geq n\\}$ for
every $n\in\omega$. Then $A_{n}\notin\mathcal{R}$, $A_{0}=[\omega]^{2}$ and
$A_{n}\setminus A_{n+1}=\\{\\{n,i\\}:i>n\\}\in\mathcal{R}$. Suppose, for sake
of contradiction, that there is $B\notin\mathcal{R}$ such that
$B\subseteq^{*}A_{n}$ for every $n\in\omega$. Let $H=\\{h_{n}:n\in\omega\\}$
be an infinite set such that $[H]^{2}\subseteq B$ and $h_{n}<h_{n+1}$ for
every $n\in\omega$. Since $[H]^{2}\subseteq^{*}A_{h_{1}}$, there is a finite
set $F$ such that $[H]^{2}\setminus F\subseteq A_{h_{1}}$. Since $F$ is
finite, there is $k>0$ such that $\\{h_{0},h_{n}\\}\notin F$ for every $n\geq
k$. Then $\\{\\{h_{0},h_{n}\\}:n\geq k\\}\subseteq[H]^{2}\setminus F$ and
$\\{\\{h_{0},h_{n}\\}:n\geq k\\}\cap A_{h_{1}}=\emptyset$, a contradiction.

_Case of $\mathcal{I}=\mathcal{D}$._ Let $A_{k}=\\{2^{k}(2n+1):n\in\omega\\}$
for each $k\in\omega$. In [58, item (2) in the proof of Theorem 2.1], the
author showed that $A_{k}\in\mathcal{D}$ for every $k\in\omega$, whereas in
[58, item (1) in the proof of Theorem 2.1] it is shown that for every
$B\notin\mathcal{D}$ there is $k\in\omega$ such that $B\cap A_{k}$ is
infinite. Thus, the family $\\{A_{k}:k\in\omega\\}$ witnesses the fact that
$\mathcal{D}$ is not $P^{-}(\omega)$. ∎

The following easy observation will be useful in our considerations.

######  Proposition 6.8.

If $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ is partition regular with
$\mathcal{F}\subseteq[\Omega]^{\omega}$, then the following conditions are
equivalent.

  1. (1)

$\rho$ is $P^{-}$ ($P^{-}(\Lambda)$, resp.).

  2. (2)

For every countable family $\mathcal{B}\subseteq\mathcal{I}_{\rho}$ with
$\bigcup\mathcal{B}\notin\mathcal{I}_{\rho}$ ($\bigcup\mathcal{B}=\Lambda$,
resp.) there exists $F\in\mathcal{F}$ such that
$\rho(F)\subseteq\bigcup\mathcal{B}$ and for every finite subfamily
$\mathcal{C}\subseteq\mathcal{B}$ there is a finite $K\subseteq\Omega$ such
that $\rho(F\setminus K)\cap\bigcup\mathcal{C}=\emptyset$.

###### Proof.

We will assume that $\rho$ is $P^{-}$, as the proof in the case of
$P^{-}(\Lambda)$ is similar.

$(1)\implies(2)$. Let $\mathcal{B}=\\{B_{n}:n\in\omega\\}$, where
$\bigcup\mathcal{B}\notin\mathcal{I}_{\rho}$ and $B_{n}\in\mathcal{I}_{\rho}$
for every $n\in\omega$. For each $n\in\omega$, we define
$A_{n}=\bigcup\mathcal{B}\setminus\bigcup\\{B_{i}:i<n\\}$. Since $\rho$ is
$P^{-}$, there exists $F\in\mathcal{F}$ such that
$\rho(F)\subseteq^{\rho}A_{n}$ for each $n\in\omega$. Let
$\mathcal{C}\subseteq\mathcal{B}$ be a finite subfamily. Let $n\in\omega$ be
such that $\mathcal{C}\subseteq\\{B_{i}:i<n\\}$. Then
$\bigcup\mathcal{C}\subseteq\bigcup\\{B_{i}:i<n\\}$. Let $K\subseteq\Omega$ be
a finite set such that $\rho(F\setminus K)\subseteq A_{n}$. Then
$\rho(F\setminus K)\cap\bigcup\\{B_{i}:i<n\\}=\emptyset$, so $\rho(F\setminus
K)\cap\bigcup\mathcal{C}=\emptyset$.

$(2)\implies(1)$. Let $A_{n}\in\mathcal{I}_{\rho}^{+}$ be such that
$A_{n}\supseteq A_{n+1}$ and $A_{n}\setminus A_{n+1}\in\mathcal{I}_{\rho}$ for
each $n\in\omega$. For each $n\in\omega$ we define $B_{n}=A_{n}\setminus
A_{n+1}$. Let $\mathcal{B}=\\{B_{n}:n\in\omega\\}$. Then there exists
$F\in\mathcal{F}$ such that $\rho(F)\subseteq\bigcup\mathcal{B}$ and for every
finite subfamily $\mathcal{C}\subseteq\mathcal{B}$ there is a finite
$K\subseteq\Omega$ such that $\rho(F\setminus
K)\cap\bigcup\mathcal{C}=\emptyset$. Thus for any $n\in\omega$, we find a
finite set $K\subseteq\Omega$ such that $\rho(F\setminus
K)\cap\bigcup\\{B_{i}:i<n\\}=\emptyset$. Hence $\rho(F\setminus
K)\subseteq\bigcup\\{B_{i}:i\geq n\\}=A_{n}$, so
$\rho(F)\subseteq^{\rho}A_{n}$. ∎

###  7\. Katětov order

####  7.1. Katětov order between ideals

We say that an _ideal $\mathcal{I}_{1}$ on $\Lambda_{1}$ is above an ideal
$\mathcal{I}_{2}$ on $\Lambda_{2}$ in the Katětov order_ (in short:
$\mathcal{I}_{2}\leq_{K}\mathcal{I}_{1}$) [53] if there exists a function
$\phi:\Lambda_{1}\to\Lambda_{2}$ such that $\phi[A]\notin\mathcal{I}_{2}$ for
each $A\notin\mathcal{I}_{1}$. If $\Lambda_{1}=\Lambda_{2}$ and
$\mathcal{I}_{2}\subseteq\mathcal{I}_{1}$, then obviously the identity
function on $\Lambda_{1}$ witnesses that
$\mathcal{I}_{2}\leq_{K}\mathcal{I}_{1}$.

There are known relationships between Katětov order, P-like properties and
topological complexity.

######  Proposition 7.1 ([43, Theorem 3.8]).

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

  1. (1)

$\mathcal{I}$ is $P^{-}(\Lambda)$ $\iff$
$\mathrm{Fin}^{2}\not\leq_{K}\mathcal{I}$.

  2. (2)

$\mathcal{I}$ is $P^{-}$ $\iff$
$\mathrm{Fin}^{2}\not\leq_{K}\mathcal{I}\restriction A$ for every
$A\in\mathcal{I}^{+}$.

######  Proposition 7.2.

  1. (1)

$\mathrm{Fin}^{2}\leq_{K}\mathcal{I}$ for
$\mathcal{I}\in\\{\mathcal{D},\mathcal{H},\mathcal{R}\\}$.

  2. (2)

If $\mathcal{I}$ is an $G_{\delta\sigma\delta}$ ideal, then
$\mathrm{Fin}^{2}\not\leq_{K}\mathcal{I}\restriction A$ for every
$A\in\mathcal{I}^{+}$ In particular, $\mathrm{Fin}^{2}\not\leq_{K}\mathcal{W}$
and $\mathrm{Fin}^{2}\not\leq_{K}\mathcal{I}_{1/n}$.

###### Proof.

(1) Using Proposition 7.1(1), we need to show that $\mathcal{D}$,
$\mathcal{H}$ and $\mathcal{R}$ are not $P^{-}(\Lambda)$ ideals, but this
follows from Proposition 6.7(4). (For $\mathcal{I}=\mathcal{R}$, this item was
earlier proved by Meza-Alcántara [65, Lemma 1.6.25].)

(2) It follows from Theorem 6.2(1) and Propositions 7.1(2) and 6.7(1). ∎

####  7.2. Katětov order between partition regular operations

The following notion will be crucial for showing when a class of sequentially
compact spaces defined by $\rho_{1}$ is contained in a class of sequentially
compact spaces defined by $\rho_{2}$.

######  Definition 7.3.

Let $\rho_{i}:\mathcal{F}_{i}\to[\Lambda_{i}]^{\omega}$ be partition regular
(with $\mathcal{F}_{i}\subseteq[\Omega_{i}]^{\omega}$) for each $i=1,2$. We
say that _$\rho_{1}$ is above $\rho_{2}$ in the Katětov order_ (in short:
$\rho_{2}\leq_{K}\rho_{1}$) if there is a function
$\phi:\Lambda_{1}\to\Lambda_{2}$ such that

$\forall F_{1}\in\mathcal{F}_{1}\,\exists F_{2}\in\mathcal{F}_{2}\,\forall
K_{1}\in[\Omega_{1}]^{<\omega}\,\exists
K_{2}\in[\Omega_{2}]^{<\omega}\,(\rho_{2}(F_{2}\setminus
K_{2})\subseteq\phi[\rho_{1}(F_{1}\setminus K_{1})])$

or equivalently:

$\forall F_{1}\in\mathcal{F}_{1}\,\exists F_{2}\in\mathcal{F}_{2}\,\forall
K_{1}\in[\Omega_{1}]^{<\omega}\,(\rho_{2}(F_{2})\subseteq^{\rho_{2}}\phi[\rho_{1}(F_{1}\setminus
K_{1})]).$

The following proposition reveals some basic properties of this new order on
partition regular functions.

######  Proposition 7.4.

  1. (1)

The relation $\leq_{K}$ is a preorder (a.k.a. quasi order) i.e. it is
reflexive and transitive.

  2. (2)

The preorder $\leq_{K}$ is upward and downward directed.

  3. (3)

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ (with
$\mathcal{F}\subseteq[\Omega]^{\omega}$) be partition regular.

    1. (a)

$\rho\leq_{K}\rho\restriction\rho(F)$ for every $F\in\mathcal{F}$.

    2. (b)

$\rho_{\mathrm{Fin}(\Lambda)}\leq_{K}\rho$.

###### Proof.

(1) Reflexivity of $\leq_{K}$ is obvious. To show transitivity, fix
$\mathcal{F}_{i}\subseteq[\Omega_{i}]^{\omega}$ and
$\rho_{i}:\mathcal{F}_{i}\to[\Lambda_{i}]^{\omega}$, $i=1,2,3$, and suppose
that $\rho_{1}\leq_{K}\rho_{2}$ is witnessed by $f$ and
$\rho_{2}\leq_{K}\rho_{3}$ is witnessed by $g$. We claim that
$\rho_{1}\leq_{K}\rho_{3}$ is witnessed by $h:\Lambda_{3}\to\Lambda_{1}$ given
by $h(x)=f(g(x))$ for all $x\in\Lambda_{3}$. Let $F_{3}\in\mathcal{F}_{3}$.
Then we can find $F_{2}\in\mathcal{F}_{2}$ such that for every
$K\in[\Omega_{3}]^{<\omega}$ there exists $L_{K}\in[\Omega_{2}]^{<\omega}$
such that $\rho_{2}(F_{2}\setminus L_{K})\subseteq
g\left[\rho_{3}(F_{3}\setminus K)\right].$ Then for $F_{2}$ we can find
$F_{1}\in\mathcal{F}_{1}$ such that for every $L\in[\Omega_{2}]^{<\omega}$
there exists $M_{L}\in[\Omega_{1}]^{<\omega}$ such that
$\rho_{1}(F_{1}\setminus M_{L})\subseteq f\left[\rho_{2}(F_{2}\setminus
L)\right].$ Now for a given $K\in[\Omega_{3}]^{<\omega}$ we have
$\rho_{1}(F_{1}\setminus M_{L_{K}})\subseteq f\left[\rho_{2}(F_{2}\setminus
L_{K})\right]\subseteq f\left[g\left[\rho_{3}(F_{3}\setminus
K)\right]\right]=h\left[\rho_{3}(F_{3}\setminus K)\right],$ so the proof is
finished.

(2) Let $\rho_{i}:\mathcal{F}_{i}\to[\Lambda_{i}]^{\omega}$ with
$\mathcal{F}_{i}\subseteq[\Omega_{i}]^{\omega}$ be partition regular for
$i=0,1$. We define the following partition regular functions
$\pi:\\{F_{0}\times
F_{1}:F_{0}\in\mathcal{F}_{0},F_{1}\in\mathcal{F}_{1}\\}\to[\Lambda_{0}\times\Lambda_{1}]^{\omega}$
by $\pi(F_{0}\times F_{1})=\rho_{0}(F_{0})\times\rho_{1}(F_{1})$ and
$\sigma:\mathcal{F}_{0}\oplus\mathcal{F}_{1}\to[\Lambda_{0}\oplus\Lambda_{1}]^{\omega}$
by
$\sigma((F_{0}\times\\{0\\})\cup(F_{1}\times\\{1\\}))=(\rho_{0}(F_{0})\times\\{0\\})\cup(\rho_{1}(F_{1})\times\\{1\\}).$

Then $\sigma\leq_{K}\rho_{i}$ ($i=0,1$) is witnessed by a function
$\phi_{i}:\Lambda_{i}\to\Lambda_{0}\oplus\Lambda_{1}$ given by
$\phi_{i}(x)=(x,i)$, whereas $\rho_{i}\leq_{K}\pi$ ($i=0,1$) is witness by a
function $\psi_{i}:\Lambda_{0}\times\Lambda_{1}\to\Lambda_{i}$ given by
$\psi_{i}(x_{0},x_{1})=x_{i}$.

(3a) Let $F\in\mathcal{F}$. We claim that $\phi:\rho(F)\to\Lambda$ given by
$\phi(\lambda)=\lambda$ is a witness for
$\rho\leq_{K}\rho\restriction\rho(F)$. Let
$F_{1}\in\mathcal{F}\restriction\rho(F)$. Then $F_{2}=F_{1}$ is such that for
every finite set $K_{1}\subseteq\Omega$ we take $K_{2}=K_{1}$ and see that
$\rho(F_{2}\setminus K_{2})\subseteq\phi(\rho(F_{1}\setminus K_{1}))$.

(3b) We claim that $\phi:\Lambda\to\Lambda$ given by $\phi(\lambda)=\lambda$
is a witness for $\rho_{\mathrm{Fin}(\Lambda)}\leq_{K}\rho$. Let
$F\in\mathcal{F}$. Let $\Omega=\\{o_{n}:n\in\omega\\}$. Since
$\rho(F\setminus\\{o_{i}:i<n\\})$ is infinite for every $n\in\omega$, we can
pick a one-to-one sequence $(a_{n}:n\in\omega)$ such that
$a_{n}\in\rho(F\setminus\\{o_{i}:i<n\\})$ for every $n\in\omega$. Then
$A=\\{a_{n}:n\in\omega\\}\in\mathrm{Fin}(\Lambda)^{+}$ is an infinite set. For
a finite set $K\subseteq\Omega$ there is $n\in\omega$ such that
$K\subseteq\\{o_{i}:i<n\\}$. Then $L=\\{a_{i}:i<n\\}$ is finite subset of
$\Lambda$ and $A\setminus
L\subseteq\rho(F\setminus\\{o_{i}:i<n\\})\subseteq\rho(F\setminus K)$. ∎

Now we compare the relation $\leq_{K}$ between partition regular operations
with the relation $\leq_{K}$ between ideals.

######  Proposition 7.5.

Let $\rho_{i}:\mathcal{F}_{i}\to[\Lambda_{i}]^{\omega}$ for each $i=1,2$ and
$\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular. Let
$\mathcal{I}$ be ideal.

  1. (1)

$\rho_{2}\leq_{K}\rho_{1}\implies\mathcal{I}_{\rho_{2}}\leq_{K}\mathcal{I}_{\rho_{1}}$
with the same witnessing function.

  2. (2)

    1. (a)

If $\rho_{2}$ is $P^{+}$ (in particular, if $\rho_{2}=\rho_{\mathcal{I}}$ and
$\mathcal{I}$ is $P^{+}$), then
$\rho_{2}\leq_{K}\rho_{1}\iff\mathcal{I}_{\rho_{2}}\leq_{K}\mathcal{I}_{\rho_{1}}$;

    2. (b)

$\rho\leq_{K}\rho_{\mathcal{I}}\iff\mathcal{I}_{\rho}\leq_{K}\mathcal{I}$;

###### Proof.

(1) Let $\mathcal{F}_{i}\subseteq[\Omega_{i}]^{\omega}$ for $i=1,2$. Let
$\phi$ be a witness for $\rho_{2}\leq_{K}\rho_{1}$. We claim that $\phi$ is
also a witness for $\mathcal{I}_{\rho_{2}}\leq_{K}\mathcal{I}_{\rho_{1}}$. Let
$A\notin\mathcal{I}_{\rho_{1}}$. Then there is $F_{1}\in\mathcal{F}_{1}$ with
$\rho_{1}(F_{1})\subseteq A$. Since $\rho_{2}\leq_{K}\rho_{1}$, there is
$F_{2}\in\mathcal{F}_{2}$ and a finite set $K_{2}\subseteq\Omega_{2}$ such
that $\rho_{2}(F_{2}\setminus
K_{2})\subseteq\phi[\rho_{1}(F_{1}\setminus\emptyset)]=\phi[\rho_{1}(F_{1})]$.
Since $F_{2}\setminus K_{2}\in\mathcal{F}_{2}$ and $\rho_{2}(F_{2}\setminus
K_{2})\subseteq\phi[A]$, we obtain that $\phi[A]\notin\mathcal{I}_{\rho_{2}}$.
Thus the proof of this item is finished.

(2a) The “in particular” part follows from Propositions 3.3(2) and 6.5(2).

We only have to show the implication “$\impliedby$”, because the reversed
implication is true by item (1). Let $\phi:\Lambda_{1}\to\Lambda_{2}$ be a
witness for $\mathcal{I}_{\rho_{2}}\leq_{K}\mathcal{I}_{\rho_{1}}$. We claim
that $\phi$ is also a witness for $\rho_{2}\leq_{K}\rho_{1}$. Let
$F_{1}\in\mathcal{F}_{1}$, $\Omega_{1}=\\{o_{n}:n\in\omega\\}$ and
$B_{n}=\phi[\rho_{1}(F_{1}\setminus\\{o_{i}:i<n\\})]$ for each $n\in\omega$.
Then $B_{n}\notin\mathcal{I}_{\rho_{2}}$, $B_{n}\supseteq B_{n+1}$ for each
$n\in\omega$, and since $\mathcal{I}_{\rho_{2}}$ is $P^{+}$, there is
$F_{2}\in\mathcal{F}_{2}$ such that for each $n\in\omega$ there is a finite
set $L_{n}\subseteq\Omega_{2}$ with $\rho_{2}(F_{2}\setminus L_{n})\subseteq
B_{n}$. Now, for any finite set $K_{1}\subseteq\Omega_{1}$ there is
$n\in\omega$ such that $K_{1}\subseteq\\{o_{i}:i<n\\}$. Let $K_{2}=L_{n}$.
Then $\rho_{2}(F_{2}\setminus K_{2})\subseteq
B_{n}\subseteq\phi[\rho_{1}(F_{1}\setminus K_{1})]$. Thus the proof of this
item is finished.

(2b) The implication “$\implies$” follows from item (1) and Proposition
3.3(2), so below we show the reverse implication.

Suppose that $\mathcal{I}$ is an ideal on $\Lambda$ and
$\mathcal{F}\subseteq[\Omega]^{\omega}$. Let $\phi:\Lambda\to\Lambda$ be a
witness of $\mathcal{I}_{\rho}\leq_{K}\mathcal{I}$. We claim that the same
$\phi$ is also a witness for $\rho\leq_{K}\rho_{\mathcal{I}}$. Indeed, for
$A\notin\mathcal{I}$ we find $E\in\mathcal{F}$ such that
$\rho(E)\subseteq\phi[A]$. Using Proposition 3.2, we can find a set
$F\in\mathcal{F}$ such that $F\subseteq E$ and for any finite set
$K\subseteq\Lambda_{1}$ there exists a finite set $L\subseteq\Omega$ with
$\rho(F\setminus L)\subseteq\rho(F)\setminus\phi[K]$. Consequently,
$\rho(F\setminus L)\subseteq\phi[A]\setminus\phi[K]\subseteq\phi[A\setminus
K]$. ∎

The following example shows that in general $\rho_{2}\leq_{K}\rho_{1}$ and
$\mathcal{I}_{\rho_{2}}\leq_{K}\mathcal{I}_{\rho_{1}}$ are not equivalent.

######  Example 7.6.

$\mathrm{Fin}^{2}\leq_{K}\mathcal{H}$, but
$\rho_{\mathrm{Fin}^{2}}\not\leq_{K}\operatorname{FS}$.

###### Proof.

By Proposition 7.2(1) we know that
$\mathcal{I}_{\rho_{\mathrm{Fin}^{2}}}=\mathrm{Fin}^{2}\leq_{K}\mathcal{H}=\mathcal{I}_{\operatorname{FS}}$.
Thus, we only need to show that
$\rho_{\mathrm{Fin}^{2}}\not\leq_{K}\operatorname{FS}$.

Suppose that $\rho_{\mathrm{Fin}^{2}}\leq_{K}\operatorname{FS}$ and let
$\phi:\omega\to\omega^{2}$ be a witness for this. For each $n\in\omega$, we
define $A_{n}=\phi^{-1}[(\omega\setminus n)\times\omega]$. Then
$A_{0}=\omega$, $A_{n}\supseteq A_{n+1}$ and $A_{n}\setminus
A_{n+1}\subseteq\phi^{-1}[\\{n\\}\times\omega]\in\mathcal{H}$ for each
$n\in\omega$ by Proposition 7.5(1). Since $\operatorname{FS}$ is
$P^{-}(\omega)$ by Proposition 6.7(3), there is $F\in[\omega]^{\omega}$ such
that for every $n\in\omega$ there is a finite set $K_{n}\subseteq\omega$ with
$\operatorname{FS}(F\setminus K_{n})\subseteq A_{n}$. Now, using the fact that
$\rho_{\mathrm{Fin}^{2}}\leq_{K}\operatorname{FS}$, we find
$B\notin\mathrm{Fin}^{2}$ such that for every $n\in\omega$ there is a finite
set $L_{n}\subseteq\omega^{2}$ with $B\setminus
L_{n}\subseteq\phi[\operatorname{FS}(F\setminus
K_{n})]\subseteq\phi[A_{n}]\subseteq(\omega\setminus n)\times\omega$. In
particular, sets $B\cap(\\{n\\}\times\omega)$ are finite for every $n$, so
$B\in\mathrm{Fin}^{2}$, a contradiction. ∎

######  Remark.

The partition regular function $\rho_{\mathrm{Fin}^{2}}$ from Example 7.6 is
not $P^{-}$. In Example 15.5 we will show that there are partition regular
functions $\rho_{1}$ and $\rho_{2}$ which are $P^{-}$ and have small
accretions such that $\mathcal{I}_{\rho_{2}}\subseteq\mathcal{I}_{\rho_{1}}$
(in particular, $\mathcal{I}_{\rho_{2}}\leq_{K}\mathcal{I}_{\rho_{1}}$), but
$\rho_{2}\not\leq_{K}\rho_{1}$.

####  7.3. Katětov order between $\operatorname{FS},r,\Delta,\mathcal{W}\text{
and }\mathcal{I}_{1/n}$

######  Theorem 7.7.

  1. (1)

$\mathcal{H}\not\leq_{K}\mathcal{R}$. In particular,
$\operatorname{FS}\not\leq_{K}r$.

  2. (2)

$\mathcal{R}\not\leq_{K}\mathcal{H}$. In particular,
$r\not\leq_{K}\operatorname{FS}$.

  3. (3)

$\Delta\leq_{K}\operatorname{FS}$ and $\mathcal{D}\subseteq\mathcal{H}$. In
particular, $\mathcal{D}\leq_{K}\mathcal{H}$.

  4. (4)

$\Delta\leq_{K}r$. In particular, $\mathcal{D}\leq_{K}\mathcal{R}$.

  5. (5)

$\mathcal{R}\not\leq_{K}\mathcal{D}$. In particular, $r\not\leq_{K}\Delta$.

  6. (6)

$\mathcal{H}\not\leq_{K}\mathcal{D}$. In particular,
$\operatorname{FS}\not\leq_{K}\Delta$.

  7. (7)

$\mathcal{I}_{1/n}\not\leq_{K}\mathcal{R}$. In particular,
$\rho_{\mathcal{I}_{1/n}}\not\leq_{K}r$.

  8. (8)

$\mathcal{I}_{1/n}\not\leq_{K}\mathcal{H}$. In particular,
$\rho_{\mathcal{I}_{1/n}}\not\leq_{K}\operatorname{FS}$.

  9. (9)

$\mathcal{I}_{1/n}\not\leq_{K}\mathcal{D}$. In particular,
$\rho_{\mathcal{I}_{1/n}}\not\leq_{K}\Delta$.

  10. (10)

$\mathcal{I}_{1/n}\not\leq_{K}\mathcal{W}$. In particular,
$\rho_{\mathcal{I}_{1/n}}\not\leq_{K}\rho_{\mathcal{W}}$.

  11. (11)

$\mathcal{D}\not\leq_{K}\mathcal{I}_{1/n}$. In particular,
$\Delta\not\leq_{K}\rho_{\mathcal{I}_{1/n}}$.

  12. (12)

$\mathcal{H}\not\leq_{K}\mathcal{I}_{1/n}$. In particular,
$\operatorname{FS}\not\leq_{K}\rho_{\mathcal{I}_{1/n}}$.

  13. (13)

$\mathcal{R}\not\leq_{K}\mathcal{I}_{1/n}$. In particular,
$r\not\leq_{K}\rho_{\mathcal{I}_{1/n}}$.

  14. (14)

$\mathcal{D}\not\leq_{K}\mathcal{W}$. In particular,
$\Delta\not\leq_{K}\rho_{\mathcal{W}}$.

  15. (15)

$\mathcal{H}\not\leq_{K}\mathcal{W}$. In particular,
$\operatorname{FS}\not\leq_{K}\rho_{\mathcal{W}}$.

  16. (16)

$\mathcal{R}\not\leq_{K}\mathcal{W}$. In particular,
$r\not\leq_{K}\rho_{\mathcal{W}}$.

###### Proof.

The “in particular” parts follow from Proposition 7.5(1).

The proofs of items (1), (2), (7), (8) and (10) can be found in [21].

(3) The inclusion is proved in [71, Proposition 4.2.1] (see also [20,
Propositions 4.2]). Below, we show that $\Delta\leq_{K}\operatorname{FS}$.

We claim that the identity function $\phi:\omega\to\omega$, $\phi(n)=n$ for
every $n\in\omega$ is a witness for $\Delta\leq_{K}\operatorname{FS}$.

For any infinite set $A\subseteq\omega$, we define an infinite set
$B=\\{\sum_{i\leq n}a_{i}:n\in\omega\\}$, where $\\{a_{n}:n\in\omega\\}$ is
the increasing enumeration of $A$. Next, for any finite set $K$, we define a
finite set $L=\\{0,1,\dots,\sum_{i\leq k}a_{i}\\}$, where
$k=\max\\{i\in\omega:a_{i}\in K\\}$ (for $K=\emptyset$ we take $k=0$).
Finally, we observe that $\Delta(B\setminus
L)\subseteq\operatorname{FS}(A\setminus K)=\phi[\operatorname{FS}(A\setminus
K)]$, so the proof is finished.

(4) We claim that $\phi:[\omega]^{2}\to\omega$ given by the formula
$\phi(\\{n,k\\})=n-k$, where $n>k$, is a witness for $\Delta\leq_{K}r$. For
any infinite set $A\subseteq\omega$, we take $B=A$. Then for any finite set
$K\subseteq\omega$, we take $L=K$. Next, we notice that $\Delta(B\setminus
L)=\Delta(A\setminus K)=\phi[[A\setminus K]^{2}]=\phi[r(A\setminus K)]$, so
the proof is finished.

(5) It follows from items (3) and (2).

(6) It follows from items (4) and (1).

(9) It follows from items (8) and (3).

(11) Suppose otherwise: $\mathcal{D}\leq_{K}\mathcal{I}_{1/n}$. By Proposition
7.2(1) $\mathrm{Fin}^{2}\leq_{K}\mathcal{D}$, so
$\mathrm{Fin}^{2}\leq_{K}\mathcal{I}_{1/n}$. By Proposition 7.1(1), we obtain
that $\mathcal{I}_{1/n}$ is not a $P^{-}(\omega)$ ideal, a contradiction with
Proposition 6.7(1).

(12) It follows from items (3) and (11).

(13) It follows from items (4) and (11).

(14) Suppose otherwise: $\mathcal{D}\leq_{K}\mathcal{W}$. Using Proposition
7.2(1) we get that $\mathrm{Fin}^{2}\leq_{K}\mathcal{D}$, so
$\mathrm{Fin}^{2}\leq_{K}\mathcal{W}$. However, since $\mathcal{W}$ is
$F_{\sigma}$ (see [23, Example 4.12]),
$\mathrm{Fin}^{2}\not\leq_{K}\mathcal{W}$ by [13, Theorems 7.5 and 9.1]. A
contradiction.

(15) It follows from items (3) and (14).

(16) It follows from items (3) and (14).

∎

######  Question 7.8.

Is $\mathcal{W}\leq_{K}\mathcal{I}$ for
$\mathcal{I}\in\\{\mathcal{I}_{1/n},\mathcal{H},\mathcal{R},\mathcal{D}\\}$?

######  Remark.

The positive answer to Question 7.8 for $\mathcal{I}=\mathcal{I}_{1/n}$ is
implied by the inclusion $\mathcal{W}\subseteq\mathcal{I}_{1/n}$ that is known
as the Erdős conjecture on arithmetic progressions (a.k.a. the Erdős–Turán
conjecture) which can be rephrased in the following manner: if the sum of the
reciprocals of the elements of a set $A\subseteq\omega$ diverges, then $A$
contains arbitrarily long finite arithmetic progressions.

###  8\. Tallness and homogeneity

####  8.1. Tallness of partition regular functions

An ideal $\mathcal{I}$ on $\Lambda$ is _tall_ if for every infinite set
$A\subseteq\Lambda$ there exists an infinite set $B\subseteq A$ such that
$B\in\mathcal{I}$ ([62, p. 210], see also [63, Definition 0.6]). It is not
difficult to see that $\mathcal{I}$ is not tall $\iff$
$\mathcal{I}\leq_{K}\mathcal{J}$ for every ideal $\mathcal{J}$ $\iff$
$\mathcal{I}\leq_{K}\mathrm{Fin}$ $\iff$ $\mathcal{I}\restriction
A=\mathrm{Fin}(A)$ for some $A\in\mathcal{I}^{+}$.

The following proposition characterizes tallness of the ideal
$\mathcal{I}_{\rho}$ in terms of $\rho$ and serves as a definition of tallness
of partition regular functions.

######  Proposition 8.1.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular (with
$\mathcal{F}\subseteq[\Omega]^{\omega}$). The following conditions are
equivalent.

  1. (1)

$\mathcal{I}_{\rho}$ is tall.

  2. (2)

There exists a partition regular function $\tau$ such that
$\rho\not\leq_{K}\tau$.

  3. (3)

$\mathcal{I}_{\rho}\restriction\rho(F)\neq\mathrm{Fin}(\rho(F))$ for every
$F\in\mathcal{F}$.

  4. (4)

$\rho\not\leq_{K}\rho_{\mathrm{Fin}(\Lambda)}$.

###### Proof.

(1)$\implies$(2) If $\mathcal{I}_{\rho}$ is tall, there is an ideal
$\mathcal{J}$ such that $\mathcal{I}_{\rho}\not\leq_{K}\mathcal{J}$. Then
$\rho\not\leq_{K}\rho_{\mathcal{J}}$ by Proposition 7.5(1).

(2)$\implies$(3) Suppose that there is $F\in\mathcal{F}$ such that
$\mathcal{I}_{\rho}\restriction\rho(F)=\mathrm{Fin}(\rho(F))$. We will show
that $\rho\leq_{K}\tau$ for every partition regular function $\tau$ .

Take any partition regular function $\tau:\mathcal{G}\to[\Sigma]^{\omega}$
with $\mathcal{G}\subseteq[\Gamma]^{\omega}$.

Let $\phi:\Sigma\to\Lambda$ be a one-to-one function such that
$\phi[\Sigma]=\rho(F)$. We claim that $\phi$ is a witness for
$\rho\leq_{K}\tau$.

Let $G\in\mathcal{G}$ and $\Gamma=\\{\gamma_{n}:n\in\omega\\}$. Since
$\phi[\tau(G\setminus\\{\gamma_{i}:i<n\\})]$ is infinite for every
$n\in\omega$, we can pick a one-to-one sequence $(b_{n}:n\in\omega)$ such that
$b_{n}\in\phi[\tau(G\setminus\\{\gamma_{i}:i<n\\})]$ for each $n\in\omega$.
Define $B=\\{b_{n}:n\in\omega\\}$. Since $B$ is infinite, $B\subseteq\rho(F)$
and $\mathcal{I}_{\rho}\restriction\rho(F)=\mathrm{Fin}(\rho(F))$, there is
$H\in\mathcal{F}$ such that $\rho(H)\subseteq B$. Using Proposition 3.2, there
is $E\in\mathcal{F}$ with $E\subseteq H$ such that for any $n\in\omega$ there
is a finite set $L\subseteq\Omega$ such that $\rho(E\setminus
L)\subseteq\rho(E)\setminus\\{b_{i}:i<n\\}$. Consequently, for any finite set
$K\subseteq\Gamma$ there is $n\in\omega$ such that
$K\subseteq\\{\gamma_{i}:i<n\\}$, so we can find a finite set
$L\subseteq\Omega$ such that $\rho(E\setminus
L)\subseteq\rho(E)\setminus\\{b_{i}:i<n\\}\subseteq
B\setminus\\{b_{i}:i<n\\}\subseteq\phi[\tau(G\setminus\\{\gamma_{i}:i<n\\})]\subseteq\phi[\tau(G\setminus
K)]$.

(3)$\implies$(4) Let $\phi:\Lambda\to\Lambda$ be a witness for
$\rho\leq_{K}\rho_{\mathrm{Fin}(\Lambda)}$. Since
$\phi^{-1}[\\{\lambda\\}]\in\mathrm{Fin}(\Lambda)$ for every
$\lambda\in\Lambda$ and $\phi[\Lambda]$ is infinite, there is an infinite set
$A\subseteq\Lambda$, such that $\phi\restriction A$ is one-to-one. Then we can
find $F\in\mathcal{F}$ such that $\rho(F)\subseteq\phi[A]$. We claim that
$\mathcal{I}_{\rho}\restriction\rho(F)=\mathrm{Fin}(\rho(F))$. Indeed, let
$B\subseteq\rho(F)$ be infinite and observe that $\phi^{-1}[B]$ is infinite,
so $B=\phi[\phi^{-1}[B]]\notin\mathcal{I}_{\rho}$.

(4)$\implies$(1) If $\rho\not\leq_{K}\rho_{\mathrm{Fin}(\Lambda)}$ then by
Proposition 7.5(2b), $\mathcal{I}_{\rho}\not\leq_{K}\mathrm{Fin}(\Lambda)$,
and consequently $\mathcal{I}_{\rho}$ is tall. ∎

######  Definition 8.2.

We say that a partition regular function $\rho$ is _tall_ if any item of
Proposition 8.1 holds.

######  Proposition 8.3.

The ideals $\mathcal{H}$, $\mathcal{R}$, $\mathcal{D}$, $\mathcal{W}$ and
$\mathcal{I}_{1/n}$ are tall (hence, $FS$, $r$, $\Delta$, $\rho_{\mathcal{W}}$
and $\rho_{\mathcal{I}_{1/n}}$ are tall).

###### Proof.

For the case of $\mathcal{W}$ and $\mathcal{I}_{1/n}$ see [7, p. 3-4]. For
other cases, see [65, under Lemma 1.6.24] and [20, Proposition 4.3 and text
above Lemma 3.2]). Tallness of the listed partition regular functions follows
then from Proposition 8.1. ∎

####  8.2. Homogeneity of partition regular functions

Let $\mathcal{I}_{i}$ be an ideal on $\Lambda_{i}$ for each $i=1,2$. Ideals
$\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are _isomorphic_ (in short:
$\mathcal{I}_{1}\approx\mathcal{I}_{2}$) if there exists a bijection
$\phi:\Lambda_{1}\to\Lambda_{2}$ such that
$A\in\mathcal{I}_{1}\iff\phi[A]\in\mathcal{I}_{2}$ for each
$A\subseteq\Lambda_{1}$. An ideal $\mathcal{I}$ on $\Lambda$ is _homogeneous_
if the ideals $\mathcal{I}$ and $\mathcal{I}\restriction A$ are isomorphic for
every $A\in\mathcal{I}^{+}$ ([60, Definition 1.3], see also [32]). We say that
$\mathcal{I}$ is _K-homogeneous_ if $\mathcal{I}\restriction
A\leq_{K}\mathcal{I}$ for every $A\in\mathcal{I}^{+}$ (in [44, p. 37], the
author uses the name _K-uniform_ in this case). Note that we always have
$\mathcal{I}\leq_{K}\mathcal{I}\restriction A$ for every $A\in\mathcal{I}^{+}$
(see e.g. [44, p. 46]).

######  Definition 8.4.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular. We say that
$\rho$ is _K-homogeneous_ if $\rho\restriction A\leq_{K}\rho$ for every
$A\in\mathcal{I}_{\rho}^{+}$ (note that we always have
$\rho\leq_{K}\rho\restriction A$ for every $A\in\mathcal{I}_{\rho}^{+}$ by
Proposition 7.4(3a)).

######  Proposition 8.5.

  1. (1)

If a partition regular function $\rho$ is K-homogeneous then
$\mathcal{I}_{\rho}$ is K-homogeneous.

  2. (2)

An ideal $\mathcal{I}$ is K-homogeneous $\iff$ $\rho_{\mathcal{I}}$ is
K-homogeneous.

###### Proof.

(1) It follows from Propositions 7.5(1) and 4.1.

(2) Observe that if $A\in\mathcal{I}^{+}$ then $\rho_{\mathcal{I}\restriction
A}=\rho_{\mathcal{I}}\restriction A$. Thus, it follows from Propositions
7.5(2b) and 4.1. ∎

We need the following lemma to show that $FS$ and $r$ are K-homogeneous.

######  Lemma 8.6.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular (with
$\mathcal{F}\subseteq[\Omega]^{\omega}$). If $\mathcal{I}_{\rho}$ is
homogeneous and $\rho$ is $P^{-}$ and has small accretions then $\rho$ is
K-homogeneous.

###### Proof.

Let $A\in\mathcal{I}_{\rho}^{+}$. Since $\mathcal{I}_{\rho}$ is homogeneous,
$\mathcal{I}_{\rho}\restriction A$ and $\mathcal{I}_{\rho}$ are isomorphic.
Let $f:\Lambda\to A$ be a bijection witnessing it. We claim that $f$ witnesses
$\rho\restriction A\leq_{K}\rho$.

Let $F\in\mathcal{F}$. Since $\rho$ has small accretions, there is
$G\in\mathcal{F}$ such that $G\subseteq F$ and $G$ has small accretions.
Enumerate $\Omega=\\{o_{n}:n\in\omega\\}$ and define $K_{n}=\\{o_{i}:i\leq
n\\}$ and $A_{n}=f[\rho(G\setminus K_{n})]$ for all $n\in\omega$. Then
$A_{n}\supseteq A_{n+1}$. Since $G$ has small accretions and $f$ is a
bijection and witnesses that $\mathcal{I}_{\rho}\restriction A$ and
$\mathcal{I}_{\rho}$ are isomorphic, $A_{n}\in(\mathcal{I}_{\rho}\restriction
A)^{+}$ and $A_{n}\setminus A_{n+1}\subseteq f[\rho(G\setminus
K_{n})\setminus\rho(G\setminus K_{n+1})]\subseteq
f[\rho(G)\setminus\rho(G\setminus K_{n+1})]\in\mathcal{I}_{\rho}\restriction
A$. Using the fact that $\rho$ is $P^{-}$, we can find
$H\in\mathcal{F}\restriction A$ such that
$\rho(H)\subseteq^{\rho}A_{n}=f[\rho(G\setminus K_{n})]$ for all $n\in\omega$.
Hence, given any finite set $K\subseteq\Omega$ there are $n\in\omega$ and
finite $L\subseteq\Omega$ such that $K\subseteq K_{n}$ and $\rho(H\setminus
L)\subseteq A_{n}=f[\rho(G\setminus K_{n})]\subseteq f[\rho(G\setminus K)]$. ∎

######  Proposition 8.7.

  1. (1)

The ideals $\mathcal{H}$, $\mathcal{R}$ and $\mathcal{W}$ are homogeneous
(hence, K-homogeneous).

  2. (2)

The functions $\operatorname{FS}$ and $r$ are K-homogeneous.

###### Proof.

(1) See [60, Examples 2.5 and 2.6].

(2) It follows from item (1), Lemma 8.6 and Propositions 6.7(3) and 4.3. ∎

######  Question 8.8.

  1. (1)

Is the function $\Delta$ K-homogeneous?

  2. (2)

Is the ideal $\mathcal{I}_{1/n}$ K-homogeneous?

##  Part II $\operatorname{FinBW}$ spaces

In this part we define the main object of our studies – classes of
sequentially compact spaces defined with the aid of partition regular
functions (Definition 10.1). Next, we prove some general results about those
classes of spaces (Theorem 10.5).

###  9\. A convergence with respect to partition regular functions

######  Definition 9.1.

Let $X$ be a topological space. Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be
partition regular with $\mathcal{F}\subseteq[\Omega]^{\omega}$.

  1. (1)

For $F\in\mathcal{F}$, a function $f:\rho(F)\to X$ is called a _$\rho$
-sequence_ in $X$.

  2. (2)

A $\rho$-sequence $f:\rho(F)\to X$ is _$\rho$ -convergent_ to a point $x\in X$
if for every neighbourhood $U$ of $x$ there is a finite set $K\subseteq\Omega$
such that

$f[\rho(F\setminus K)]\subseteq U.$

######  Remark.

Various kinds of convergences considered in the literature can be described in
terms of $\rho$-convergence.

  1. (1)

If $\rho=\operatorname{FS}$, then $\rho$-convergence coincides with _$IP$
-convergence_ (see [33], [34] or [55]).

  2. (2)

If $\rho=r$, then $\rho$-convergence coincides with the _$\mathcal{R}$
-convergence_ (see [5], [6], [59, Definition 2.1]).

  3. (3)

If $\rho=\Delta$, then $\rho$-convergence coincides with the _differential
convergence_ (see [71, Definition 4.2.4] or [20, p. 2010]).

  4. (4)

If $\mathcal{I}$ is an ideal on $\Lambda$ and $\rho_{\mathcal{I}}$ is defined
as in Proposition 3.3(2), then $\rho_{\mathcal{I}}$-convergence coincides with
the ordinary convergence.

The following proposition reveals relationships between $\rho$-convergence and
convergence.

######  Proposition 9.2.

  1. (1)

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular with
$\mathcal{F}\subseteq[\Omega]^{\omega}$. Let $F\in\mathcal{F}$ and
$f:\rho(F)\to X$.

    1. (a)

If $f$ is convergent to $L$, then $f\restriction\rho(E)$ is $\rho$-convergent
to $L$ for some $E\in\mathcal{F}\restriction\rho(F)$.

    2. (b)

If $f$ is $\rho$-convergent to $L$, then $f\restriction A$ is convergent to
$L$ for some infinite set $A\subseteq\rho(F)$.

  2. (2)

Let $\mathcal{I}$ be an ideal on $\Lambda$ and $f:A\to X$ for some
$A\in\mathcal{I}^{+}$. Then $f$ is convergent to $L$ $\iff$ $f$ is
$\rho_{\mathcal{I}}$-convergent to $L$.

###### Proof.

(1a) Let $E\in\mathcal{F}$ with $E\subseteq F$ be as in Proposition 3.2 and
let $U$ be a neighborhood of $L$. Then there exists a finite set $K$ such that
$f(n)\in U$ for every $n\in\rho(F)\setminus K$. There is a finite set $L$ such
that $\rho(E\setminus L)\subseteq\rho(E)\setminus K\subseteq\rho(F)\setminus
K$. Consequently, $f(n)\in U$ for every $n\in\rho(E\setminus L)$.

(1b) Let $\Omega=\\{o_{n}:n\in\omega\\}$. For each $n\in\omega$, we pick
$\lambda_{n}\in\rho(F\setminus\\{o_{i}:i<n\\})\setminus\\{\lambda_{i}:i<n\\}$.
Let $A=\\{\lambda_{n}:n\in\omega\\}$. We claim that $f\restriction A$ is
convergent to $L$. Indeed, if $U$ is a neighborhood of $L$, then there is a
finite set $K\subseteq\Omega$ such that $f[\rho(F\setminus K)]\subseteq U$.
Let $n\in\omega$ be such that $K\subseteq\\{o_{i}:i<n\\}$. Then
$f[A\setminus\\{\lambda_{i}:i<n\\}]\subseteq
f[\rho(F\setminus\\{o_{i}:i<n\\})]\subseteq f[\rho(F\setminus K)]\subseteq U$.

(2) It is straightforward. ∎

###  10\. $\operatorname{FinBW}$ spaces

Let $\mathcal{I}$ be an ideal on a countable infinite set $\Lambda$. The
following classes of topological spaces were extensively examined in the
literature (see e.g. [24, 29, 61]):

  1. (1)

$\operatorname{FinBW}(\mathcal{I})$ is the class of all topological spaces $X$
such that for every sequence $f:\Lambda\to X$ there exists
$A\in\mathcal{I}^{+}$ such that $f\restriction A$ converges (in [29], spaces
from $\operatorname{FinBW}(\mathcal{I})$ are called _$\mathcal{I}$ -spaces_);

  2. (2)

$\operatorname{hFinBW}(\mathcal{I})$ is the class of all topological spaces
$X$ such that for every $B\in\mathcal{I}^{+}$ and every sequence $f:B\to X$
there exists $A\in\mathcal{I}^{+}$ such that $A\subseteq B$ and $f\restriction
A$ converges.

######  Remark.

The classes $\operatorname{FinBW}(\mathcal{I}_{\rho})$ for
$\rho\in\\{\mathcal{W},\mathcal{I}_{1/n}\\}$ were examined in the literature
under other names:

  1. (1)

in [56, Definition 3], spaces from $\operatorname{FinBW}(\mathcal{W})$ are
called _van der Waerden spaces_ ;

  2. (2)

in [29, Definition 2.1], spaces from $\operatorname{FinBW}(\mathcal{I}_{1/n})$
are called _$\mathcal{I}_{1/n}$ -spaces_.

######  Definition 10.1.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be a partition regular function.

  1. (1)

$\operatorname{FinBW}(\rho)$ is the class of all topological spaces $X$ such
that for every sequence $f:\Lambda\to X$ there exists $F\in\mathcal{F}$ such
that $f\restriction\rho(F)$ $\rho$-converges.

  2. (2)

$\operatorname{hFinBW}(\rho)$ is the class of all topological spaces $X$ such
that for every $\rho$-sequence $f:\rho(E)\to X$ there exists $F\in\mathcal{F}$
such that $\rho(F)\subseteq\rho(E)$ and $f\restriction\rho(F)$
$\rho$-converges.

######  Remark.

The classes $\operatorname{FinBW}(\rho)$ for
$\rho\in\\{\operatorname{FS},r,\Delta\\}$ were examined in the literature
under other names:

  1. (1)

in [55, Definition 4], spaces from $\operatorname{FinBW}(\operatorname{FS})$
are called _Hindman spaces_ ;

  2. (2)

in [6] (see also [59, Definition 2.1]), spaces from $\operatorname{FinBW}(r)$
are called _spaces with the Ramsey property_ , and we will call them _Ramsey
spaces_ in short;

  3. (3)

in [71, Definition 4.2.4] (see also [20, p. 2010]), spaces in
$\operatorname{FinBW}(\Delta)$ are called _differentially compact spaces_.

######  Remark.

Recall that if $(\Lambda,<)$ is a directed set, then any function
$f:\Lambda\to X$ is called a _net_ in $X$. A net $f:\Lambda\to X$ in a
topological space $X$ _converges_ to $x\in X$ if for every neighborhood $U$ of
$x$ there is $\lambda_{0}\in\Lambda$ such that $f(\lambda)\in U$ for every
$\lambda>\lambda_{0}$ (see e.g. [15, p. 49]). In [18, Remark 2.6], the authors
notice that if $\mathcal{B}$ is a coideal basis on $(\Lambda,<)$, then
$(B,<\cap(B\times B))$ is a directed set and $f\restriction B$ is a subnet of
$f$ for every $B\in\mathcal{B}$. Furthermore, they examine topological spaces
$X$ having the property that every net $f:\Lambda\to X$ has a convergent
subnet $f\restriction B$ with some $B\in\mathcal{B}$ ([18, p. 418]). It is not
difficult to see that the class of spaces they examine coincides with the
class $\operatorname{FinBW}(\rho_{\mathcal{B}})$ with $\rho_{\mathcal{B}}$
defined as in Proposition 3.10(2).

The following proposition reveals relationships between
$\operatorname{FinBW}$-like spaces defined with the aid of partition regular
functions and ideals.

######  Proposition 10.2.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular with
$\mathcal{F}\subseteq[\Omega]^{\omega}$. Let $\mathcal{I}$ be an ideal on
$\Lambda$.

  1. (1)

    1. (a)

$\operatorname{hFinBW}(\rho)=\bigcap\\{\operatorname{FinBW}(\rho\restriction\rho(F)):F\in\mathcal{F}\\}$.

    2. (b)

$\operatorname{hFinBW}(\mathcal{I})=\bigcap\\{\operatorname{FinBW}(\mathcal{I}\restriction
A):A\in\mathcal{I}^{+}\\}$.

  2. (2)

    1. (a)

$\operatorname{hFinBW}(\rho)\subseteq\operatorname{FinBW}(\rho)$.

    2. (b)

$\operatorname{hFinBW}(\mathcal{I})\subseteq\operatorname{FinBW}(\mathcal{I})$.

  3. (3)

$\operatorname{FinBW}(\mathcal{I}_{\rho})\subseteq\operatorname{FinBW}(\rho)$
and
$\operatorname{hFinBW}(\mathcal{I}_{\rho})\subseteq\operatorname{hFinBW}(\rho)$.

  4. (4)

$\operatorname{FinBW}(\mathcal{I})=\operatorname{FinBW}(\rho_{\mathcal{I}})$
and
$\operatorname{hFinBW}(\mathcal{I})=\operatorname{hFinBW}(\rho_{\mathcal{I}})$.

  5. (5)

    1. (a)

If $\rho$ is K-homogeneous, then
$\operatorname{hFinBW}(\rho)=\operatorname{FinBW}(\rho)$.

    2. (b)

If $\mathcal{I}$ is K-homogeneous, then
$\operatorname{hFinBW}(\mathcal{I})=\operatorname{FinBW}(\mathcal{I})$.

###### Proof.

(1) and (2) Straightforward.

(3) It follows from Proposition 9.2(1a) (the other inclusion does not follow
from Proposition 9.2(1b) as it gives us only an infinite set $A$, not
necessarily $A\in\mathcal{I}_{\rho}^{+}$).

(4) It follows from Proposition 9.2(2).

(5a) We only need to show
$\operatorname{FinBW}(\rho)\subseteq\operatorname{hFinBW}(\rho)$. Let
$X\in\operatorname{FinBW}(\rho)$ and $f:\rho(E)\to X$ be a $\rho$-sequence in
$X$ for some $E\in\mathcal{F}$. Let $\phi:\Lambda\to\rho(E)$ be a witness for
$\rho\restriction(\mathcal{F}\restriction\rho(E))\leq_{K}\rho$. Since
$f\circ\phi:\Lambda\to X$, there is $F\in\mathcal{F}$ such that
$f\circ\phi\restriction\rho(F)$ is $\rho$-convergent to some $x\in X$. Since
$\rho\restriction(\mathcal{F}\restriction\rho(E))\leq_{K}\rho$, there is
$G\in\mathcal{F}\restriction\rho(E)$ such that for every finite set
$K\subseteq\Omega$ there is a finite set $L\subseteq\Omega$ with
$\rho(G\setminus L)\subseteq\phi[\rho(F\setminus K)]$. We claim that
$f\restriction\rho(G)$ is $\rho$-convergent to $x$. Let $U$ be a neighborhood
of $x$. Then there is a finite set $K\subseteq\Omega$ such that
$(f\circ\phi)[\rho(F\setminus K)]\subseteq U$. We pick a finite set
$L\subseteq\Omega$ such that $\rho(G\setminus L)\subseteq\phi[\rho(F\setminus
K)]$. Then $f[\rho(G\setminus L)]\subseteq f[\phi[\rho(F\setminus
K)]]\subseteq U$, so the proof is finished.

(5b) It follows from items (5a) and (4) and Proposition 8.5(2). ∎

######  Remark.

In Theorem 10.2(3), we cannot replace inclusion with equality in general
because in [55, Theorems 3 and 10] the author proved that
$\operatorname{FinBW}(\mathcal{H})$ contains only finite Hausdorff spaces,
whereas $\operatorname{FinBW}(\operatorname{FS})$ contains infinite (even
uncountable) ones.

######  Corollary 10.3.

  1. (1)

[56, Proposition 4]
$\operatorname{hFinBW}(\mathcal{W})=\operatorname{FinBW}(\mathcal{W})$, and
consequently the product of two van der Waerden spaces is van der Waerden.

  2. (2)

[55, Lemma 8]
$\operatorname{hFinBW}(\operatorname{FS})=\operatorname{FinBW}(\operatorname{FS})$,
and consequently the product of two Hindman spaces is Hindman.

  3. (3)

[59, Theorem 3.4] $\operatorname{hFinBW}(r)=\operatorname{FinBW}(r)$, and
consequently the product of two Ramsey spaces is Ramsey.

###### Proof.

It follows from Theorem 10.2(5) and Proposition 8.7. ∎

######  Question 10.4 ([28, 30] and [71, Question 4.2.3]).

  1. (1)

    1. (a)

Does
$\operatorname{FinBW}(\mathcal{I}_{1/n})=\operatorname{hFinBW}(\mathcal{I}_{1/n})$?

    2. (b)

Is the product of two $\mathcal{I}_{1/n}$-spaces an $\mathcal{I}_{1/n}$-space?

  2. (2)

    1. (a)

Does $\operatorname{FinBW}(\Delta)=\operatorname{hFinBW}(\Delta)$?

    2. (b)

Is the product of two differentially compact spaces a differentially compact
space?

Note that the positive answer to the question in item (1a) gives the positive
answer to the question in item (1b), and similarly for the questions in the
second item. Moreover, the positive answer to the Question 8.8(1) (Question
8.8(2), resp.) gives the positive answer to Question 10.4(2a) (Question
10.4(1a), resp.).

Let us now turn to one of the main result of this paper.

######  Theorem 10.5.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular with
$\mathcal{F}\subseteq[\Omega]^{\omega}$.

  1. (1)

$\operatorname{FinBW}(\rho)$ contains all finite spaces and is a subclass of
the class of all sequentially compact spaces.

  2. (2)

$\rho$ is not tall $\iff$ $\operatorname{FinBW}(\rho)$ coincides with the
class of all sequentially compact spaces.

  3. (3)

The following conditions are equivalent.

    1. (a)

$\rho$ is $P^{-}(\Lambda)$.

    2. (b)

There are Hausdorff compact spaces of arbitrary cardinality that belong to
$\operatorname{FinBW}(\rho)$.

    3. (c)

There exists an infinite Hausdorff topological space
$X\in\operatorname{FinBW}(\rho)$.

  4. (4)

The following conditions are equivalent.

    1. (a)

$\rho$ is $P^{-}$.

    2. (b)

There are Hausdorff compact spaces of arbitrary cardinality that belong to
$\operatorname{hFinBW}(\rho)$.

    3. (c)

There exists an infinite Hausdorff topological space
$X\in\operatorname{hFinBW}(\rho)$.

  5. (5)

If $\rho$ is $P^{-}$, then

    1. (a)

the uncountable non-compact Hausdorff space $X=\omega_{1}$ with the order
topology belongs to $\operatorname{FinBW}(\rho)$,

    2. (b)

assuming Continuum Hypothesis (CH) there are Hausdorff compact and separable
spaces of cardinality $\mathfrak{c}$ that belong to
$\operatorname{FinBW}(\rho)$.

  6. (6)

If $\rho$ is weak $P^{+}$, then every compact metric space is in
$\operatorname{hFinBW}(\rho)$.

  7. (7)

If $\rho$ is $P^{+}$, then every Hausdorff topological space with the property
$(*)$ belongs to $\operatorname{hFinBW}(\rho)$.

(A topological space $X$ has the _property (*)_ if for every countable set
$D\subseteq X$ the closure $\operatorname{cl}_{X}(D)$ is compact and first-
countable – see [56].)

###### Proof.

(1) It follows from Proposition 9.2(1b).

(2) The implication “$\impliedby$” will follow from Theorem 13.2(1). To prove
the implication “$\implies$” we only need to show that every sequentially
compact space belongs to $\operatorname{FinBW}(\rho)$. Fix a sequentially
compact space $X$ and $f:\Lambda\to X$. Let $\phi:\Lambda\to\Lambda$ be a
witness for $\rho\leq_{K}\rho_{\mathrm{Fin}(\Lambda)}$. Since
$f\circ\phi:\Lambda\to X$, there is an infinite set $A\subseteq\Lambda$ such
that $(f\circ\phi)\restriction A$ is convergent to some $x\in X$. Then there
is $F\in\mathcal{F}$ with $\rho(F)\subseteq\phi[A]$. We claim that
$f\restriction\rho(F)$ is $\rho$-convergent to $x$. Let $U$ be any
neighbourhood of $x$. Then there is a finite set $L\subseteq\Lambda$ such that
$f[\phi[A\setminus L]]\subseteq U$. Now, we can find a finite set
$K\subseteq\Omega$ such that $\rho(F\setminus K)\subseteq\phi[A\setminus L]$.
Thus, $f[\rho(F\setminus K)]\subseteq U$.

(3) (a)$\implies$(b) Let $\kappa$ be an infinite cardinal number. Let
$X=\kappa\cup\\{\infty\\}$ be the Alexandroff one-point compactification of
the discrete space $\kappa$. Then $X$ is Hausdorff, compact and has
cardinality $\kappa$. Moreover, open neighborhoods of $\infty$ are of the form
$X\setminus S$ where $S$ is a compact (hence finite) subset of $\kappa$. We
show that $X$ is in $\operatorname{FinBW}(\rho)$. Let $f:\Lambda\to X$. If
there is $x\in X$ with $f^{-1}[\\{x\\}]\notin\mathcal{I}_{\rho}$, then we take
$F\in\mathcal{F}$ such that $\rho(F)\subseteq f^{-1}[\\{x\\}]$ and see that
$f\restriction\rho(F)$ is $\rho$-convergent to $x$. Now, we assume that
$f^{-1}[\\{x\\}]\in\mathcal{I}_{\rho}$ for every $x\in X$. By Proposition 6.8,
there is $F\in\mathcal{F}$ such that for every finite set $S\subseteq X$ there
is a finite set $K_{S}$ such that $\rho(F\setminus K_{S})\cap
f^{-1}[S]=\emptyset$. We claim that $f\restriction\rho(F)$ is
$\rho$-convergent to $\infty$. Let $U$ be an open neighborhood of $\infty$.
Let $S\subseteq\kappa$ be a finite set with $U=X\setminus S$. Then
$f[\rho(F\setminus K_{S})]\subseteq X\setminus S=U$.

(b)$\implies$(c) Obvious.

(c)$\implies$(a) Suppose that $\rho$ is not $P^{-}(\Lambda)$ and let
$A_{n}\in\mathcal{I}^{+}_{\rho}$ be the witnessing sequence, i.e.,
$A_{0}=\Lambda$, $A_{n+1}\subseteq A_{n}$, $A_{n}\setminus
A_{n+1}\in\mathcal{I}_{\rho}$ and for each $F\in\mathcal{F}$ there is
$n\in\omega$ such that $\rho(F)\not\subseteq^{\rho}A_{n}$. Note that
$\bigcap_{n\in\omega}A_{n}\in\mathcal{I}_{\rho}$.

Let $X$ be an infinite Hausdorff topological space. We will show that
$X\notin\operatorname{FinBW}(\rho)$. If $X$ is not sequentially compact, then
$X\notin\operatorname{FinBW}(\rho)$ by item (1). If $X$ is sequentially
compact, then find any one-to-one sequence $\\{x_{n}:n\in\omega\\}$ in $X$
converging to some $x\in X$. Without loss of generality we may assume that
$x\neq x_{n}$ for all $n\in\omega$. Define $f:\Lambda\to X$ by
$f\restriction\bigcap_{n\in\omega}A_{n}=x_{0}$ and $f(\lambda)=x_{n+1}$, where
$n$ is such that $\lambda\in A_{n}\setminus A_{n+1}$. Suppose for the sake of
contradiction that there are $L\in X$ and $F\in\mathcal{F}$ such that
$f\restriction\rho(F)$ $\rho$-converges to $L$. By Proposition 9.2(1b) we get
that either $L=x_{n}$ for some $n\in\omega$ or $L=x$.

If $L=x_{n}$ for some $n\in\omega$, find open $U$ and $V$ such that $x_{n}\in
U$, $x\in V$ and $U\cap V=\emptyset$. Since $x_{m}\in V$ (so $x_{m}\notin U$)
for almost all $m\in\omega$ and $f^{-1}[\\{x_{m}\\}]\in\mathcal{I}_{\rho}$ for
all $m\in\omega$, $f^{-1}[U]\in\mathcal{I}_{\rho}$. Hence,
$f\restriction\rho(F)$ cannot $\rho$-converge to $x_{n}$.

If $L=x$, we can find $n\in\omega$ such that
$\rho(F)\not\subseteq^{\rho}A_{n}$. Since $X$ is Hausdorff, there is an open
neighbourhood $U$ of $L$ such that $x_{i+1}\notin U$ for all $i<n$. Since
$f\restriction\rho(F)$ $\rho$-converges to $L$, there should be a finite
$K\subseteq\Omega$ such that $f[\rho(F\setminus K)]\subseteq U$, however
$\rho(F\setminus K)\setminus A_{n}\neq\emptyset$ (by
$\rho(F)\not\subseteq^{\rho}A_{n}$), so $f[\rho(F\setminus
K)]\cap\\{x_{i+1}:i<n\\}\neq\emptyset$, which contradicts $x_{i+1}\notin U$
for all $i<n$.

(4) (a)$\implies$(b) Notice that if $X$ is the space defined in the proof of
the implication $(\ref{thm:compact-metric-implies-FinBW(rho):Pminus-is-
necessary}a)\implies(\ref{thm:compact-metric-implies-FinBW(rho):Pminus-is-
necessary}b)$ then $X\in\operatorname{FinBW}(\rho)$ for every $\rho$ that is
$P^{-}(\Lambda)$ (the definition of $X$ did not depend on $\rho$). Thus, if
$\rho$ is $P^{-}$ then $\rho\restriction\rho(F)$ is $P^{-}(\rho(F))$ for every
$F\in\mathcal{F}$ and consequently
$X\in\bigcap_{F\in\mathcal{F}}\operatorname{FinBW}(\rho\restriction\rho(F))=\operatorname{hFinBW}(\rho)$
(by Proposition 10.2(1a)).

(b)$\implies$(c) Obvious.

(c)$\implies$(a) If $\rho$ is not $P^{-}$ then $\rho\restriction\rho(F)$ is
not $P^{-}(\rho(F))$ for some $F\in\mathcal{F}$. Hence, by item (3),
$\operatorname{FinBW}(\rho\restriction\rho(F))$ contains only finite Hausdorff
spaces. Since
$\operatorname{hFinBW}(\rho)\subseteq\operatorname{FinBW}(\rho\restriction\rho(F))$
by Proposition 10.2(1a), $\operatorname{hFinBW}(\rho)$ also contains only
finite Hausdorff spaces.

(5a) Let $f:\Lambda\to\omega_{1}$. If there is $\alpha<\omega_{1}$ with
$f^{-1}[\\{\alpha\\}]\notin\mathcal{I}_{\rho}$, then we take $F\in\mathcal{F}$
such that $\rho(F)\subseteq f^{-1}[\\{\alpha\\}]$ and see that
$f\restriction\rho(F)$ is $\rho$-convergent to $\alpha$. Now, we assume that
$f^{-1}[\\{\alpha\\}]\in\mathcal{I}_{\rho}$ for every $\alpha<\omega_{1}$.
Since $\Lambda$ is countable and the cofinality of $\omega_{1}$ is
uncountable, there is $\alpha<\omega_{1}$ with
$f^{-1}[\alpha]\notin\mathcal{I}_{\rho}$. Let $\alpha_{0}$ be the smallest
$\alpha$ such that $f^{-1}[\alpha]\notin\mathcal{I}_{\rho}$. Note that
$\alpha_{0}$ is a limit ordinal. Indeed, if $\alpha_{0}=\alpha+1$, then
$\alpha<\alpha_{0}$ and $f^{-1}[\alpha]=f^{-1}[\alpha_{0}]\setminus
f^{-1}[\\{\alpha\\}]\notin\mathcal{I}_{\rho}$, a contradiction. Since
$\alpha_{0}$ is a countable limit ordinal, there is an increasing sequence
$\\{\beta_{n}:n\in\omega\\}$ such that
$\sup\\{\beta_{n}:n\in\omega\\}=\alpha_{0}$. By Proposition 6.8, there is
$F\in\mathcal{F}$ such that $\rho(F)\subseteq f^{-1}[\alpha_{0}]$ and for each
$n\in\omega$ there is a finite set $K_{n}$ such that $\rho(F\setminus
K_{n})\cap f^{-1}[\beta_{n}]=\emptyset$. We claim that $f\restriction\rho(F)$
is $\rho$-convergent to $\alpha_{0}$. Indeed, let $U$ be a neighborhood of
$\alpha_{0}$. Without loss of generality, we can assume that
$U=(\alpha_{0}+1)\setminus\beta_{n}$ for some $n\in\omega$. Then
$f[\rho(F\setminus K_{n})]\subseteq\alpha_{0}\setminus\beta_{n}\subseteq U$.

(5b) Spaces with these properties are constructed in Theorem 14.5.

(6) Let $f:\rho(E)\to X$ be a $\rho$-sequence in a metric compact space $X$.

Since $\rho$ is weak $P^{+}$, there exists $F\in\mathcal{F}$ such that
$\rho(F)\subseteq\rho(E)$ and for every sequence
$\\{F_{n}:n\in\omega\\}\subseteq\mathcal{F}$ such that
$\rho(F)\supseteq\rho(F_{n})\supseteq\rho(F_{n+1})$ for each $n\in\omega$
there exists $G\in\mathcal{F}$ such that $\rho(G)\subseteq\rho(F)$ and
$\rho(G)\subseteq^{\rho}\rho(F_{n})$ for each $n\in\omega$.

For $x\in X$ and $r>0$ we write $B(x,r)$ and $\overline{B}(x,r)$ to denote an
open and closed ball of radius $r$ centered at a point $x$, respectively.

Since $X$ is compact metric, there are finitely many $x_{i}^{0}\in X$,
$i<n_{0}$ such that $X=\bigcup\\{B(x_{i}^{0},1):i<n_{0}\\}$. Then there exists
$i_{0}<n_{0}$ such that $\rho(F)\cap
f^{-1}[B(x^{0}_{i_{0}},1)]\notin\mathcal{I}_{\rho}$, and consequently there is
$F_{0}\in\mathcal{F}$ such that $\rho(F_{0})\subseteq\rho(F)\cap
f^{-1}[B(x^{0}_{i_{0}},1)]$.

Since $\overline{B}(x^{0}_{i_{0}},1)$ is compact metric, there are finitely
many $x_{i}^{1}\in X$, $i<n_{1}$ such that
$\overline{B}(x^{0}_{i_{0}},1)\subseteq\bigcup\\{B(x_{i}^{1},\frac{1}{2}):i<n_{1}\\}$.
Then there exists $i_{1}<n_{1}$ such that $\rho(F_{0})\cap
f^{-1}[B(x^{1}_{i_{1}},\frac{1}{2})]\notin\mathcal{I}_{\rho}$, and
consequently there is $F_{1}\in\mathcal{F}$ such that
$\rho(F_{1})\subseteq\rho(F_{0})\cap f^{-1}[B(x^{1}_{i_{1}},\frac{1}{2})]$.

If we continue the above procedure, we obtain $F_{n}\in\mathcal{F}$ and
$x^{n}_{i_{n}}\in X$ such that $\rho(F_{n})\subseteq\rho(F_{n-1})\cap
f^{-1}[B(x^{n}_{i_{n}},\frac{1}{n+1})]$ for each $n\in\omega$ (assuming that
$F_{-1}=F$).

Let $x\in\bigcap\\{\overline{B}(x^{n}_{i_{n}},\frac{1}{n+1}):n\in\omega\\}$.

Since $\rho$ is weak $P^{+}$, we have $G\in\mathcal{F}$ such that
$\rho(G)\subseteq\rho(F)$ and $\rho(G)\subseteq^{\rho}\rho(F_{n})$ for each
$n\in\omega$.

We claim that $f\restriction\rho(G)$ is $\rho$-convergent to $x$. Let $U$ be a
neighborhood of $x$. Since the sequence $(x^{n}_{i_{n}})_{n\in\omega}$ is
convergent to $x$, there is $n_{0}\in\omega$ such that
$B(x^{n}_{i_{n}},\frac{1}{n+1})\subseteq U$ for every $n\geq n_{0}$.
Consequently, there is $n\in\omega$ with
$B(x^{n}_{i_{n}},\frac{1}{n+1})\subseteq U$. Let $K\subseteq\Omega$ be a
finite set such that $\rho(G\setminus K)\subseteq\rho(F_{n})$. Then
$f[\rho(G\setminus K)]\subseteq f[\rho(F_{n})]\subseteq
B(x^{n}_{i_{n}},\frac{1}{n+1})\subseteq U,$ so the proof is finished.

(7) Let $E\in\mathcal{F}$ and $f:\rho(E)\to X$ be a sequence in a Hausdorff
topological space $X$ having the property $(*)$. Since the set
$D=\\{f(\lambda):\lambda\in\rho(E)\\}$ is countable, the closure
$\operatorname{cl}_{X}(D)$ is compact and first-countable. We claim that there
exists $L\in\operatorname{cl}_{X}(D)$ such that
$f^{-1}[U]\in\mathcal{I}_{\rho}^{+}$ for every neighborhood $U$ of $L$.

Suppose, for sake of contradiction, that for every
$x\in\operatorname{cl}_{X}(D)$ there is a neighborhood $U_{x}$ of $x$ such
that $f^{-1}[U_{x}]\in\mathcal{I}_{\rho}$. Since $\operatorname{cl}_{X}(D)$ is
compact, there are finitely many $x_{i}\in\operatorname{cl}_{X}(D)$ for $i<n$
with $\operatorname{cl}_{X}(D)\subseteq\bigcup\\{U_{x_{i}}:i<n\\}.$ Then
$\rho(E)=\bigcup\\{f^{-1}[U_{x_{i}}]:i<n\\}\in\mathcal{I}_{\rho},$ a
contradiction, so the claim is proved.

Let $\\{U_{n}:n\in\omega\\}$ be a base at $L$. Without loss of generality, we
can assume that $U_{n}\supseteq U_{n+1}$ for each $n\in\omega$. For each
$n\in\omega$, we define $A_{n}=\\{\lambda\in\rho(E):f(\lambda)\in U_{n}\\}$.
Since $A_{n}\in\mathcal{I}_{\rho}^{+}$ and $A_{n}\supseteq A_{n+1}$ for each
$n\in\omega$, using the fact that $\rho$ is $P^{+}$, there exists
$F\in\mathcal{F}$ such that $\rho(F)\subseteq\rho(E)$ and
$\rho(F)\subseteq^{\rho}A_{n}$ for each $n\in\omega$. We claim that
$f\restriction\rho(F)$ is $\rho$-convergent to $L$.

Take any neighborhood $U$ of $L$. Then there exists $n_{0}\in\omega$ with
$U_{n_{0}}\subseteq U$. Since $\rho(F)\subseteq^{\rho}A_{n_{0}}$, there exists
a finite set $K\subseteq\Omega$ such that $\rho(F\setminus K)\subseteq
A_{n_{0}}$. Thus $f[\rho(F\setminus K)]\subseteq U_{n_{0}}\subseteq U$, so the
proof is finished. ∎

The following series of corollaries shows that many known earlier results can
be easily derived from Theorem 10.5.

######  Corollary 10.6 ([26, Proposition 2.4]).

If an ideal $\mathcal{I}$ is not tall, then
$\operatorname{FinBW}(\mathcal{I})$ coincides with the class of all
sequentially compact spaces.

###### Proof.

It follows from Theorem 10.5(2) and Propositions 10.2(4) and 8.1. ∎

######  Corollary 10.7 ([61, Theorem 6.5]).

$\mathrm{Fin}^{2}\leq_{K}\mathcal{I}$ $\iff$
$\operatorname{FinBW}(\mathcal{I})$ coincides with the class of all finite
spaces in the realm of Hausdorff spaces.

###### Proof.

It follows from Theorem 10.5(3) and Propositions 7.1(1),6.5(2),10.2(4). ∎

######  Corollary 10.8.

Every metric compact space belongs to $\operatorname{hFinBW}(\rho)$ in case
when

  1. (1)

$\rho=\rho_{\mathcal{I}}$ and $\mathcal{I}$ is $P^{+}$ ideal,

  2. (2)

[29, Theorem 2.3] $\rho=\rho_{\mathcal{I}}$ and $\mathcal{I}$ is an
$F_{\sigma}$ ideal,

  3. (3)

[56, Theorem 10] $\rho=\rho_{\mathcal{I}}$ and $\mathcal{I}=\mathcal{W}$,

  4. (4)

[29, Theorem 2.3] $\rho=\rho_{\mathcal{I}}$ and
$\mathcal{I}=\mathcal{I}_{1/n}$,

  5. (5)

[34, Theorem 2.5] $\rho=\operatorname{FS}$,

  6. (6)

[6, Theorem 1] (see also [5, Theorem 1.16]) $\rho=r$,

  7. (7)

[20, Corollary 4.8] $\rho=\Delta$.

###### Proof.

It follows from Theorem 10.5(6), Propositions 6.7 and 6.5. ∎

######  Corollary 10.9.

Every Hausdorff space with the property $(*)$ belongs to
$\operatorname{hFinBW}(\rho)$ in case when

  1. (1)

$\rho=\rho_{\mathcal{I}}$ and $\mathcal{I}$ is $P^{+}$ ideal,

  2. (2)

[29, Theorem 2.3] $\rho=\rho_{\mathcal{I}}$ and $\mathcal{I}$ is an
$F_{\sigma}$ ideal,

  3. (3)

[56, Theorem 10] $\rho=\rho_{\mathcal{I}}$ and $\mathcal{I}=\mathcal{W}$,

  4. (4)

[29, Theorem 2.3] $\rho=\rho_{\mathcal{I}}$ and
$\mathcal{I}=\mathcal{I}_{1/n}$.

###### Proof.

It follows from Theorem 10.5(7), Propositions 6.7 and 6.5. ∎

In [55, Theorem 11] ([59, Corollary 3.2], resp.), the authors proved that
every Hausdorff space with the property $(*)$ belongs to
$\operatorname{hFinBW}(\operatorname{FS})$ ($\operatorname{hFinBW}(r)$,
resp.). However, their proofs use properties very specific to
$\operatorname{FS}$ and $r$. For instance, the proof for $\operatorname{FS}$
uses idempotent ultrafilters whereas the proof for $r$ uses the bounding
number $\mathfrak{b}$.

######  Problem 10.10.

Find a property $W$ of partition regular functions such that both
$\operatorname{FS}$ and $r$ have the property $W$ and if $\rho$ has the
property $W$ then every Hausdorff space with the property $(*)$ belongs to
$\operatorname{hFinBW}(\rho)$.

In [20, Corollary 4.8], the author proved that every Hausdorff space with the
property $(*)$ belongs to $\operatorname{FinBW}(\Delta)$.

######  Question 10.11.

Does every Hausdorff space with the property $(*)$ belong to
$\operatorname{hFinBW}(\Delta)$?

Note that the positive answer to Question 10.4(2a) gives the positive answer
to Question 10.11.

###  11\. Inclusions between $\operatorname{FinBW}$ classes

######  Theorem 11.1.

Let $\rho_{i}:\mathcal{F}_{i}\to[\Lambda_{i}]^{\omega}$ be partition regular
with $\mathcal{F}_{i}\subseteq[\Omega_{i}]^{\omega}$ for each $i=1,2$. Let
$\mathcal{I}$ be an ideal on $\Lambda$.

  1. (1)

$\rho_{2}\leq_{K}\rho_{1}\implies\operatorname{FinBW}(\rho_{1})\subseteq\operatorname{FinBW}(\rho_{2})$.

  2. (2)

    1. (a)

If $\rho_{2}$ is $P^{+}$, then

$\mathcal{I}_{\rho_{2}}\leq_{K}\mathcal{I}_{\rho_{1}}\implies\operatorname{FinBW}(\rho_{1})\subseteq\operatorname{FinBW}(\rho_{2}).$

    2. (b)

$\mathcal{I}_{\rho_{2}}\leq_{K}\mathcal{I}\implies\operatorname{FinBW}(\mathcal{I})\subseteq\operatorname{FinBW}(\rho_{2})$.

###### Proof.

(1) Let $\phi:\Lambda_{1}\to\Lambda_{2}$ be a witness for
$\rho_{2}\leq_{K}\rho_{1}$. Let $X\in\operatorname{FinBW}(\rho_{1})$. If
$f:\Lambda_{2}\to X$, then $f\circ\phi:\Lambda_{1}\to X$, so there is
$F_{1}\in\mathcal{F}_{1}$ such that $\rho_{1}(F_{1})\subseteq\Lambda_{1}$ and
$(f\circ\phi)\restriction\rho_{1}(F_{1})$ is $\rho_{1}$-convergent to some
$x\in X$.

Let $F_{2}\in\mathcal{F}_{2}$ be such that for every finite
$K_{1}\subseteq\Omega_{1}$ there is a finite $K_{2}\subseteq\Omega_{2}$ with
$\rho_{2}(F_{2}\setminus K_{2})\subseteq\phi[\rho_{1}(F_{1}\setminus K_{1})]$.

We claim that $f\restriction\rho(F_{2})$ is $\rho_{2}$-convergent to $x$. Let
$U$ be a neighborhood of $x$. Since $(f\circ\phi)\restriction\rho_{1}(F_{1})$
is $\rho_{1}$-convergent to $x$, then there is a finite set
$K_{1}\subseteq\Omega_{1}$ such that $(f\circ\phi)[\rho_{1}(F_{1}\setminus
K_{1})]\subseteq U$. Hence, we can find a finite $K_{2}\subseteq\Omega_{2}$
with $\rho_{2}(F_{2}\setminus K_{2})\subseteq\phi[\rho_{1}(F_{1}\setminus
K_{1})]$. Then $f[\rho_{2}(F_{2}\setminus K_{2})]\subseteq
f[\phi[\rho_{1}(F_{1}\setminus K_{1})]]\subseteq U$. That finishes the proof.

(2a) It follows from Proposition 7.5(2a) and item (1).

(2b) It follows from item (1) and Propositions 7.5(2b) and 10.2(4). ∎

The following series of corollaries shows that many known earlier results as
well as some new one can be easily derived from Theorem 11.1.

######  Corollary 11.2.

  1. (1)

[71, p. 39] Every Hindman space is differentially compact.

  2. (2)

Every Ramsey space is differentially compact.

  3. (3)

Let $\mathcal{I}$ be a $P^{+}$ ideal.

    1. (a)

[22, Proposition 2.6] If $\mathcal{I}\leq_{K}\mathcal{H}$ then every Hindman
space is in $\operatorname{FinBW}(\mathcal{I})$.

    2. (b)

If $\mathcal{I}\leq_{K}\mathcal{R}$ then every Ramsey space is in
$\operatorname{FinBW}(\mathcal{I})$.

    3. (c)

If $\mathcal{I}\leq_{K}\mathcal{D}$ then every differentially compact space is
in $\operatorname{FinBW}(\mathcal{I})$.

  4. (4)

[61, Corollary 10.2(a)] If $\mathcal{I}_{i}$ are ideals for $i=1,2$ and
$\mathcal{I}_{2}\leq_{K}\mathcal{I}_{1}$, then
$\operatorname{FinBW}(\mathcal{I}_{1})\subseteq\operatorname{FinBW}(\mathcal{I}_{2})$.

###### Proof.

(1) It follows from Theorems 11.1(1) and 7.7(3).

(2) It follows from Theorems 11.1(1) and 7.7(4).

(3) It follows from Theorem 11.1(2a) and Propositions 6.5(2) and 10.2(4).

(4) It follows from Theorem 11.1(2b) and Proposition 10.2(4). ∎

######  Corollary 11.3.

  1. (1)

Let $\rho:\mathcal{F}_{i}\to[\Lambda]^{\omega}$ be a partition regular
function.

    1. (a)

If $\rho\leq_{K}\rho^{\prime}$ for some weak $P^{+}$ partition regular
function $\rho^{\prime}$, then every compact metric space belongs to
$\operatorname{FinBW}(\rho)$.

    2. (b)

If $\rho\leq_{K}\rho^{\prime}$ for some $P^{+}$ partition regular function
$\rho^{\prime}$, then every Hausdorff topological space with the property
$(*)$ belongs to $\operatorname{FinBW}(\rho_{2})$.

  2. (2)

Let $\mathcal{I}$ be an ideal. If an ideal $\mathcal{I}$ can be extended to a
$P^{+}$ ideal, then:

    1. (a)

every compact metric space belongs to $\operatorname{FinBW}(\mathcal{I})$;

    2. (b)

[25, Corollary 5.6] every Hausdorff topological space with the property $(*)$
belongs to $\operatorname{FinBW}(\mathcal{I})$.

###### Proof.

(1) It follows from Theorems 11.1(1), 10.5(6)(7) and Proposition 10.2(2).

(2) It follows from item (1), Propositions 6.5(2), 7.5(2b), 10.2(4). ∎

The following proposition shows that when comparing classes
$\operatorname{FinBW}(\rho_{1})$ and $\operatorname{FinBW}(\rho_{2})$ for
distinct functions $\rho_{1}$ and $\rho_{2}$ we can in fact assume that both
$\rho_{1}$ and $\rho_{2}$ “live” on the same sets $\Omega$ and $\Lambda$.

######  Proposition 11.4.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular with
$\mathcal{F}\subseteq[\Omega]^{\omega}$. Suppose that $\Gamma$ and $\Sigma$
are countable infinite sets and $\phi:\Omega\to\Gamma$ and
$\psi:\Lambda\to\Sigma$ are bijections. Let
$\mathcal{G}=\\{\phi[F]:F\in\mathcal{F}\\}$ and
$\tau:\mathcal{G}\to[\Sigma]^{\omega}$ be given by
$\tau(G)=\psi[\rho(\phi^{-1}[G])]$. Then

  1. (1)

$\tau$ is partition regular,

  2. (2)

$\rho$ and $\tau$ are Katětov equivalent: $\rho\leq_{K}\tau$ and
$\tau\leq_{K}\rho$,

  3. (3)

$\rho$ and $\tau$ are hereditary Katětov equivalent:

    1. (a)

$\forall F\in\mathcal{F}\,\exists
G\in\mathcal{G}\,(\rho\restriction\rho(F)\leq_{K}\tau\restriction\tau(G)),$

    2. (b)

$\forall G\in\mathcal{G}\,\exists
F\in\mathcal{F}\,(\tau\restriction\tau(G)\leq_{K}\rho\restriction\rho(F)),$

  4. (4)

$\operatorname{FinBW}(\tau)=\operatorname{FinBW}(\rho)$ and
$\operatorname{hFinBW}(\tau)=\operatorname{hFinBW}(\rho)$.

###### Proof.

Items (1)–(3) are straightforward, whereas item (4) follows from previous
items and Proposition 11.1. ∎

##  Part III Distinguishing between $\operatorname{FinBW}$ classes

In this part we are interested in finding spaces that are in
$\operatorname{FinBW}(\rho_{1})$, but are not in
$\operatorname{FinBW}(\rho_{2})$. Similar investigations concerning the
classes $\operatorname{FinBW}(\mathcal{I})$ were conducted in [61]. In that
paper all the examples (showing that under some set-theoretic assumption
$\operatorname{FinBW}(\mathcal{I})\setminus\operatorname{FinBW}(\mathcal{J})\neq\emptyset$
for some ideals $\mathcal{I}$ and $\mathcal{J}$) were inspired by [57] and are
of one specific type – they are defined using _maximal_ almost disjoint
families. It turns out (see Theorem 13.2) that, in general, we cannot use
_maximal_ almost disjoint families to distinguish between
$\operatorname{FinBW}(\rho)$ classes with the aid of spaces defined as in
[61]. Fortunately, we managed to use _not necessary maximal_ almost disjoint
families to prove two main results of this part (Theorems 14.3 and 15.2),
which give us
$\operatorname{FinBW}(\rho_{1})\setminus\operatorname{FinBW}(\rho_{2})\neq\emptyset$
for certain $\rho_{1}$ and $\rho_{2}$. Our methods were inspired by [59].

###  12\. Mrówka spaces and their compactifications

For an _infinite_ almost disjoint family $\mathcal{A}$ on a countable set
$\Lambda$, we define a set

$\Psi(\mathcal{A})=\Lambda\cup\mathcal{A}$

and introduce a topology on $\Psi(\mathcal{A})$ as follows: the points of
$\Lambda$ are isolated and a basic neighborhood of $A\in\mathcal{A}$ has the
form $\\{A\\}\cup(A\setminus F)$ with $F$ finite.

Topological spaces of the form $\Psi(\mathcal{A})$ were introduced by
Alexandroff and Urysohn in [1, chapter V, paragraph 1.3] (as noted in [15, p.
182], [14, p. 1380] and [45, p. 605]) and its topology is known as the
_rational sequence topology_ (see [72, Example 65]; the same topology was
later described by Katětov in [52, p. 74]). Spaces $\Psi(\mathcal{A})$ with
_maximal_ (with respect to the inclusion) almost disjoint families
$\mathcal{A}$ were first examined by Mrówka (see [67]) and Isbell (as noted in
[37, p. 269]). It seems that the notation $\Psi$ for these kind of spaces was
used for the first time in [37, Problem 5I, p. 79].

Spaces of the form $\Psi(\mathcal{A})$ are known under many names, including
_$\Psi$ -spaces_, _Isbell-Mrówka spaces_ and _Mrówka spaces_. Recent surveys
on these spaces and their numerous applications can be found in [45, 41].

It is known that $\Psi(\mathcal{A})$ is Hausdorff, regular, locally compact,
first countable and separable, but it is not compact nor sequentially compact
(see [67] or [74, Section 11]). It is not difficult to see that $A\cup\\{A\\}$
is compact in $\Psi(\mathcal{A})$ for every $A\in\mathcal{A}$ and for every
compact set $K\subseteq\Psi(\mathcal{A})$ both sets $K\cap\mathcal{A}$ and
$(K\cap\Lambda)\setminus\bigcup\\{A:A\in K\cap\mathcal{A}\\}$ are finite. In
particular, for every compact set $K\subseteq\Psi(\mathcal{A})$ there are
finitely many sets $A_{i}\in\mathcal{A}$ and a finite set $F$ such that
$K\subseteq\\{A_{i}:i<n\\}\cup\bigcup\\{A_{i}\cup F:i<n\\}$.

Let

$\Phi(\mathcal{A})=\Psi(\mathcal{A})\cup\\{\infty\\}=\Lambda\cup\mathcal{A}\cup\\{\infty\\}$

be the Alexandroff one-point compactification of $\Psi(\mathcal{A})$ (recall
that open neighborhoods of $\infty$ are of the form
$\Phi(\mathcal{A})\setminus K$ for compact sets
$K\subseteq\Psi(\mathcal{A})$). It is not difficult to see that
$\Phi(\mathcal{A})$ is Hausdorff, compact, sequentially compact, separable and
first countable at every point of $\Phi(\mathcal{A})\setminus\\{\infty\\}$.

Topological spaces of the form $\Phi(\mathcal{A})$ with _maximal_ almost
disjoint families $\mathcal{A}$ were first used by Franklin [31, Example 7.1]
where the author used the notation $\Psi^{*}$ instead of $\Phi$. Later, these
spaces were considered in [36] where the authors use the notation
$\mathcal{F}(\mathcal{A})$ and call them the _Franklin compact spaces_
associated to $\mathcal{A}$, whereas in [35] the authors use the notation
$Fr(\mathcal{A})$ and call them the _Franklin spaces of $\mathcal{A}$_. The
notation $\Phi(\mathcal{A})$ for these spaces is used in the following papers
[20, 22, 58, 61] Recently, spaces of the form $\Phi(\mathcal{A})$ were also
considered for _non_ maximal almost disjoint families [59, 11].

It also makes sense to define $\Phi(\mathcal{A})$ for infinite families
$\mathcal{A}$ that are _not_ almost disjoint, but then $\Phi(\mathcal{A})$ is
no longer Hausdorff (almost disjointness of $\mathcal{A}$ is a necessary and
sufficient condition for a space $\Phi(\mathcal{A})$ to be Hausdorff).

The following lemma (which will be used repeatedly in the sequel) shows that a
sequence in a space $\Phi(\mathcal{A})$ may fail to have a $\rho$-convergent
$\rho$-subsequence only in one specific case. Hence, checking whether
$\Phi(\mathcal{A})\in\operatorname{FinBW}(\rho)$ will be reduced to
considering only sequences of this one specific kind.

######  Lemma 12.1.

Let $\rho:\mathcal{F}\to[\Lambda]^{\omega}$ be partition regular with
$\mathcal{F}\subseteq[\Omega]^{\omega}$. Let $\mathcal{A}$ be an infinite
almost disjoint family on $\Lambda$. For every sequence
$f:\Lambda\to\Phi(\mathcal{A})$, the following five cases can only occur:

  1. (1)

$f^{-1}(\infty)\notin\mathcal{I}_{\rho}$,

  2. (2)

$f^{-1}[\mathcal{A}]\notin\mathcal{I}_{\rho}$,

  3. (3)

$f^{-1}(\infty)\in\mathcal{I}_{\rho}$,
$f^{-1}[\mathcal{A}]\in\mathcal{I}_{\rho}$,
$f^{-1}[\Lambda]\in\mathcal{I}_{\rho}^{*}$ and

    1. (a)

$f^{-1}(\lambda)\notin\mathcal{I}_{\rho}$ for some $\lambda\in\Lambda$,

    2. (b)

$f^{-1}(\lambda)\in\mathcal{I}_{\rho}$ for every $\lambda\in\Lambda$ and
$f^{-1}[A]\notin\mathcal{I}_{\rho}$ for some $A\in\mathcal{A}$,

    3. (c)

$f^{-1}(\lambda)\in\mathcal{I}_{\rho}$ for every $\lambda\in\Lambda$ and
$f^{-1}[A]\in\mathcal{I}_{\rho}$ for every $A\in\mathcal{A}$.

If $\rho$ is $P^{-}$, then in cases (1), (2), (3a) and (3b) there is
$F\in\mathcal{F}$ such that $f\restriction\rho(F)$ is $\rho$-convergent.

###### Proof.

_Case (1)._ There is $F\in\mathcal{F}$ such that $f\restriction\rho(F)$ is
constant (with the value $\infty$), hence it is $\rho$-convergent.

_Case (2)._ We find $F\in\mathcal{F}$ with $\rho(F)\subseteq
f^{-1}[\mathcal{A}]$. Then we enumerate $f[\rho(F)]=\\{A_{n}:n\in\omega\\}$
and define $E_{n}=f^{-1}[\\{A_{n}\\}]$ for each $n\in\omega$.

If there is $n_{0}\in\omega$ such that $E_{n_{0}}\notin\mathcal{I}_{\rho}$,
then we find $F^{\prime}\in\mathcal{F}$ with $\rho(F^{\prime})\subseteq
E_{n_{0}}$, and we see that $f\restriction\rho(F^{\prime})$ is constant, so it
is $\rho$-convergent.

Now assume that $E_{n}\in\mathcal{I}_{\rho}$ for each $n\in\omega$. Since
$\rho(F)\subseteq\bigcup\\{E_{n}:n\in\omega\\}$, we can use Proposition 6.8 to
find $E\in\mathcal{F}$ such that for each $n\in\omega$ there is a finite set
$K\subseteq\Omega$ with $\rho(E\setminus
K)\subseteq\rho(F)\cap\bigcup\\{E_{i}:i\geq n\\}$. We claim that
$f\restriction\rho(E)$ is $\rho$-convergent to $\infty$. Let $U$ be a
neighborhood of $\infty$. Without loss of generality, we can assume that
$U=\Phi(\mathcal{A})\setminus(\\{A_{i}:i<n\\}\cup\bigcup_{i<n}A_{i})$ for some
$n\in\omega$. Let $K\subseteq\Omega$ be a finite set such that
$\rho(E\setminus K)\subseteq\rho(F)\cap\bigcup\\{E_{i}:i\geq n\\}$. Then
$f[\rho(E\setminus K)]\cap\\{A_{i}:i<n\\}=\emptyset$, and consequently
$f[\rho(E\setminus K)]\subseteq U$.

_Case (3a)._ There is $F\in\mathcal{F}$ such that $f\restriction\rho_{1}(F)$
is constant, hence it is $\rho$-convergent.

_Case (3b)._ Let $A\in\mathcal{A}$ be such that
$f^{-1}[A]\notin\mathcal{I}_{\rho}$. Using Proposition 6.8, we find