|  Universidad Nacional Autónoma de México y Universidad Michoacana de San Nicolás de Hidalgo |
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Posgrado Conjunto en Ciencias Matemáticas UMSNH-UNAM

T E S I S
que para obtener el grado de

_Doctor en Ciencias Matemáticas_

presenta

<EMAIL_ADDRESS>

Asesores:

Osvaldo Guzmán González

Michael Hrušák

Morelia, Michoacán, México

Junio, 2024

## Agradecimientos

Puedo decir sin lugar a dudas que estoy donde estoy, y soy quien soy, gracias
a toda la gente que me ha amado y apoyado a lo largo de los años. Entre dicha
gente, el lugar mas especial lo ocupa mi familia en constante crecimiento .
Les reitero mi gratitud, ya que sin ustedes nada de esto hubiera sido posible.

Fue por culpa Manuel Lara y Jonas Martínez que conocí y me enamoré de la
Teoría de Conjuntos. Gracias a Manuel, tambien conocí, al menos de nombre, a
mis tres mayores heroes en el mundo de la Teoría de Conjuntos: Jonathan
Cancino, Osvaldo Guzmán y Michael Hrušák. A mitad de mi licenciatura tuve la
oportunidad de tomar clase con Osvaldo y Jonathan, y desde ese momento
reafirme mi deseo, no solo de hacer un posgrado en Teoría de conjuntos, sino
de hacerlo bajo la tutela de Michael. Afortunadamente sucedio. Durante estos
ultimos 4 años, Michael y Osvaldo fueron mi asesores en el Doctorado. Les
agradezco profundamente por su paciencia, comprension, empatía y por todo lo
que pude aprender de ustedes a lo largo de este tiempo.

Un Agradecimiento especial merecen tambien mis sinodales: David Fernández,
Fernando Hernández y Reynaldo Rojas. Sus valiosas aportaciones mejoraron esta
tesis sustancialmente. De igual manera, siento la necesidad de agradecer a
aquellos profesoras y profesores que más impacto tuvieron en mi formación
matemática: Natalia Jonard, Jorge Marcos Martínez, Gabriela Campero, Ángel
Tamariz, Diana Avella, David Meza, Luis Paredes, Pierre Bayard, Javier Páez,
Luis Turcio, Rodrigo Hernández, Ariet Ramos, Leonardo Salmerón y Luis Nava.

Creo firmemente que el One Piece si son los amigos que hicimos en el camino.
Le agradezco a Richi por su incondicional amistad a lo largo de todos estos
años, al igual que por la interminable ayuda que me ha brindado. Él, sin duda,
ocupa un lugar muy especial dentro de estos agradecimientos. En igual medida
le agradezco a Marcos y Uhthoff. Me llena de felicidad saber que las amistades
pueden perdurar tanto tiempo. Durante mi estancía en el $CCM$ conoci a grandes
amigos. David Valencía, por ejemplo, llegó al punto de llevarme atole a mi
casa una vez que estaba triste. Por todo esto y más, le estoy muy agradecido.
Definitivamente, no hubiera divertido tanto como lo hice si no hubiera sido
por los amigos que conoci durante esta etapa de mi vida: Norberto, Miguel,
Mario, Yhon, Carlos, Emmanuel, Kevin, Yulu, Sonia, Angel, Francisco, Tero,
Jesus, Daniel, Tristan, Tatsuya, Julia, y muchos más. Otro agradecimiento
especial se lo debo a Cesar, pues además de darme su amistad, fue mi guia y
ejemplo a seguir durante gran parte de mi posgrado. Él, Thelma, Mati y Dony me
adoptaron cuando era un vagabundo en las frias calle de Toronto. Por un mes,
me hicieron parte de su familia, me dieron comida y techo, y jamás me pidieron
nada a cambio. A ustedes cuatro les estaré eternamente agradecido.

Agradezco de nuevo a Tero, ya que sin su ayuda este trabajo no tendría la
calidad que tiene. El me pasó la plantilla con la que hizo su tesis, y durante
estos dos últimos años no dudó en resolver al instante cualquier duda que
llegue a tener relacionada a Latex.

Por último, me gustaría agradecer a Stevo Todorčević. Primero, por las bellas
matemáticas que ha desarrollado a lo largo de los años, sin la cuales, esta
tesis jamás habría existido. Tambien le agradezco por la oportunidad que me
dio de poder trabajar junto con él y Osvaldo en esquemas de construcción.
Durante el desarrollo de este proyecto, Stevo no solo ha sido coautor, también
ha sido un gran mentor y maestro. Muchas gracias Stevo.

La investigación de este trabajo se realizó en parte gracias al apoyo
económico brindado por el gobierno de México, por parte del Consejo Nacional
de Humanidades, Ciencias y Tecnologías (CONAHCyT). Por esto y más, les estoy
eternamente agradecido.

## Abstract

### Resumen

> En esta tesis se desarrollará un análisis estructural de los esquemas de
> construción introducidos en [112]. La importancia de este estudio se verá
> reflejada al construir una gran cantidad de objetos combinatorios distintos
> que han sido de gran interés en las matemáticas. Tambien se continuará con
> el estudio de los axiomas de captura asociados a esquemas de construcción.
> De dichos axiomas se construirán varios objetos cuya existencia se sabe
> independiente de los axiomas usuales de la teoría de conjuntos.

> Palabras Clave: uncountable, construction scheme, diamond principle,
> morasses, capturing, trees, gap, coloring.

### Abstract

> In this thesis, a structural analysis of construction schemes (as introduced
> in [112]) is developed. The importance of this study will be justified by
> constructing several distinct combinatorial objects which have been of great
> interest in mathematics. We then continue the study of capturing axioms
> associated to construction schemes. From them, we construct several
> uncountable structures whose existence is known to be independent from the
> usual axioms of Set Theory.

> Keywords: uncountable, construction scheme, diamond principle, morasses,
> capturing, trees, gap, coloring.

###### Contents

  1. Abstract
  2. Introduction
  3. 1 Preliminaries and Notation
    1. 1.1 Set-Theoretic Notation
    2. 1.2 Order-Theoretic Notation
    3. 1.3 Boolean algebras and filters
    4. 1.4 Forcing
    5. 1.5 P-ideal Dichotomy
    6. 1.6 Diamond principle
  4. 2 Ordinal metrics and construction schemes
    1. 2.1 Ordinal metrics
    2. 2.2 Construction schemes
    3. 2.3 Two more canonical functions
    4. 2.4 Capturing Axioms
  5. 3 Applications
    1. 3.1 Gaps, towers and almost disjoint families
      1. 3.1.1 Donut-separability
      2. 3.1.2 The gap cohomology group
      3. 3.1.3 Destructibility of gaps
    2. 3.2 Trees and lines
      1. 3.2.1 Countryman lines and Aronszajn trees
      2. 3.2.2 Suslin trees
      3. 3.2.3 Suslin lattices
      4. 3.2.4 Entangled sets
    3. 3.3 Colorings at the uncountable
      1. 3.3.1 ccc-polychromatic coloring
    4. 3.4 Oscillation theory of $2$-capturing schemes
      1. 3.4.1 A sixth Tukey type
      2. 3.4.2 A Suslin tower
      3. 3.4.3 S and L spaces
      4. 3.4.4 Failure of Baumgartner’s Axiom
    5. 3.5 On $\sigma$-monotone spaces
  6. 4 Fragments of Martin’s axiom
    1. 4.1 The principle $\mathfrak{m}^{n}_{\mathcal{F}}$
    2. 4.2 The $n$-projection filter
    3. 4.3 $n$-capturing with partitions
    4. 4.4 Ramsey ultrafilters from construction schemes
  7. 5 On the existence of construction and capturing schemes
    1. 5.1 The forcing $\mathbb{P}(\mathcal{F})$ and schemes in ZFC
      1. 5.1.1 Construction schemes in ZFC
    2. 5.2 Diamond principle and FCA(part)
      1. 5.2.1 Forcing the property $IH_{2}$
    3. 5.3 P-ideal Dichotomy and capturing axioms
  8. 6 Open problems
    1. 6.1 Problems about gaps
    2. 6.2 Problems about trees and lines
    3. 6.3 More problems regarding applications
    4. 6.4 Problems about construction schemes

## Introduction

Throughout history, infinity has captivated and amazed humanity, sparking
curiosity and contemplation among generations. A fabulous trip lasting more
than 2000 years lead, in the 19th century, to beautiful results regarding this
concept. Georg Cantor delved deeper into the nature of infinity by proving,
among other things, that infinite sets may have different sizes. Formally, he
showed that there can not be a bijective function from the natural to the real
numbers. Since then, a great part of effort has been put into understanding
the mysteries lying in the realm surrounding the two aforementioned
infinities. In this thesis, we will explore $\omega_{1}$, the first
uncountable cardinal lying in between $\omega$ and the hill of treasures that
are the real numbers.

The purpose of this thesis is to study $\omega_{1}$ by further developing the
theory of construction schemes as introduced by Stevo Todorčević in [112]. By
means of descriptive set-theoretic results, it is often impossible to deduce
the existence of interesting objects of size $\omega_{1}$ by presenting them
throughout nice and simple definitions. For that reason, there has been a lot
of interest in presenting and studying methods for constructing such objects:

  * $\blacksquare$

Probably the most straightforward approach is to build them by transfinite
recursion in $\omega_{1}$ many steps. Typical examples of this method are the
construction of an Aronszajn tree from [54] or [64] and the construction of a
Hausdorff gap from [91] or [125].

  * $\blacksquare$

A second approach would be to show the consistency of an uncountable object
with the desired properties, analyze the complexity of the sentences defining
such properties, and then appeal to Jerome H. Keisler’s completeness theorem
for $L^{\omega}_{\omega}(\mathcal{Q})$ (see [61]) to conclude that such an
object in fact exists. An example of this method is the coherent family of
functions from [36]. A variation of this method involving the
$\Diamond$-principle was presented by Menachem Magidor and Jerome Malitz in
[72].

  * $\blacksquare$

A third approach is by using the method of walks on ordinals and ordinal
metrics as introduced by Tordorčević (see [110]). Historically, this method
has been proved to be one of the most useful when dealing with constructions
of uncountable objects. Examples of this are the solution of the famous
$L$-space problem from [76] and the construction of a rainbow coloring from
[120].

  * $\blacksquare$

A fourth approach, which is arguably the most related to the present work, is
to construct the desirable uncountable object by finite approximations through
the use of simplified morasses as introduced by Ronald Jensen (see [26]) and
later simplified by Daniel Velleman in [128].

Just as simplified morasses, construction schemes are objects that let us
build uncountable objects through finite approximations. Roughly speaking, a
construction scheme $\mathcal{F}$ is a collection of finite subsets of
$\omega_{1}$ with strong coherent properties (see Definition 2.2.2). These
coherent properties allow us to build directed families of finite structures
indexed by the elements of the scheme. Such structures are built recursively
by amalgamation processes dictated by the construction scheme. In the end, the
uncountable object that we construct is in some sense the direct limit of the
family that we defined. The main difference between construction schemes and
morasses is that in morasses we only amalgamate two structures at a given
time, whereas in construction schemes, the number of amalgamations may vary.
That same attribute which sets appart construction schemes and morasses, allow
construction schemes to consistently have further properties, powerful enough
to imply the existence of a large amount of objects which are known to exist
under extra set-theoretic assumptions. These properties, which we call
$n$-capturing, capturing and fully capturing, can be described as
finitizations (relative to a construction scheme) of the well-known
$\Diamond$-prinicple (see 2.4.8). The capturing axioms $CA_{n}$, $CA$ and
$FCA$ assert the existence of construction schemes which are $n$-capturing,
capturing and fully capturing respectively. All of these axioms are implied by
the $\Diamond$-principle and also hold in any forcing extension of the
universe obtained by adding at least $\omega_{1}$ many Cohen reals.
Given an $n$-capturing construction scheme $\mathcal{F}$ we also can define
the $n$-parameterized Martin’s number associated to it as
$\mathfrak{m}^{n}_{\mathcal{F}}$. This cardinal invariant is defined as the
Martin’s number corresponding to the family of $ccc$-forcing notions which
force our construction scheme to be $n$-capturing. Even though this cardinal
is strictly related to a combinatorial structure over $\omega_{1}$ (the
construction scheme), the assertion
$\mathfrak{m}^{n}_{\mathcal{F}}>\omega_{1}$ is strong enough to imply the
existence of objects such as Ramsey ultrafilters over $\omega$. Furthermore,
as we will show, there are meaningful statements about uncountable objects
which Martin’s axiom can not decide, that are true under
$\mathfrak{m}^{n}_{\mathcal{F}}>\omega_{1}$, and which are false under axioms
such as $PFA$, and $CH$ (so in particular, $\Diamond$-principle).

### Structure of the thesis

In this thesis we will develop to a great extent the theory of construction
schemes and their structural properties, and we will obtain important results
about set-theoretic assumptions associated with them. The importance of this
study will be justified by constructing several distinct combinatorial objects
which have been of a great interest to set theorists and topologists. A large
amount of these objects will be constructed in a rather different form from
the original constructions, and some of them will make their first appearance
here.

In Chapter 1 we will present some of the preliminaries, and fix the notation
that will be used through out the thesis.

Chapter 2 is intended to serve as an introduction to the theory of ordinal
metrics and construction schemes over $\omega_{1}$. In this chapter, we will
define these two notions and analyze their general behavior. In particular, we
will prove that construction schemes are in a one-to-one correspondence
between certain kinds of ordinal metrics. We will also define the notions of
$n$-capturing, capturing, and fully capturing construction schemes, as well as
the capturing axioms $CA_{n}$, $CA$, and $FCA$ associated to them. Finally, we
will introduce the parametrized Martin’s axioms
$\mathfrak{m}^{n}_{\mathcal{F}}$ associated to $n$-capturing construction
schemes. Along the way, we will present the main theorems regarding the
existence of distinct types of construction schemes. Nevertheless, such
theorems will be proved and discussed in further detail until Chapters 4 and
5.

In Chapter 3 we will provide a handful of applications of construction schemes
both in $ZFC$ and under extra set-theoretic assumptions. These applications
are of interest in set theory, topology, infinite combinatorics, algebra, and
analysis. The constructions are mostly independent of each other, so that the
reader can start from the ones that she or he finds more interesting. For the
convenience of the reader, we list here some of the constructions that appear
in this chapter. References for further study and historic remarks will be
provided as we encounter them.

  1. 1.

Hausdorff gaps An interesting feature of the Boolean algebra
$\mathscr{P}(\omega)/\text{FIN}$ is that it is not complete. The easiest way
to see this is as follows: Let $\\{A_{\alpha}\,:\,\alpha\in
2^{\omega}\\}\subseteq[\omega]^{\omega}$ be an $AD$ family. For every
$H\subseteq 2^{\omega}$, consider the set
$\mathcal{A}_{H}=\\{[A_{\alpha}]\,:\,\alpha\in H\\}$ (where $[A_{\alpha}]$ is
the class of $A_{\alpha}$ in $\mathscr{P}(\omega)/\text{FIN}$). It follows
form a simple counting argument that there must be an $H\subseteq 2^{\omega}$
such that $\mathcal{A}_{H}$ has no supremum. Although this is a very simple
argument, we are often interested in more concrete examples of the
incompleteness of $\mathscr{P}(\omega)/\text{FIN}$. The nicest examples are
provided by gaps. Moreover, gaps are important because they represent
obstructions that we may encounter when embedding structures in
$\mathscr{P}(\omega)/\text{FIN}$ (a very illustrative example of this
situation is Theorem 8.8 of [121]). The classic construction of a Hausdorff
gap requires a very clever argument to take care of $2^{\omega}$ many tasks in
only $\omega_{1}$ many steps. With construction schemes, we will be able to
provide a very simple construction of a Hausdorff gap, which under extra set-
theoretic assumptions satisfies some interesting and (until now) unstudied
combinatorial properties.

  2. 2.

Luzin-Jones almost disjoint families. This topic is about almost disjoint
families (AD) and their separation properties. A Luzin family is an $AD$
family of size $\omega_{1}$ in which no two uncountable subfamilies can be
separated. On the other hand, a Jones family is an $AD$ family with the
property that every countable subfamily of it can be separated from its
complement. It is easy to prove that both of this kind of families exist.
However, building a family that is both Luzin and Jones at the same time is
much more complicated, since there is a tension between this two properties. A
very difficult and highly complex construction of a Luzin-Jones family appears
in [42]. With construction schemes, we will be able to build an almost
disjoint family satisfying strong properties related to a concept that we will
call the Luzin representation of a partial order. From these properties we
will not only be able to show that the constructed family is in fact Luzin-
Jones, but from it, we will derive the existence of a handful of objects
related to gaps. In particular, we will generalize, in strong way, the results
of Hausdorff regarding the existence of gaps in
$\mathscr{P}(\omega)/\text{FIN}$. It is worth pointing out that Luzin-Jones
families can also be used to build interesting examples in functional analysis
(see [39]).

  3. 3.

Donut-inseparable gaps. Naively speaking, we say that gap over
$\mathscr{P}(\omega)\backslash\text{FIN}$ is donut-inseparable if it can be
reconstructed from two disjoint inseparable subfamilies of an $AD$ family. We
will show, without assuming any extra axioms, that there are some gaps which
are donut-inseparable and some which are not. This result suggest the
following weakining of the previously defined notion: A gap is strongly donut-
separable if each cofinal subgap is donut-separable. With the help of
construction schemes and their parametrized Martin’s axioms, we will show that
the statement There are no strongly donut-separable gaps is independent from
$ZFC$.

  4. 4.

Gap cohomology groups. In [105], Talayco defined what it means for two
$(\omega_{1},\omega_{1})$-gaps to be cohomologically different. This notion
allowed him to define the gap cohomology group of an $\omega_{1}$-tower. We
will generalize the notion of the gap cohomology group to a greater class of
substructures of $\mathscr{P}(\omega)/\text{FIN}$ and show that all these
groups can be as big as possible.

  5. 5.

Luzin coherent family of functions. We now look at a generalization of the
Hausdorff gaps discussed previously. A Luzin coherent family of functions is a
coherent system of functions supported by a pretower, in which we impose a
strong non triviality condition. The importance of these families is that they
provide many cohomologically different gaps. They were first studied by
Talayco in [105]. Later Farah proved that such families exist (see [36]). The
proof of Farah is highly non-constructive and indirect, since it appeals to
Keisler’s completeness Theorem. We will build such families using a
construction scheme. No previous direct construction was known.

  6. 6.

Independent coherent family of functions. We now return to the study of gaps
that are obtained from a coherent family of functions. However, this time we
want our family of gaps to be independent. This means that we can either fill
or freeze any subfamily without filling or freezing any of the remaining gaps
in the family. A similar result was obtained by Yorioka assuming the
$\Diamond$-principle in [133] (the analogue result for Suslin trees was proved
by Abraham and Shelah in [3]).

  7. 7.

Countryman lines. Let $(X,<)$ be a total linear order. Except for the trivial
cases $X^{2}$ is not a linear order. In this way, it makes sense to ask how
many chains we need to cover it. A Countryman line is an uncountable linear
order whose square can be covered with only countably many chains. These
orders seem so paradoxical at first glance that Countryman conjectured they do
not exist. However, it was first proved by Shelah that Countryman lines do
exist (see [93]).

  8. 8.

Aronszajn trees. Aronszajn trees are the most well-known examples of the
incompactness of $\omega_{1}$. An Aronszajn tree is a tree of height
$\omega_{1}$, its levels are countable, yet it has no cofinal branches. A
simple way in which we can guarantee that a tree (of height $\omega_{1}$) has
no cofinal branches is to make it special, which means that it can be covered
with countably many antichains (equivalently, they can be embedded in the
rational numbers). Aronszajn proved this kind of trees exists.

  9. 9.

Suslin trees. A Suslin tree is an Aronszajn tree in which every antichain is
countable. The Suslin Hypothesis ($SH$) is the statement that there are no
Suslin trees. We now know that $SH$ is independent from $ZFC$. A related
concept, the Suslin lines, were introduced by Suslin while studying the
ordering of the real numbers. Kurepa was the one to realize that there is a
Suslin line if and only if there is a Suslin tree. Applications and
constructions from Suslin trees are abundant in the literature. We will use
capturing schemes to build two types of these trees: Coherent Suslin and full
Suslin trees. These two families of trees are diametrically opposed. While
forcing with a Coherent tree, it completely destroys the $ccc$-ness of it,
while with a full Suslin tree, many subtrees of it remain $ccc$.

  10. 10.

Suslin lower semi-lattices. If in the definition of a Suslin tree, we relax
the condition of being a tree to just a being a lower semi-lattice, we get the
notion of a Suslin lower semi-lattice. They were introduced by Dilworth, Odell
and Sari (see [28]) in the context of Banach spaces. They were then studied by
Raghavan and Yorioka (see [86]). Among other things they proved that the
$\Diamond$-principle implies that $\mathscr{P}(\omega)$ contains a Suslin
lower semi-lattice. We were able to obtain the same result from a
$2$-capturing scheme.

  11. 11.

Entangled sets. A well-known theorem of Cantor is that any two countable dense
linear orders with no end points are isomorphic. The straightforward
generalization to linear orders of size $\omega_{1}$ is false. For this reason
we want to restrict to suborders of the real numbers. We say that an
uncountable $B\subseteq\mathbb{R}$ is $\omega_{1}$-dense if
$|U\cap\mathbb{B}|=\omega_{1}$ whenever $U$ is an open interval whose
intersection with $U$ is nonempty. A remarkable theorem of Baumgartner is that
$PFA$ implies that any two $\omega_{1}$-dense sets of reals are isomorphic
(see [11]). This statement is now known as Baumgartner axiom
($BA(\omega_{1})$). An entangled set is a subset of $\mathbb{R}$ with very
strong combinatorial properties. The existence of an entangled set implies the
failure of the Baumgartner axiom. Entangled sets were introduced by Abraham
and Shelah in order to show that $BA(\omega_{1})$ does not follow from
Martin’s axiom (see [7]). We will use a capturing scheme to build an entangled
set. In this way, the existence of certain capturing schemes contradict the
Baumgartner axiom.

  12. 12.

$ccc$ destructible $2$-bounded coloring without injective sets. A coloring
$c:[\omega_{1}]^{2}\longrightarrow\omega_{1}$ is called $2$-bounded if every
color appears at most $2$ times. A set $A\subseteq\omega_{1}$ is $c$-injective
if no color appears twice in $[A]^{2}$. Galvin was the first to wonder if
there is a $2$-bounded coloring without an uncountable injective set. He
proved that such coloring exists assuming the Countinuum Hypothesis. On the
other hand, the third author proved that no such coloring exist under $PFA$.
years later, Abraham, Cummings and Smyth proved that $MA$ is consistent with
the existence of a $2$-bounded coloring without uncountable injective sets
(see [2]). After hearing this result, Friedman asked for a concrete example of
a $2$-bounded coloring without an uncountable injective set, but that such set
can be added with a $ccc$ partial order. In [2] such example is constructed
assuming $CH$ and the failure of the Suslin hypothesis. We will find an
example with a $3$-capturing scheme.

  13. 13.

Oscillation theory of $2$-capturing construction schemes In the book [121],
Todorčević developed an oscillation theory which is based on an unbounded
family of functions. A plethora of applications of this theory have been found
throughout the years. For example, the oscillation theory is key in the proof
that $PFA$ implies that the continuum is $\omega_{2}$ (see [12]). We will
develop a similar theory using a $2$-capturing scheme. An important difference
between the classic oscillation theory and the one from capturing schemes, is
that this new one is based on a bounded family of functions. Using this new
oscillation theory, we can prove the existence of the following objects:

    1. (a)

Sixth Tukey type. The Tukey ordering is a useful tool to compare directed
partial orders. Its purpose is to study how a directed partial order behave
cofinally. It was introduced by Tukey in [124] in order to study convergence
in topology. The Tukey classification of countable directed partial orders is
very simple: Every countable directed partial order is Tukey equivalent to $1$
or to $\omega$. The Tukey classification of directed sets of size $\omega_{1}$
becomes much more interesting. We now have at leas five Tukey types: $1$,
$\omega$, $\omega_{1}$, $\omega\times\omega_{1}$ and $[\omega_{1}]^{<\omega}$.
We may wonder if there is a directed partial order of size $\omega_{1}$ that
is not Tukey equivalent to one if this five. In [40], Isbell proved that $CH$
entails the existence of a sixth Tukey type. This was greatly improved by
Todorčević in [117], where he proved that $CH$ implies that there are
$2^{\omega_{1}}$ distinct Tukey types. On the other hand, in the same paper he
showed that $PFA$ implies that there are no sixth Tukey types. Here, we found
a sixth Tukey type from a capturing scheme.

    2. (b)

$\omega_{1}\not\rightarrow(\omega_{1},\omega+2)^{2}_{2}$. Given
$\alpha<\omega_{1}$, the partition
$\omega_{1}\rightarrow(\omega_{1},\alpha)^{2}_{2}$ means that for every
$c:[\omega_{1}]^{2}\longrightarrow 2$, either there is an uncountable
$0$-monochromatic set, or there is a $1$-monochromatic set of order type
$\alpha$. A celebrated result of Erdös and Rado (extending a theorem by
Dushnik, Miller and Erdös) is that
$\omega_{1}\rightarrow(\omega_{1},\omega+1)^{2}_{2}$. We may wonder if this
theorem can be improved. This turns out to be independent from $ZFC$. The
Proper Forcing Axiom implies that
$\omega_{1}\rightarrow(\omega_{1},\alpha)^{2}_{2}$ for every
$\alpha<\omega_{1}$ (see [111]), while $\mathfrak{b}=\omega_{1}$ implies
$\omega_{1}\not\rightarrow(\omega_{1},\omega+2)^{2}_{2}$. We will prove a
similar result from our oscillation theory.

    3. (c)

Non productivity of $ccc$ partial orders. When is the product of two $ccc$
partial orders again $ccc$? This question has been of interest to set
theorists for a long time. On one hand, Martin’s axiom implies that the
product of $ccc$ partial orders is $ccc$. On the other hand, the failure of
the Suslin Hypothesis implies the opposite. Consistent examples of two $ccc$
partial orders whose product is not $ccc$ have been constructed by Galvin
under $CH$ (see [64]) and by Todorčević under $\mathfrak{b}=\omega_{1}$ in
[121]. As an applications of the oscillation theory that we developed, we
encounter new situations in which there are $ccc$ partial orders whose product
is not $ccc$.

    4. (d)

Suslin towers. In [108], Todorčević developed an analogue of his oscillation
theory, now this time based on non-meager towers. While studying this
oscillation, Borodulin-Nadzieja and Chodounský introduced the notion of a
Suslin tower. A tower $\mathcal{T}=\\{T_{\alpha}\,:\,\alpha\in\omega_{1}\\}$
is Suslin if every uncountable subset of $\mathcal{T}$ contains two pairwise
$\subseteq$-incomparable elements. The existence of a Suslin tower is
independent from $ZFC$. It can be proved that the Open Graph Axiom forbids the
existence of such families, while a Suslin tower can be constructed assuming
$\mathfrak{b}=\omega_{1}$ (see [13]). We will build a Suslin tower as an
application of the oscillation theory obtained from a $2$-capturing scheme.

    5. (e)

S-spaces. Hereditarily separable and hereditarily Lindelöf are two properties
that in some sense are dual to each other. We may wonder if they always
coincide. The question is only of interest in the realm of regular spaces. An
$S$-space is a regular hereditarily separable, non Lindelöf space. The study
of $S$-spaces (and the dual notion, $L$-spaces) used to be one of the most
active areas of set-theoretic topology. $S$-spaces can be constructed under
several set-theoretic hypothesis (like $CH$ or the negation of the Suslin
Hypothesis). It was Todorčević who for the first time succeeded in proving
that it is consistent that there may be no $S$-spaces (see [121]). Here, we
will apply the oscillation theory obtained from a $2$-capturing scheme to
construct several $S$-spaces.

  14. 14.

$\sigma$-monotone spaces. Monotone and $\sigma$-monotone spaces are a
particular kind of metric spaces which were defined in [83] by Aleš Nekvinda
and Ondřej Zindulka. In [135], Zindulka used such spaces to prove the
existence of universal measure zero sets of large Hausdorff dimension. Here,
we will show that the capturing axiom $CA$ implies the existence of a metric
space of cardinality $\omega_{1}$ which has no uncountable monotone subspaces.

In Chapter 4 we will study the effect that forcing with $ccc$ posets has on
the capturing properties related to construction schemes. This is done for two
purposes: The first one is to prove that the notions of capturing and
capturing with partitions are different, and the second one is to show that
the inequality $\mathfrak{m}_{\mathcal{F}}>\omega_{1}$ is consistent with
$ZFC$. Lastly, we will prove that under this later assumption, we can deduce
the existence of Ramsey ultafilters over $\omega$. A particularity of these
results is that such ultrafilters can be explicitly defined from a
combinatorial structure over $\omega_{1}$ (i.e., a construction scheme).

In Chapter 5 we will define the forcing $\mathbb{P}(\mathcal{F})$ and use it
to construct construction schemes without assuming any extra axioms. Later we
will prove that Jensen’s $\Diamond$-principle implies the capturing axiom
$FCA(part)$. This was first claimed in [112]. However, the sketch of the proof
given there is incomplete. Finally, we will show that $PID$ implies the
nonexistence of $2$-capturing construction schemes.

Finally, in Chapter 6, we briefly discuss some open problems and future lines
of research.

The following chart summarizes many of the constructions which can be carried
out with a construction scheme, and that will appear in this thesis. All of
the results stated in the chart are new contributions.

Object |  Extra set-theoretic assumption |  Result number
---|---|---
Donut-separable gap |  - |  3.1.25
Donut-inseparable gap |  - |  3.1.22
Almost disjoint family coding any $\omega_{1}$-like order |  - |  3.1.13 and 3.1.20
$(X,Y)$-gap for all $\omega_{1}$-like orders |  - |  3.1.21
$*$-Lower semilattices with large gap cohomology groups |  - |  3.1.69
Not strongly donut-separable Hausdorff gap |  $\mathfrak{m}_{\mathcal{F}}>\omega_{1}$ |  3.1.36
Ramsey ultrafilter |  $\mathfrak{m}_{\mathcal{F}}>\omega_{1}$ |  4.4.1 and 4.4.8
Rainbow coloring $o^{*}$ |  $CA_{2}$ |  3.4.9
Sixth Tukey type |  $CA_{2}$ |  3.4.20
$\omega_{1}\not\rightarrow(\omega_{1},\omega+2)^{2}_{2}$ |  $CA_{2}$ |  3.3.5
Non productivity of $ccc$ |  $CA_{2}$ |  3.4.10
Suslin towers |  $CA_{2}$ |  3.4.22
$S$-spaces |  $CA_{2}$ |  3.4.29, 3.4.32 and 3.4.46
Suslin lower semi-lattices |  $CA_{2}$ |  3.2.26
Failure of $BA(\omega_{1})$ |  $CA_{2}$ |  3.4.54
$ccc$ destructible $2$-bounded coloring |  $CA_{3}$ |  3.3.9
Uncountable metric space without uncountable monotone subspaces |  $CA$ |  3.5.7
Coherent Suslin Tree |  $FCA(part)$ |  3.2.18
Full Suslin Tree |  $FCA$ |  3.2.22
Entangled sets |  $FCA$ |  3.2.33
Independent coherent family of functions |  $FCA$ |  3.1.88

In addition to applications, the main contributions of this work are listed
below. All of the undefined notions will be defined throught the thesis.

Result |  Extra set-theoretic assumption |  Result number
---|---|---
There are no strongly donut-separable gaps Either |  $PFA$ or $CH$ |  3.1.39 and 3.1.53
MA is independent from the statement There is a strongly donut-separable gap |  - |  3.1.54
There are no destructible gaps |  $\mathfrak{m}_{\mathcal{F}}>\omega_{1}$ |  3.1.82
There are no Suslin trees |  $\mathfrak{m}_{\mathcal{F}}>\omega_{1}$ |  3.2.24
$\mathfrak{c}=\mathfrak{m}_{\mathcal{F}}^{n}>\omega_{1}$ is consistent with an arbitrarily large continuum |  - |  4.1.8
Consistency of the existence of a construction scheme $\mathcal{F}$ which is $n+1$-capturing, $\mathcal{P}$-capturing, but not $\mathcal{P}^{\prime}$-capturing |  - |  4.3.9
There are no $2$-capturing construction schemes |  Either $\mathfrak{m}>\omega_{1}$ or $PID$ |  4.3.4 and 5.3.12
Full capturing with partitions axiom ($FCA(part)$) |  $\Diamond$ |  5.2.31

Most of the results presented here were obtained in a joint work with Osvaldo
Guzmán and Stevo Todorčević. Part of them appear in [24] which has already
been submitted.

##  Chapter 1 Preliminaries and Notation

We devote this chapter to fix most of the preliminaries, notation and
terminology that will be used throughout the rest of this work. Broadly
speaking, the results appearing in this text can be separated into two
categories. The ones regarding the development of the theory of construction
schemes and the ones regarding applications of this theory to set theory and
topology. Of course, familiarity with the basics of these two branches of
mathematics is assumed. Nothing more is asked for the reader solely interested
in applications. However, familiarity with the forcing method is mandatory for
the reader interested in learning more about the development of the theory.
For set theory, the texts Halbeisen [44], Jech [54] and Kunen [64] provide, in
particular, all the prerequisites needed. The texts Engelking [34], Nagata
[82] and Willard [131] do the same for the topological background.

###  1.1 Set-Theoretic Notation

Ordinals are usually denoted by the first lower case Greek letters
$\alpha,\beta,\gamma,\delta,\dots$, whereas the middle letters
$\kappa,\lambda,\mu,\dots$ are reserved for cardinals (commonly infinite). The
set of natural numbers is denoted by $\omega$ and its elements are mainly
denoted by the lowercase letters in the English alphabet. $\omega_{1}$ stands
for the first uncountable ordinal. The set of all limit ordinals smaller than
$\omega_{1}$ is denoted by Lim.

We denote the power set of $X$ by $\mathscr{P}(X)$. The cardinality of $X$ is
denoted by $|X|$. Given a cardinal $\kappa$, we let $[X]^{\kappa}$ be the set
of all subsets of $X$ of size $\kappa$ and $[X]^{<\kappa}$ denotes
$\bigcup_{\alpha<\kappa}[X]^{\alpha}$. The set $[X]^{\leqslant\kappa}$ is
defined in a similar way. By $\text{FIN}(X)$ we mean
$[X]^{<\omega}\backslash\\{\emptyset\\}$. $\text{FIN}(\omega)$ is just denoted
as FIN. We write $X\subseteq^{*}Y$ if $X\backslash Y$ is finite and $X=^{*}Y$
if $X\subseteq^{*}Y$ and $Y\subseteq^{*}X$. Following this notation, we write
$X\subsetneq^{*}Y$ whenever $X\subseteq^{*}Y$ but $X\not=^{*}Y$.

######  Definition 1.1.1 ($\Delta$-system).

A family $\mathcal{D}$ is called a $\Delta$-system with root $R$ if
$|\mathcal{D}|\geqslant 2$ and $X\cap Y=R$ whenever $X,Y\in\mathcal{D}$ are
different.

Let $f:X\longrightarrow Y$ be a function. As usual, for $x\in X$ we denote the
image of $x$ under $f$ as $f(x)$. For $A\subseteq X$ and $B\subseteq Y$ the
direct image of $A$ under $f$, that is, $\\{f(x)\,:\,x\in A\\}$, is denoted as
$f[A]$ and the inverse image of $B$, that is, $\\{x\in X\,:\,f(x)\in B\\}$ is
denoted by $f^{-1}[B]$. Sometimes we write $dom(f)$ instead of $X$ $im(f)$
instead of $f[X]$. Whenever we write $g;X\longrightarrow Y$ we mean that $g$
is a function with $dom(g)\subseteq X$ and $im(g)\subseteq Y$. The set of all
functions from $X$ to $Y$ is denoted by $X^{Y}$. For an ordinal $\beta$ we
define $X^{<\beta}$ and $X^{\leqslant\beta}$ as
$\bigcup_{\alpha<\kappa}X^{\alpha}$ and $X^{<\beta}\cup X^{\beta}$
respectively.

###  1.2 Order-Theoretic Notation

Let $(X,<)$ be a partial order and $x,y\in X$. We say $x$ and $y$ are
comparable if either $x\leqslant y$ or $y\leqslant x$. We say $x$ and $y$ are
compatible and write it as $x\parallel y$ if there is $r\in X$ with both
$r\leqslant x$ and $r\leqslant y$. The incompatibility relation between $x$
and $y$ is written as $x\perp y$. We define the following sets:

$\displaystyle(-\infty,x)_{X}$ $\displaystyle=\\{z\in X\,:\,z<x\\}$
$\displaystyle(-\infty,x]_{X}$ $\displaystyle=\\{z\in X:\,z\leqslant x\\}$
$\displaystyle(x,\infty)_{X}$ $\displaystyle=\\{z\in X\,:\,x<z\\}$
$\displaystyle[x,\infty)_{X}$ $\displaystyle=\\{z\in X\,:\,x\leqslant z\\}$
$\displaystyle(x,y)_{X}$ $\displaystyle=(x,\infty)\cap(-\infty,y)$
$\displaystyle(x,y]_{X}$ $\displaystyle=(x,\infty)\cap(-\infty,y]$
$\displaystyle[x,y)_{X}$ $\displaystyle=[x,\infty)\cap(-\infty,y)$
$\displaystyle[x,y]_{X}$ $\displaystyle=[x,\infty)\cap(-\infty,y]$

We say that $A\subseteq X$ is an interval in $X$ whenever $(x,y)\subseteq A$
for all $x,y\in A$. $A$ is an initial segment of $X$ if $(-\infty,x)\subseteq
A$ for every $x\in A$. $A$ is a final segment of $X$ if $(x,\infty)\subseteq
A$ whenever $x\in A$. $A$ is cofinal in $X$ if for any $[x,\infty)\cap A$ is
non-empty for all $x\in X$. $A$ is a chain if any two elements of $A$ are
comparable. On the other hand, if any two elements of $A$ are incomparable, we
call $A$ a pie. Finally, $A$ is an antichain in $X$ if any two elements in $A$
are incompatible.

Let $B\subseteq X$ and $x\in B$. $x$ is said to be a minimal element of $B$ if
there is no $y\in B$ with $y<x$. Furthermore, if $x\leqslant y$ for any $y\in
B$ we say that $x$ is the minimum of $B$ and denote it as $\min(B)$.
Analogously, $x$ is said to be a maximal element of $B$ if there is no $y\in
B$ with $y>x$. Moreover, $x\geqslant y$ for any $y\in B$ we call $x$ the
maximum of $B$ and denote it as $\max(B)$. An element $z\in X$ is an upper
bound (resp. lower bound) of $B$ if $z=\max(B\cup\\{z\\})$ (resp.
$z=\min(B\cup\\{z\\})$. In case the set of upper bounds of $B$ has a minimum,
it is denoted as $\sup B$. In a similar way, if the set of lower bounds of $B$
has a maximum, it is denoted as $\inf B$. Lastly, for an element $x\in X$ we
denote by

$suc(x)=\\{y\in(x,\infty)\,:\,y\textit{ is a minimal element of
}(x,\infty)\,\\},$ $pred(x)=\\{y\in(-\infty,x)\,:\,y\textit{ is a maximal
element of }(-\infty,x)\,\\}.$

Note that $y\in suc(x)$ if $x<y$ and there is no $z\in X$ with $x<z<y$.
Analogously $y\in pred(x)$ if $y<x$ and there is no $z\in X$ such that
$y<z<x$. Let $A,B\subseteq X$ be arbitrary subsets of $X$. We say $A<B$ if $B$
is non-empty and $a<b$ for all $a\in A$ and $b\in B$. We say that $B$ end-
extends $A$ (or $A$ is end-extended by $B$) and write it as $A\sqsubseteq B$
if $A$ is an initial segment of $B$.

######  Definition 1.2.1 (Root-tail-tail $\Delta$-system).

A $\Delta$-system $\mathcal{D}\subseteq\mathscr{P}$ with root $R$ is said to
be root-tail-tail if $R\sqsubseteq A$ and either $A\backslash R<B\backslash R$
or $B\backslash R<A\backslash R$ for any two distinct $A,B\in\mathcal{D}$. In
this same situation, if $\mathcal{D}$ is indexed by some ordinal $\gamma$ as,
say, $\langle D_{\alpha}\rangle_{\alpha<\gamma}$ we will make the implicit
assumption that $D_{\alpha}\backslash R<D_{\beta}\backslash R$ whenever
$\alpha<\beta.$

$X$ is said to be well-founded if any non-empty subset of $X$ has a minimal
element. In this case, there is a unique function $rank:X\longrightarrow Ord$
satisfying:

$rank(x)=\sup(\,rank(y)+1\,:\,y<x\,)$

Note that this function is order preserving and its image is an ordinal. For
each ordinal $\alpha$, we define the level $\alpha$ of $X$ as
$X_{\alpha}=\\{x\in X\,:\,rank(x)=\alpha\\}$ and the height of $X$ as
$Ht(X)=\min(\,\alpha\in Ord\,:\,X_{\alpha}=\emptyset\,)=rank[X]$. $X$ is said
to be a well-order if any non-empty subset of $X$ has a minimum. In this case,
there is a unique ordinal $\beta$ and a unique order preserving function
$\phi:\alpha\longrightarrow X$ called the enumeration of $X$. We call $\beta$
the order type of $X$ and denote it as $ot(X)$.

Very important notational remark In the previous situation, following what
some people may call abuse of notation, we identify $X$ with $\phi$. In this
way, $X(\alpha)$ denotes $\phi(\alpha)$ and $X^{-1}(x)$ denotes $\phi^{-1}(x)$
whenever $\alpha\in\beta$ and $x\in X$. In the same way, $X[S]$ denotes
$\phi[S]$ and $X^{-1}[Y]$ denotes $\phi^{-1}[Y]$ for all $S\subseteq\alpha$
and $Y\subseteq X.$

######  Definition 1.2.2 (Product order and lexicographical order).

Whenever $(X,\leqslant_{X})$ and $(Y,\leqslant_{Y})$ are two partial orders,
we define the following two partial orders over the product $X\times Y$; the
order product $\leqslant$ given by $(x,y)\leqslant(w,z)$ if $x\leqslant_{X}w$
and $y\leqslant_{Y}z$ and the lexicographical product $\leqslant_{lex}$ given
by $(x,y)\leqslant_{lex}(w,z)$ if either $x<_{X}w$ or $x=w$ and
$y\leqslant_{Y}z$.

######  Definition 1.2.3 (Lower semi-lattice).

$(X,\leqslant,\wedge)$ is said to be a lower semi-lattice if $(X,\leqslant)$
is a partial order and $\wedge:X^{2}\longrightarrow X$ is such that for all
$x,y\in X$, $\inf(x,y)$ exists and it is equal to $x\wedge y$.

###  1.3 Boolean algebras and filters

Boolean algebras are one of the central objects of study in modern set theory
and set-theoretic topology. Strictly speaking, all the results appearing in
this thesis can be presented by avoiding any explicit mention to these
objects. However, I think that, at least in this case, they provide a perfect
setting for the motivation and analysis of some of the main topics which will
be discussed eventually. We will briefly review some of the basic definitions
and theorems. A detailed treatment of the subject can be found in Koppelberg
[62].

######  Definition 1.3.1.

A Boolean algebra is a structure
$(\mathbb{A},+,\cdot,-,\mathbb{0},\mathbb{1})$ with two binary
operations111Here we decided to adopt the algebraic notation for the
operations in Boolean algebras. Many authors prefer the usage of the symbols
$\vee$, $\wedge$, and $\neg$ to refer to these operations (see Givant and
Halmos [41]). $+$ and $\cdot$, a unary operation $-$, and two (distinct)
distinguished elements $\mathbb{0}$ and $\mathbb{1}$ such that for all
$x,y,z\in A$ the following conditions hold:

  * $\blacksquare$

$x+(y+z)=(x+y)+z,$

  * $\blacksquare$

$x+y=y+x,$

  * $\blacksquare$

$x\cdot(y+z)=x\cdot y+x\cdot z,$

  * $\blacksquare$

$x+y\cdot x=x,$

  * $\blacksquare$

$x+(-x)=\mathbb{1}$,

  * $\blacksquare$

$x\cdot(y\cdot z)=(x\cdot y)\cdot z,$

  * $\blacksquare$

$x\cdot y=y\cdot x,$

  * $\blacksquare$

$x+(y\cdot z)=(x+y)\cdot(x+z),$

  * $\blacksquare$

$x\cdot(y+x)=x,$

  * $\blacksquare$

$x\cdot(-x)=\mathbb{0}.$

The most tipical example of a Boolean algebra is
$(\mathscr{P}(X),\cup,\cap,\,^{c},\emptyset,X)$ where c represents the
complementation with respect to $X$.

Any Boolean algebra has an associated partial order given by

$x\leqslant y\text{ if }x\cdot y=x\text{(equivalently }x+y=y\text{)}.$

In this order, $\mathbb{0}$ and $\mathbb{1}$ are the minimum and maximum of
$\mathbb{A}$, respectively. In addition, Boolean operations can be completely
recovered by the order in the sense that $\sup\\{x,y\\}=x+y$ and
$\inf\\{x,y\\}=x\cdot y$. In the case of $\mathscr{P}(X)$ the order induced by
the boolean operations coincide with $\subseteq$.

######  Definition 1.3.2.

Let $\mathbb{A}$ be a Boolean algebra and let
$\mathcal{F},\mathcal{I}\subseteq\mathbb{A}$. We say that

$\mathcal{F}$ is a filter if:

  * $\blacksquare$

$\mathbb{1}\in\mathcal{F}$ and $\mathbb{0}\notin\mathcal{F}$,

  * $\blacksquare$

$x\cdot y\in\mathcal{F}$ whenever $x,y\in\mathcal{F}$,

  * $\blacksquare$

If $x\in\mathcal{F}$ and $y\geqslant x$ then $y\in\mathcal{F}$.

$\mathcal{I}$ is an ideal if:

  * $\blacksquare$

$\mathbb{0}\in\mathcal{I}$ and $\mathbb{1}\notin\mathcal{I}$,

  * $\blacksquare$

$x+y\in\mathcal{I}$ whenever $x,y\in\mathcal{I}$,

  * $\blacksquare$

If $x\in\mathcal{I}$ and $y\leqslant x$ then $y\in\mathcal{I}$.

A maximal filter is called an ultrafilter, and a maximal ideal is called a
prime ideal.

In this text, we will work mainly with filters over Boolean algebras of the
form $\mathscr{P}(X)$. In this context, a filter $\mathcal{F}$ over
$\mathscr{P}(X)$ (or simply a filter over $X$) is a non-empty family of non-
empty subsets of $X$ which is closed upwards and under finite intersections.
We call such $\mathcal{F}$ principal if there is $A\subseteq X$ for which
$\mathcal{F}=\\{B\subseteq X\,:\,A\subseteq B\\}.$ If there is no such $A$,
$\mathcal{F}$ is called non-principal. Analogously, an ideal $\mathcal{I}$
over $\mathscr{P}(X)$ (or simply an ideal over $X$) is a family of proper
subsets of $X$ that is closed downards and under finite unions.
Let $\mathcal{F}$ be a filter over some set $X$. We define the family of
positive sets with respect to $\mathcal{F}$, that is, $\mathcal{F}^{+}$, as
the set of all $A\subseteq X$ which intersect each member of $\mathcal{F}$. In
the same way, if $\mathcal{I}$ is an ideal in some set $X$, we define
$\mathcal{I}^{+}$ as $\mathscr{P}(X)\backslash\mathcal{I}.$

######  Definition 1.3.3 (Quotient algebras).

Let $\mathbb{A}$ be a Boolean algebra and let $\mathcal{I}\subseteq\mathbb{A}$
be an ideal. $\mathcal{I}$ naturally induces an equivalence relation $\sim$ on
$\mathbb{A}$ given by

$x\sim y\text{ if and only if }x\cdot(-y)+y\cdot(-x)\in\mathcal{I}.$

We denote the equivalence class of $x$ under such relation as $[x]$. The set
of equivalence classes is denoted as $\mathbb{A}/\mathcal{I}$. The operations
on $\mathbb{A}/\mathcal{I}$ given by $-[x]=[x]$, $[x]+[y]=[x+y]$, and
$[x]\cdot[y]=[x\cdot y]$ are well-defined. Furthermore,
$(\mathbb{A}/\mathcal{I},+,\cdot,-,[\mathbb{0}],[\mathbb{1}])$ is a Boolean
algebra.

The Boolean algebra in which we will focus most of our attention is
$\mathscr{P}(\omega)/\text{FIN}$. Note that in this case, if
$X,Y\in\mathscr{P}(\omega)$ then $[X]\leqslant[Y]$ if and only if
$X\subseteq^{*}Y$.

###  1.4 Forcing

Throughout this thesis, we will force downwards. Thus, a forcing notion (or
simply a  forcing) is triplet $(\mathbb{P},\leqslant,\mathbb{1}_{\mathbb{P}})$
so that $(\mathbb{P},\leqslant)$ is a partial order with
$\mathbb{1}_{\mathbb{P}}$ as its maximum. Usually, we will denote
$(\mathbb{P},\leqslant,\mathbb{1}_{\mathbb{P}})$ simply as $\mathbb{P}$ unless
there is any risk of confusion. A subset $D$ of $\mathbb{P}$ is said to be
dense if for any $q\in\mathbb{P}$ there is $p\in D$ so that $p\leqslant q$.
Lastly, we call a subset $G$ of $\mathbb{P}$ a filter if $G$ is non-empty, it
is closed upwards and for any $p,q\in G$ there is $r\in G$ so that $r\leqslant
p,q$.

######  Definition 1.4.1 (Cohen forcing).

Given $\kappa$ an infinite cardinal, we define the $\kappa$-Cohen forcing as

$\mathbb{C}_{\kappa}=\\{\,p;\kappa\longrightarrow 2\,:\,p\textit{ is
finite}\\}$

and order with the reverse inclusion. The forcing $\mathbbb{C}_{\omega}$ is
simply called Cohen forcing and denoted as $\mathbb{C}$.

Let $V$ be a transitive model of $ZFC$ and $\mathbb{P}\in V$. We say that a
filter $G$ is $\mathbb{P}$-generic over $V$ if $G$ intersects any dense set of
$\mathbb{P}$ lying in $V$. In this situation, there is a minimal transitive
model of $ZFC$ which extends $V$ and has $G$ as an element. This model is
denoted as $V[G].$
The forcing relation is denoted as $\Vdash$. Given a formula $\phi$, we write
$\mathbb{P}\Vdash\text{\say{$\phi$}}$ instead of
$\mathbb{1}_{\mathbb{P}}\Vdash\text{\say{$\phi$}}$. We adopt the convention of
denoting $\mathbb{P}$-names with dots above them. However, if $X$ is an
element of the universe $V$, we will make an abuse of notation and denote the
canonical $\mathbb{P}$-name of $X$ simply as $X$. Whenever $\dot{Y}$ is a
$\mathbb{P}$-name and $G$ is a $\mathbb{P}$-generic filter over $V$, we denote
the interpretation of $\dot{Y}$ with respect to $G$ as $\dot{Y}^{G}$. Given an
ordinal $\gamma$, finite support iterations of length $\gamma$ are denoted as

$(\langle\mathbb{P}_{\xi}\rangle_{\xi\leqslant\gamma},\langle\dot{\mathbb{Q}}_{\xi}\rangle_{\xi<\gamma}).$

For us, elements $p\in\mathbb{P}_{\gamma}$ are finite partial functions with
domain contained in $\gamma$.

We now define some classes of forcing notions which will be important in
further discussions.

######  Definition 1.4.2 ($ccc$ forcings).

A forcing $\mathbb{P}$ is said to be $ccc$ if any antichain in $\mathbb{P}$ is
at most countable.

######  Definition 1.4.3 (Knaster forcings).

Let $2\leqslant n\in\omega$. Given a forcing $\mathbb{P}$, we say that
$\mathcal{B}\subseteq\mathbb{P}$ is $n$-linked if for any
$p_{0},\dots,p_{n-1}\in\mathcal{B}$ there is $p\in\mathbb{P}$ (not necessarily
in $\mathcal{B}$) so that $p\leqslant p_{i}$ for each $i<n$. $\mathbb{P}$ is
said to be $n$-Knaster if for any $\mathcal{A}\in[\mathbb{P}]^{\omega_{1}}$
there is $\mathcal{B}\in[\mathcal{A}]^{\omega_{1}}$ which is $n$-linked.
$2$-Knaster forcings are simply called Knaster.

######  Definition 1.4.4 ($\sigma$-centered forcings.).

Given a forcing $\mathbb{P}$, a subset $\mathcal{C}$ of $\mathbb{P}$ is said
to be centered if it is $n$-linked for each $2\leqslant n\in\omega$. We say
that $\mathbb{P}$ is $\sigma$-centered if
$\mathbb{P}=\cup_{l\in\omega}\mathcal{C}_{l}$ where $\mathcal{C}_{l}$ is
centered for each $l$.

######  Definition 1.4.5 (Precaliber).

Let $\kappa$ be an infinite cardinal. We say that a forcing $\mathbb{P}$ has
precaliber $\kappa$ if for each sequence $\langle
p_{\alpha}\rangle_{\alpha\in\mathbb{P}}$ there is $X\in[\kappa]^{\kappa}$ for
which $\\{p_{\alpha}\,:\,\alpha\in X\\}$ is centered.

######  Remark 1.4.6.

Any $\sigma$-centered forcing has precaliber $\omega_{1}$, and each forcing
with precaliber $\omega_{1}$ is $n$-Knaster for each $2\leqslant n\in\omega$.
Furthermore, any Knaster forcing is $ccc.$

The following theorem will be frequently used.

######  Theorem 1.4.7.

Let $\mathbb{P}$ be an uncountable $ccc$ forcing. Then there is
$p\in\mathbb{P}$ which forces the generic filter to be uncountable.

Given a forcing $\mathbb{P}$, we define the Martin’s number of $\mathbb{P}$,
namely $\mathfrak{m}(\mathbb{P})$, as the minimal cardinal $\kappa$ for which
there is a family $\mathcal{D}$ of size $\kappa$ of dense sets of $\mathbb{P}$
so that there is no filter $G$ intersecting each member of $\mathcal{D}$. To
learn more about Martin’s numbers, see [12] and [111]. Regarding this notion,
we define the following cardinal invariants:

$\mathfrak{m}=\min(\,\mathfrak{m}(\mathbb{P})\,:\,\mathbb{P}\textit{ is
}ccc\,).$
$\mathfrak{m}_{K_{n}}=\min(\,\mathfrak{m}(\mathbb{P})\,:\,\mathbb{P}\textit{
is }n\text{-Knaster}\,).$

######  Theorem 1.4.8.

Suppose that $\mathbb{P}$ is an uncountable $ccc$ forcing so that
$\mathfrak{m}(\mathbb{P})>\omega_{1}$. Then $\mathbb{P}$ has an uncountable
filter.

Two well-known axioms regarding the Martin’s numbers of certain classes of
forcing notions will appear in some discussions, namely $MA$ and $PFA$. We
will state them bellow for the sake of completeness. The notion of property
will not be defined here. However, the reader may find a full treatment of
this subject in [95].

Martin’s Axiom [MA]: $\mathfrak{m}=\mathbb{c}>\omega_{1}$.

Martin’s Axiom for $n$-Knaster forcings: $\mathfrak{m}_{K_{n}}>\omega_{1}.$

Proper Forcing Axiom [PFA]: $\mathfrak{m}(\mathbb{P})>\omega_{1}$ for any
proper forcing notion $\mathbb{P}.$

###  1.5 P-ideal Dichotomy

Let $\mathcal{I}$ be an ideal over a set $X$. We say that $\mathcal{I}$ is a
$P$-ideal if for each countable $\mathcal{A}\subseteq\mathcal{I}$ there is
$B\in\mathcal{I}$ such that $A\subseteq^{*}B$ for any $A\in\mathcal{A}$. Any
such set $B$ is called a pseudo-union of $\mathcal{A}$. The $P$-ideal
dichotomy ($PID$) is a well-known consequence of $PFA$ regarding $P$-ideals.
Stevo Todorčević introduced it in its current form in [123]. There he showed
that $PID$ is consistent with $CH$, although $PFA$ is not. A less general
version of $PID$ was previously formulated by Uri Abraham and Todorčević in
[4].

######  Definition 1.5.1.

Let $\mathcal{I}$ be an ideal over a set $X$. We define the orthogonal ideal
of $\mathcal{I}$, namely $\mathcal{I}^{\perp}$, as the set of all $A\subseteq
X$ such that $A\cap B=^{*}\emptyset$ for any $B\in\mathcal{I}.$

$P$-ideal Dichotomy [PID]: Let $X$ be a set and $\mathcal{I}$ be a $P$-ideal
over $X$ such that
$[X]^{<\omega}\subseteq\mathcal{I}\subseteq[X]^{\leqslant\omega}$. Then one of
the two following conditions hold:

  * $\blacksquare$

There is $Y\in[X]^{\omega_{1}}$ so that $[Y]^{\omega}\subseteq\mathcal{I}$.

  * $\blacksquare$

There is $\\{Z_{n}\,:\,n\in\omega\\}\subseteq\mathcal{I}^{\perp}$ for which
$X=\bigcup\limits_{n\in\omega}Z_{n}.$

###  1.6 Diamond principle

Undoubtedly, the most important axiom regarding this work is Jensen’s
$\Diamond$-principle. This principle was introduced by Ronald Jensen in [56].
In there, he showed that this principle holds in the constructible universe
$L$. We will recall the version of the $\Diamond$-principle that we will be
using.

A $\Diamond$-sequence (over $\omega_{1}$) is a sequence $\langle
D_{\alpha}\rangle_{\alpha\in\text{Lim}}$ of countable subsets of $\omega_{1}$
for which the following properties hold:

  1. $(a)$

For each $\alpha\in\text{LIM}$, $D_{\alpha}\subseteq\alpha.$

  2. $(b)$

Given $S\subseteq\omega_{1}$, the set
$\\{\alpha\in\text{Lim}\,:\,S\cap\alpha=D_{\alpha}\,\\}$ is stationary.

Jensen’s $\Diamond$-principle [$\Diamond$]: There is a $\Diamond$-sequence.

##  Chapter 2 Ordinal metrics and construction schemes

In this chapter we introduce ordinal metrics and construction schemes as well
as study the relationship between them.

Stevo Todorčević implicitly introduced ordinal metrics in [120]111In Section 2
of such paper, it is proved in (2.3) that the function $\rho$ defined
immediately before satisfies the properties of what we now call an ordinal
metric., as a tool for showing the existence of a coloring of the pairs of
countable ordinals with uncountably many colors so that every uncountable set
contains pairs of every color. In [112], he introduced construction schemes as
a tool for building uncountable structures by finite approximations. Broadly
speaking, construction schemes provide blueprints for the domain of such
approximations, as well as a procedure to recursively improve them through the
use of finite amalgamations. This concept can be viewed as a generalisation of
the so-called $(\omega,1)$-morasses as defined by Ronald Jensen (see [26]) and
simplified by Daniel Velleman in [128]. The main difference between these two
notions is that with morasses we always amalgamate two objects when improving
approximations, whereas in construction schemes the number of amalgamations
allowed may vary.

The relationship between ordinal metrics and construction schemes has already
been highlighted in [112]. It is worth pointing out that Charles Morgan
presented in [81] a detailed analysis of the relation between ordinal metrics
and gap-1 morasses. Readers who want to know more about morasses and their
applications may look at [26], [33], [60], [92], [101], [128], [129], and
[130].

###  2.1 Ordinal metrics

For the rest of this chapter, $X$ stands for a non-empty set of ordinals.

######  Definition 2.1.1 (Closure operation).

Let $\rho:X^{2}\longrightarrow\omega$ be an arbitrary function. For each
$F\in\text{FIN}(X)$ and $k\in\omega$ we define the $k$-closure of $F$ as:

$\displaystyle(F)_{k}$ $\displaystyle=\\{\,\alpha\in X\,:\,\exists\beta\in
F\backslash\alpha\,(\rho(\alpha,\beta)\leqslant k)\,\\}$
$\displaystyle(F)^{-}_{k}$
$\displaystyle=(F)_{k}\cap\max(F)=(F)_{k}\backslash\\{\max(F)\\}$

The diameter of $F$ is defined as

$\rho^{F}=\max(\,\rho(\alpha,\beta)\,:\,\alpha,\beta\in F\,).$

$F$ is said to be $k$-closed whenever $F=(F)_{k}$. Moreover, if $F$ is
$\rho^{F}$-closed we just say that $F$ is closed. For $\alpha\in X$, we will
write $(\alpha)_{k}$ and $(\alpha)^{-}_{k}$ instead of $(\\{\alpha\\})_{k}$
and $(\\{\alpha\\})^{-}_{k}$ respectively.

######  Remark 2.1.2.

Both the $k$-closure and the diameter are monotone operations. That is, if
$F\subseteq G$ then $(F)_{k}\subseteq(G)_{k}$ and $\rho^{F}\leqslant\rho^{G}$.

Following the convention established in the preliminaries, if
$(\alpha)_{k}=\\{\beta_{0},\dots,\beta_{n}\\}$ with
$\beta_{0}<\dots<\beta_{n}$, we write $(\alpha)_{k}(i)=\beta_{i}$ for each
$i\leqslant n$. This notation will be used throughout this thesis.

######  Definition 2.1.3 ($k$-cardinality).

Let $\rho$ be as in the previous definition. Given $\alpha\in X$ and
$k\in\omega$ we define the $k$-cardinality of $\alpha$ as

$\lVert\alpha\rVert_{k}=|(\alpha)^{-}_{k}|.$

Now we define ordinal metrics following the same approach (although slightly
different notation) as in [110]. In practice, we will only be interested in
ordinal metrics whose domain is $\omega_{1}^{2}$. However, such metrics are
constructed via recursion. For that reason, it is useful to present the
definition in this generality.

######  Definition 2.1.4 (Ordinal metric).

222Some authors prefer to call ordinal metrics $T$-functions or
$\rho$-functions.

We say that $\rho:X^{2}\longrightarrow\omega$ is an ordinal metric (over $X$)
if:

  1. $(a)$

$\forall\alpha,\beta\in X\,(\,\rho(\alpha,\beta)=0\textit{ if and only if
}\alpha=\beta\,),$

  2. $(b)$

$\forall\alpha,\beta\in X\,(\,\rho(\alpha,\beta)=\rho(\beta,\alpha)\,),$

  3. $(c)$

$\forall\alpha,\beta,\gamma\in
X\,(\;\alpha\leqslant\beta,\gamma\rightarrow\rho(\alpha,\beta)\leqslant\max(\rho(\alpha,\gamma),\rho(\beta,\gamma))\;),$

  4. $(d)$

$\forall\beta\in X\,\forall k\in\omega\,(\;\lVert\beta\rVert_{k}<\omega\;)$.

As the reader may note, the previous definition resembles that of an ultra-
metric. However, one of the triangle inequalities is missing. In this way, one
may interpret $(\beta)_{k}$ as the ball of radius $k$ centered on $\beta$
intersected with $\beta+1$. In the theory of metric spaces, we use the metric
to approximate or locate points in a given space. To do that, we study the
behaviour of balls centered on a given point as they become smaller. Ordinal
metrics tend to work in a different way. Here, we want to construct a
structure whose elements are parameterized by the points of the domain of our
ordinal metric, say $X$. In order to do that, we make approximations of such
structure by analyzing the interaction between the elements $\beta\in X$ and
the points in $(\beta)_{k}$ as $k$ grows larger. For that reason it may not be
always suitable to carry the intuition of metric spaces into the context of
ordinal metrics, but rather use the word metric to help us remember the main
aspects of the definition. With the following proposition we show the
difference between these two notions in a formal sense.

######  Proposition 2.1.5.

No ordinal metric over $\omega_{1}$ is a metric.

###### Proof.

Suppose $\rho$ is as previously mentioned and let $M$ be a countable
elementary submodel of $H(\omega_{2})$ having $\rho$ as an element. We will
show that $\rho$ does not satisfy the triangle inequality of metric spaces.
Take $\gamma\in\omega_{1}\backslash M$, consider an arbitrary $\alpha\in
M\cap\omega_{1}$ and define $k$ as $\rho(\alpha,\gamma)$. Using elementarity
we can find $\beta\in M\cap\omega_{1}$ above $\max((\gamma)_{2k}\cap M)$ for
which $k=\rho(\alpha,\beta)$. By definition of the $k$-closure, we have that
$\rho(\beta,\gamma)>2k=\rho(\alpha,\beta)+\rho(\alpha,\gamma)$. This finishes
the proof. ∎

If $\rho$ is an ordinal metric over $X$ and $\alpha\in X$ then
$(-\infty,\alpha]_{X}=\bigcup_{k\in\omega}(\alpha)_{k}$. Due to the property
(d) of Definition 2 this set must be countable. Hence, the existence of such
$\rho$ implies that $ot(X)$ is at most $\omega_{1}$.

######  Remark 2.1.6.

If $\rho$ is an ordinal metric over $X$ and $F\in\text{FIN}(X)$ then
$F\subseteq(F)_{k}$ for each $k\in\omega$. Furthermore, as the $k$-closure
operation does not add points above the maximum of the set to which it is
applied, it follows that $\max((F)_{k})=\max(F)$ for any such $F$.

The following proposition summarizes the most basic properties regarding the
closure of sets in the context of ordinal metrics.

######  Proposition 2.1.7.

Let $\rho:X^{2}\longrightarrow\omega$ be and ordinal metric,
$F\in\text{FIN}(X)$ and $k\geqslant\rho^{F}$. Then:

  1. $(1)$

$\rho^{F}=\max(\,\rho(\alpha,\max(F))\,:\,\alpha\in F).$

  2. $(2)$

$(F)_{k}=\\{\,\alpha\leqslant\max(F)\,:\,\rho(\alpha,\max(F))\leqslant
k\,\\}=(\max(F))_{k}.$

  3. $(3)$

If $\beta\in F$, then $(F)_{k}\cap(\beta+1)=(F\cap(\beta+1))_{k}.$

  4. $(4)$

$(F)_{k}$ is $k$-closed and $\rho^{(F)_{k}}\leqslant k$, so in particular
$(F)_{k}$ is closed. Furthermore, if $k=\rho^{F}$, then $\rho^{(F)_{k}}=k$.

  5. $(5)$

$F$ is closed if and only if $F=(\max(F))_{\rho^{F}}$.

  6. $(6)$

If $G\in\text{FIN}(X)$ is $k$-closed, then $F\cap G\sqsubseteq F$.

###### Proof.

The points (1) and (2) are direct consequences of the property (c) of
Definition 2 and the points (4) and (5) are direct consequences of the point
(1) and (2). Thus, we will only prove the points (3) and (6).

###### Proof of $(3)$..

It is clear that $(F\cap(\beta+1))_{k}\subseteq(F)_{k}\cap(\beta+1)$ so we
will only prove that $(F)_{k}\cap(\beta+1)\subseteq(F\cap(\beta+1))_{k}$. For
this take an arbitrary $\alpha\in(F)_{k}\cap(\beta+1)$. By the point (2) we
have that $\rho(\alpha,\max(F))\leqslant k$. Using the property (c) of
Definition 2 we can conclude that

$\rho(\alpha,\beta)\leqslant\max(\rho(\alpha,\max(F)),\,\rho(\beta,\max(F)))\leqslant
k.$

Hence $\alpha\in(F\cap(\beta+1))_{k}$ and we are done. ∎

###### Proof of $(6)$.

Fix $\beta\in F\cap G$ and let $\alpha\in F$ with $\alpha\leqslant\beta$ be
arbitrary. Then $\rho(\alpha,\beta)\leqslant\rho^{F}\leqslant k$. As $G$ is
$k$-closed, $\alpha\in G$. In this way $\alpha\in F\cap G$. This finishes the
proof. ∎

∎

In this way, in order to compute $\rho^{F}$ and $(F)_{k}$, we only need to use
the maximum of $F.$

######  Corollary 2.1.8.

Let $\rho$ be an ordinal metric over $X$ and $F,G\in\text{FIN}(X)$ with
$F\sqsubseteq G$. If $G$ is closed then so is $F$.

###### Proof.

As $G$ is closed then $G=(G)_{\rho^{G}}$. Let $\beta=\max F$. Since
$F\sqsubseteq G$ then $\beta\in G$ and $F=G\cap(\beta+1)$. By the point (3) of
Proposition 2.1.7 we have that

$(F)_{\rho^{G}}=(G\cap(\beta+1))_{\rho^{G}}=(G)_{\rho^{G}}\cap(\beta+1)=G\cap(\beta+1)=F.$

This means that $F$ is $\rho^{G}$-closed. But $F\subseteq G$ so
$\rho^{F}\leqslant\rho^{G}$. Consequently $F$ is closed. ∎

We are interested in ordinal metrics which belong to three particular classes.
In the rest of this section we will study them. But first, we need the
following definition.

######  Definition 2.1.9.

Let $F\in\text{FIN}(X)$. We say that $F$ is maximally closed if it is closed
and there is no closed $G\in\text{FIN}(X)$ with $\rho^{G}=\rho^{F}$ such that
$F\subsetneq G$.

######  Definition 2.1.10 (Locally finite metric).

Let $\rho:X^{2}\longrightarrow\omega$ be an ordinal metric. We say that $\rho$
is locally finite if

$\sup(\,|F|\,:\,F\in\text{FIN}(X),F\textit{ is closed and }\rho^{F}\leqslant
k\,)$

is finite for every $k\in\omega$.

######  Remark 2.1.11.

If $\rho$ is locally finite and $F\in\text{FIN}(X)$ then there is a maximally
closed set $G\in\text{FIN}(X)$ such that $F\subseteq G$ an
$\rho^{F}=\rho^{G}$. In fact, due to the point (6) of Proposition 2.1.7 we can
conclude that $(F)_{\rho^{F}}\sqsubseteq G$ for any such $G$. In particular,
any closed set can be end-extended to a maximally closed set of the same
diameter.

The following result is an immediate consequence of the previous remark.

######  Proposition 2.1.12.

Let $\rho:X^{2}\longrightarrow\omega$ be a locally finite ordinal metric. For
each $F\in\text{FIN}(X)$ we have that

$\displaystyle\rho^{F}$ $\displaystyle=\min(\,\rho^{G}\,:\,G\textit{ is
maximally closed and }F\subseteq G\,)$
$\displaystyle=\min(\,\rho^{G}\,:\,G\textit{ is maximally closed and
}F\sqsubseteq G\,)$

######  Proposition 2.1.13.

Let $\rho:X^{2}\longrightarrow\omega$ be a locally finite ordinal metric. For
all $k\in\omega$ and every $A\in[X]^{\omega}$ there exist $\alpha,\beta\in A$
such that $\rho(\alpha,\beta)>k.$

###### Proof.

Let $k$ and $A$ be as in the hypotheses. Suppose towards a contradiction that
$\rho(\alpha,\beta)\leqslant k$ for all $\alpha,\beta\in A$. Then
$\rho^{A[n]}\leqslant k$ for each $n\in\omega$. Due to the point (6) of
Proposition 2.1.7 we conclude that $\langle(A[n])_{k}\rangle_{k\in\omega}$ is
a strictly increasing sequence of closed sets all of whose diameter is at most
$k$. This contradicts the fact that $\rho$ is locally finite. So we are done.
∎

Ordinal metrics satisfying the conclusion of the previous proposition deserve
a special name.

######  Definition 2.1.14 (unbounded metric).

Let $\rho:X^{2}\longrightarrow\omega$ be an ordinal metric. We say that $\rho$
is unbounded if for all $k\in\omega$ and every $A\in[X]^{\omega}$ there are
$\alpha,\beta\in A$ with $\rho(\alpha,\beta)>k$.

The following proposition can be found in [110] (Lemma 3.4.11, page 73). The
proof that we present here avoids the use of ultrafilter quantifiers.

######  Proposition 2.1.15.

Let $\rho:X^{2}\longrightarrow\omega$ be an unbounded ordinal metric and
$k\in\omega$. For all $l\in\omega$ and each infinite $A\subseteq[X]^{l}$
family of pairwise disjoint sets there is $B\in[A]^{\omega}$ such that for all
distinct $a,b\in B$ and every $\alpha\in a$ and $\beta\in b$, we have that
$\rho(\alpha,\beta)>k$.

###### Proof.

Let $l$ and $A$ be as in the hypotheses. Enumerate $A$ as $\langle
a_{n}\rangle_{n\in\omega}$. For any $i<l$ there are two particular colorings
that we may consider. The first one is the coloring $c_{i}:[\langle
a_{n}(i)\rangle_{n\in\omega}]^{2}\longrightarrow 2$ given by
$c_{i}(a_{m}(i),a_{n}(i))=0$ if and only if $m<n$ and $a_{m}(i)<a_{n}(i)$. The
second one is the coloring $o_{i}:[\langle
a_{n}(i)\rangle_{n\in\omega}]^{2}\longrightarrow 2$ given by
$o_{i}(a_{m}(i),a_{n}(i))=0$ if and only if $\rho(a_{m}(i),a_{n}(i))>k$. Since
there are no infinite decreasing sequences of ordinals and $\rho$ is an
unbounded metric then all infinite monochromatic sets with respect to these
two coloring choose color $0$. In this way, by applying Ramsey’s Theorem
multiple (but finitely many) times, we can get a set $B\in[A]^{\omega}$
enumerated as $\langle b_{n}\rangle_{n\in\omega}$ satisfying the following
conditions for each $i<l$ and all $m<n\in\omega$:

  * $\blacksquare$

$b_{m}(i)<b_{n}(i)$,

  * $\blacksquare$

$\rho(b_{m}(i),b_{n}(i))>k$.

and such that for all distinct $i,j<l$ either:

(A)

For all $m<n\in\omega$, $\rho(b_{m}(i),b_{n}(j))\leqslant k.$

(B)

For all $m<n\in\omega,$ $\rho(b_{m}(i),b_{n}(j))>k.$

The proof wil be finished if we are able show that (B) holds for all distinct
$i,j<l$. We divide the proof of this fact into three cases. In each of them,
it is enough to show that there are $m<n$ for which
$\rho(b_{m}(i),b_{n}(j))>k$. Indeed, this implies that (A) can not hold for
$i,j,$ or equivalently, $(B)$ holds.

Case 1: If $i<j$.

###### Proof of case..

Note that $b_{0}(i)<b_{1}(i)<b_{2}(i)<b_{2}(j)$. Then,

$k<\rho(b_{0}(i),b_{1}(i))\leqslant\max(\,\rho(b_{0}(i),b_{2}(j)),\,\rho(b_{1}(i),b_{2}(j))\,).$

In other words, either $\rho(b_{0}(i),b_{2}(j))>k$ or
$\rho(b_{1}(i),b_{2}(j))>k$. ∎

Case 2: If $i>j$ and $b_{2}(j)<b_{0}(i).$

###### Proof of case..

Here we also have that $b_{1}(j)<b_{2}(j)$. Hence,

$k<\rho(b_{1}(j),b_{2}(j))\leqslant\max(\,\rho(b_{0}(i),b_{1}(j)),\,\rho(b_{0}(i),b_{2}(j))\,).$

As in the previous case, since either $\rho(b_{0}(i),b_{1}(j))>k$ or
$\rho(b_{0}(i),b_{1}(j))>k$ then $B$ must hold. ∎

Case 3: If $i>j$ and $b_{0}(i)<b_{2}(j).$

###### Proof of case..

Here, as $b_{0}(i)<b_{1}(i)$ the inequality associated to the first case also
holds. The rest of the argument is equal to the first two cases. ∎

∎

######  Definition 2.1.16.

Let $\rho:X^{2}\longrightarrow\omega$ be an ordinal metric and let
$F,G\in\text{FIN}(X)$. We say that $F$ and $G$ are $\rho$-isomorphic if
$|F|=|G|$ and for all $i,j<|F|$,

$\rho(F(i),F(j))=\rho(G(i),G(j)).$

Equivalently, if $h:F\longrightarrow G$ is the only increasing
bijection333Note that if $|F|=|G|$ then $h$ is defined by the formula
$h(F(i))=G(i)$. then

$\rho(\alpha,\beta)=\rho(h(\alpha),h(\beta))$

for all $\alpha,\beta\in F$.

######  Definition 2.1.17 (Homogeneous metric).

Let $\rho:X^{2}\longrightarrow\omega$ be an ordinal metric. We say that $\rho$
is homogeneous if all $F,G\in\text{FIN}(X)$ maximally closed sets with the
same diameter are $\rho$-isomorphic.

######  Lemma 2.1.18.

Suppose that $\rho:X^{2}\longrightarrow\omega$ is a locally finite and
homogeneous ordinal metric. If $F,G\in\text{FIN}(X)$ are two closed sets (not
necessarily maximal) with $|F|=|G|$ and $\rho^{F}=\rho^{G}$ then $F$ and $G$
are $\rho$-isomorphic.

###### Proof.

Let $k=\rho^{F}=\rho^{G}$. As $\rho$ is locally finite, there are maximally
closed sets $\overline{F},\overline{G}\in\text{FIN}(X)$ end-extending $F$ and
$G$ respectively and such that $\rho^{\overline{F}}=\rho^{\overline{G}}=k$. In
particular, this means that $F(i)=\overline{F}(i)$ and $G(i)=\overline{G}(i)$
for each $i<|F|$. Now, since $\rho$ is homogeneous then
$\rho(F(i),F(j))=\rho(\overline{F}(i),\overline{F}(j))=\rho(\overline{F}(i),\overline{F}(j))=\rho(G(i),G(j))$
for any $i,j<|F|\leqslant|\overline{F}|$. This finishes the proof. ∎

######  Remark 2.1.19.

If $\rho$, $F$ and $G$ are as in the previous lemma and $\phi:F\longrightarrow
G$ is the only increasing bijection then $\rho^{H}=\rho^{\phi[H]}$ for each
$H\subseteq F$. As $F$ and $G$ are closed it is easy to see that $\phi[H]$ is
closed whenever $H$ is. This is because $\rho^{H}\leqslant\rho^{F}$.
Furthermore, if $H$ is maximally closed then $\phi[H]$ is also maximally
closed. This is because $\phi[H]$ can be extended to a maximally closed set
$\overline{\phi[H]}$ of diameter $\rho^{H}$. By homogeneity, such set has the
same cardinality as $H$. But then $\phi[H]=\overline{\phi[H]}$.

######  Proposition 2.1.20.

Let $\rho:X^{2}\longrightarrow\omega$ be an ordinal metric and let
$\mathcal{A}\subseteq\text{FIN}(X)$ be a family of closed sets of diameter
$k$. If $F\in\text{FIN}(X)$ is such that $\rho^{F}\leqslant k$ and
$F\subseteq\bigcup\mathcal{A}$ then there is $G\in\mathcal{A}$ for which
$F\subseteq G$.

###### Proof.

By hypothesis, there is $G\in\mathcal{A}$ with $\max(F)\in G$. As
$\rho^{F}\leqslant k=\rho^{G}$ and $G$ is closed, we know that $F\cap
G\sqsubseteq F$ due to the point (6) of Proposition 2.1.7. It follows directly
that $F\subseteq G$. ∎

As a corollary of the previous proposition we get that there are no leftovers
in the unions of maximally closed sets of the same diameter.

######  Corollary 2.1.21.

Let $\rho:X^{2}\longrightarrow\omega$ be an ordinal metric and let
$F\in\text{FIN}(X)$. If $i<\rho^{F}$ and $\mathcal{A}\subseteq\text{FIN}(X)$
is a family of maximally closed sets of diameter $i$ such that
$F=\bigcup\mathcal{A}$, then

$\mathcal{A}=\\{\,G\in\text{FIN}(X)\,:\,G\subseteq F,\,G\textit{ is maximally
closed and }\rho^{G}=i\,\\}.$

###### Proof.

We only need to prove the inclusion from right to left. For this, let
$G\in\text{FIN}(X)$ be such that $G\subseteq F$, $G$ is maximally closed and
$\rho^{G}=i$. By Proposition 2.1.20 we know there is $H\in\mathcal{A}$ with
$G\subseteq H$. As $G$ is maximally closed and $\rho^{G}=i=\rho^{H}$, then
$G=H$. This finishes the proof. ∎

######  Definition 2.1.22 (Regular metric).

Let $\rho:X^{2}\longrightarrow\omega$ be an ordinal metric. We say that $\rho$
is regular if for each $k\in\omega\backslash 1$ and each maximally closed set
$F\in\text{FIN}(X)$ of diameter $k$ there are $j_{F}\in\omega\backslash 2$ and
$F_{0},\dots,F_{j_{F}-1}\in\text{FIN}(X)$ such that:

  * $\blacksquare$

$F=\bigcup\limits_{i<j_{F}}F_{i},$

  * $\blacksquare$

For each $i<j_{F}$, $F_{i}$ is a maximal closed set with $\rho^{F_{i}}=k-1$,

  * $\blacksquare$

$\langle F_{i}\rangle_{i<j_{F}}$ forms a root-tail-tail $\Delta$-system with
root $R(F)$. That is,

$R(F)<F_{0}\backslash R(F)<\dots<F_{j_{F}-1}\backslash R(F).$

######  Remark 2.1.23.

Observe that both the number $j_{F}$ and sequence $\langle
F_{i}\rangle_{i<j_{F}}$ are unique for each $F$ due to Corollary 2.1.21. So
from now on, we will call $j_{F}$ the decomposition number of $F$ and $\langle
F_{i}\rangle_{i<j_{F}}$ its canonical decomposition.

######  Lemma 2.1.24.

Let $\rho:X^{2}\longrightarrow\omega$ be a regular, locally finite ordinal
metric. If $F\in FIN(X)$ is a closed set with positive diameter and
$i<\rho^{F}$, then there is a maximally closed set $G\in\text{FIN}(X)$ with
$\rho^{G}=i$ and $G\sqsubseteq F$.

###### Proof.

The proof is carried by induction over $k=\rho^{F}$. For this, fix
$k\in\omega$ and suppose that we have proved the lemma for each closed set
$H\in\text{FIN}(X)$ with $\rho^{H}<k$. Let $F\in\text{FIN}(X)$ be a closed set
of diameter $k$ and let $i<\rho^{F}$. Furthermore, $R(F)$ is also uniquely
determined so we will call it the root of $F$.

As $\rho$ is locally finite, there is a maximally closed set $\overline{F}$
with $F\sqsubseteq\overline{F}$ and $\rho^{\overline{F}}=k$. Since $\rho$ is
regular and $k>0$ we can consider $j_{\overline{F}}\in\omega\backslash 2$ and
$\overline{F}_{0},\dots,\overline{F}_{j_{\overline{F}}-1}\in\text{FIN}(X)$ as
in Definition 2.1.22. Now, $F\not\subseteq\overline{F}_{0}$ as
$\rho^{\overline{F}_{0}}=k-1$. As both $F$ and $\overline{F}_{0}$ are initial
segments of $\overline{F}$ it follows that $\overline{F}_{0}\sqsubseteq F$.
Observe that if $i=k-1$ then $\overline{F}_{0}$ testifies the conclusion of
the Lemma. Otherwise, if $i<k-1$ we can use the induction hypotheses to get a
maximally closed set $G$ for which $\rho^{G}=i$ and
$G\sqsubseteq\overline{F}_{0}\sqsubseteq F$. This finishes the proof. ∎

In this way, for a regular, locally finite ordinal metric, we can not skip a
distance in a closed set.

######  Corollary 2.1.25.

Let $\rho:X^{2}\longrightarrow\omega$ be a regular, locally finite ordinal
metric. If $F\in FIN(X)$ is closed and $i\leqslant\rho^{F}$, then there are
$\alpha,\beta\in F$ for which $\rho(\alpha,\beta)=i.$

######  Theorem 2.1.26.

Let $\rho:X^{2}\longrightarrow\omega$ be a regular, locally finite and
homogeneous ordinal metric. If $F,G\in FIN(X)$ are closed sets of the same
cardinality then $\rho^{F}=\rho^{G}.$

###### Proof.

Suppose towards a contradiction that $\rho^{F}\not=\rho^{G}$. Without loss of
generality we can assume that $\rho^{F}<\rho^{G}$. As $\rho$ is locally finite
there is maximally closed set $\overline{F}$ with $F\sqsubseteq\overline{F}$
and $\rho^{\overline{F}}=\rho^{F}$. Due to Lemma 2.1.24 there is also a
maximally closed set $H$ such that $H\sqsubseteq G$ and $\rho^{H}=\rho^{F}$.
Observe that $|H|<|G|$. On the other hand, by the homogeneity of $\rho$ we
have that $|\overline{F}|=|H|$. Thus, $|F|\leqslant|\overline{F}|=|H|<|G|$
which is a contradiction. This finishes the proof. ∎

######  Corollary 2.1.27.

Let $\rho:X^{2}\longrightarrow\omega$ be a regular, locally finite and
homogeneous ordinal metric. If $\alpha,\beta\in X$ and $k\in\omega$ is such
that $\lVert\alpha\rVert_{k}=\lVert\beta\rVert_{k}$ then $(\alpha)_{k}$ and
$(\beta)_{k}$ are $\rho$-isomorphic.

###### Proof.

By Theorem 2.1.26 it follows that $\rho^{(\alpha)_{k}}=\rho^{(\beta)_{k}}$ so
by Lemma 2.1.18 we are done. ∎

###  2.2 Construction schemes

By means of Lemma 2.1.18 and Theorem 2.1.26, closed sets in regular locally
finite homogeneous ordinal metrics are completely determined by their size.
Given such a metric $\rho$, let us call $m^{\rho}_{k}$ the common cardinality
of maximally closed sets of diameter $k$. The sequence $\langle
m^{\rho}_{k}\rangle_{k\in\omega}$ is strictly increasing and the first point
of such sequence, that is $m^{\rho}_{0}$, is $1$ due the point (a) of
Definition 2. A closed set will have diameter $k+1$ if and only if its
cardinality lies in the interval $(m^{\rho}_{k},m^{\rho}_{k+1}]$, and will be
maximal only when its size is exactly $m^{\rho}_{k+1}$.

Here, $F$ represents a maximally closed set of diameter $k+1$, while $G$
represents a closed (not maximal) set of diameter $k+1.$

By homogeneity we also know that for each maximally closed set $F$ of positive
diameter $k$, both the decomposition number $j_{F}$ as well as the cardinality
of the root $|R(F)|$ are uniquely determined by $k$. Hence, we can rename
these two numbers as $n^{\rho}_{k}$ and $r^{\rho}_{k}$ respectively.

Here, $F$ represents a maximally closed of diameter $k$. Each piece of its
canonical decomposition is painted into two colors. The color brown represents
the root of $F$ and the green one represents the remaining part.

From the previous picture it is clear that the numbers $m^{\rho}_{k}$,
$n^{\rho}_{k}$ and $r^{\rho}_{k}$ are related through the equation:

$m^{\rho}_{k+1}=r^{\rho}_{k+1}+(m^{\rho}_{k}-r^{\rho}_{k+1})n^{\rho}_{k+1}.$

An abstraction of the properties satisfied by the sequence $\langle
m^{\rho}_{k},n^{\rho}_{k+1},r^{\rho}_{k+1}\rangle_{k\in\omega}$ lead to the
next definition.

######  Definition 2.2.1 (Type).

We call a sequence $\tau=\langle
m_{k},n_{k+1},r_{k+1}\rangle_{k\in\omega}\subseteq\omega^{3}$ a type if:

  1. $(a)$

$m_{0}=1,$

  2. $(b)$

$\forall k\in\omega\backslash 1\big{(}\;n_{k}\geqslant 2\;\big{)},$

  3. $(c)$

$\forall k\in\omega\big{(}\;m_{k}>r_{k+1}\;\big{)},$

  4. $(d)$

$\forall k\in\omega\backslash
1\big{(}\;m_{k+1}=r_{k+1}+(m_{k}-r_{k+1})n_{k+1}\;\big{)}.$

We say that $\tau$ is good if:

  1. ($e$)

$\forall r\in\omega\;\exists^{\infty}k\in\omega\big{(}\;r_{k}=r\;\big{)},$

Additionally, we say that a partition of $\omega$, namely $\mathcal{P}$, is
compatible with $\tau$ if:

  1. ($e$’)

$\forall P\in\mathcal{P}\,\forall r\in\omega\,\exists^{\infty}k\in
P(\;r_{k}=r\;).$

Now, let $\mathcal{F}^{\rho}$ be the collection of all maximally closed sets
and consider it as ordered set with respect to $\subseteq$. First observe that
$\mathcal{F}^{\rho}$ is cofinal in $\text{FIN}(X)$ by Remark 2.1.11. Note also
that $\mathcal{F}^{\rho}$ is well-founded since it consists of finite sets.
Using Lemma 2.1.24 one can show recursively that $rank(F)=\rho^{F}$ for each
$F\in\mathcal{F}^{\rho}$.

We now define a construction scheme as a family of finite sets of $X$
satisfying the same basic properties as $\mathcal{F}^{\rho}$.444This approach
is not historically accurate. In [112], construction schemes were defined in
the second section, applied in the next three sections, and finally related to
ordinal metrics in the sixth one. But first we make a brief comment regarding
the notation. Each family $\mathcal{F}\subseteq\text{FIN}(X)$ is well-founded
with respect to $\subseteq$ (moreover, $Ht(\mathcal{F})\leqslant\omega)$. In
this way, we can define $\mathcal{F}_{k}$ as in the preliminaries for each
$k\in\omega$. That is, $\mathcal{F}_{k}=\\{F\in\mathcal{F}\,:\,rank(F)=k\\}$.

######  Definition 2.2.2 (Construction scheme).

Let $\tau=\langle m_{k},n_{k+1},r_{k+1}\rangle_{k\in\omega}$ be a type, and
$X$ be a set of ordinals. We say that $\mathcal{F}\subseteq\text{FIN}(X)$ is a
construction scheme (or simply a scheme) over $X$ of type $\tau$ if:

  1. $(a)$

$\mathcal{F}$ is cofinal in $\text{FIN}(X),$

  2. $(b)$

$\forall k\in\omega\;\forall F\in\mathcal{F}_{k}\big{(}\;|F|=m_{k}\;\big{)}$,

  3. $(c)$

$\forall k\in\omega\;\forall F,E\in\mathcal{F}_{k}\big{(}\;E\cap F\sqsubseteq
E,F\;\big{)}$,

  4. $(d)$

$\forall k\in\omega\;\forall F\in\mathcal{F}_{k+1}\;\exists
F_{0},\dots,F_{n_{k+1}-1}\in\mathcal{F}_{k}$ such that

$F=\bigcup\limits_{i<n_{k+1}}F_{i}.$

Moreover, $\langle F_{i}\rangle_{i<n_{k+1}}$ forms a $\Delta$-system with root
$R(F)$ such that $|R(F)|=r_{k+1}$ and

$R(F)<F_{0}\backslash R(F)<\dots<F_{n_{k+1}-1}\backslash R(F).$

If $n\in\omega$ and $n_{k+1}=n$ for each $k\in\omega$, we will call
$\mathcal{F}$ an n-construction scheme (or simply an
n-scheme).5552-construction schemes are the objects which we also know as
$(\omega,1)$-gap morasses.

######  Remark 2.2.3.

If $F\in\mathcal{F}_{k+1}$ and $\langle F_{i}\rangle_{i<n_{k+1}}$ is as in the
point (d) of the previous definition, then for each $i<n_{k+1}$,

$F_{i}=F[r_{k+1}]\cup F[\,[a_{i},a_{i+1})\,]$

where $a_{i}=r_{k+1}+i\cdot(m_{k}-r_{k+1})$. In particular this means that the
sequence $\langle F_{i}\rangle_{i<n_{k+1}}$ is uniquely determined. For this
reason, we will call it the canonical decomposition of $F$.

By the previous discussion and results in the first section of this chapter we
conclude:

######  Proposition 2.2.4.

Let $\rho:X^{2}\longrightarrow\omega$ be a regular, locally finite and
homogeneous ordinal metric. The set

$\mathcal{F}^{\rho}=\\{\,F\in\text{FIN}(X)\,:\,F\textit{ is maximally
closed}\,\\}$

is a construction scheme over $X$ of type $\langle
m^{\rho}_{k},n^{\rho}_{k+1},r^{\rho}_{k+1}\rangle_{k\in\omega}$.

The following Theorem was proved in [112]. Another proof can be found in [69].
We will provide a complete proof of it in the Subsection 5.1.1 for the
convenience of the reader.

######  Theorem 2.2.5.

For any good type there is a construction scheme over $\omega_{1}$ of that
type.

Our next goal is to prove the converse of Proposition 2.2.4. That is, we will
prove that every construction scheme naturally defines a regular, locally
finite and homogeneous ordinal metric. Moreover, we will show the construction
scheme coincides with the family of maximally closed sets of such metric. In
order to do that, we will need some previous results.

The next lemma is an easy consequence of the condition (d) in Definition
2.2.2.

######  Lemma 2.2.6.

Let $\mathcal{F}$ be a construction scheme over $X$, $l\in\omega$ and
$F\in\mathcal{F}_{l}$. For each $k\leqslant l$, it happens that
$F=\bigcup\\{H\in\mathcal{F}_{k}\,:\,H\subseteq F\\}$.

There is a version of condition condition (c) in Definition 2.2.2 in the case
where the sets have different rank.

######  Lemma 2.2.7.

Let $\mathcal{F}$ be a construcion scheme over $X$ and let $k\leqslant
l\in\omega$. If $G\in\mathcal{F}_{k}$ and $F\in\mathcal{F}_{l}$ then $G\cap
F\sqsubseteq G$.

###### Proof.

As a consequence of Lemma 2.2.6 we know that

$\displaystyle G\cap F$
$\displaystyle=G\cap\big{(}\bigcup\\{\,H\in\mathcal{F}_{k}\,:\,H\subseteq
F\\}\big{)}$
$\displaystyle=\bigcup\limits_{H\in\mathcal{F}_{k}\cap\mathscr{P}(F)}G\cap H.$

As each of the sets forming the union of the last equality is an initial
segment of $G$ by condition (c) of Definition 2.2.2, such union is also an
initial segment of $G$. ∎

######  Remark 2.2.8.

In general it is not true that if $k<l$, $G\in\mathcal{F}_{k}$ and
$F\in\mathcal{F}_{l}$ then $G\cap F$ is an initial segment of $F.$ An example
of this ocurrs when $G$ is the second piece of the canonical decomposition of
$F$, namely $F_{1}$.

######  Proposition 2.2.9.

Let $\mathcal{F}$ be a construction scheme over $X$ and let $k\in\omega$ be
such that $m_{k}\leqslant|X|$. Then $X=\bigcup\mathcal{F}_{k}$.

###### Proof.

Let $\alpha\in X$ and let $Y$ be a subset of $X$ of cardinality $m_{k}$. Due
to condition (a) of Definition 2.2.2 there is $l\in\omega$ and
$F\in\mathcal{F}_{l}$ for which $\\{\alpha\\}\cup Y\subseteq F$. Note that
$l\geqslant k$ as $|Y|=m_{k}$. Therefore, by Lemma 2.2.6 we conclude that
$F=\bigcup\\{G\in\mathcal{F}_{k}\,:\,G\subseteq F\\}$. In particular
$\alpha\in G$ for some $G\in\mathcal{F}_{k}$ with $G\subseteq F$. This
finishes the proof. ∎

######  Remark 2.2.10.

By applying the previous proposition to the case where $k=0$ we conclude that
$[X]^{1}=\mathcal{F}_{0}$.

######  Definition 2.2.11.

Let $\mathcal{F}$ be a construction scheme over $X$. We define
$\rho_{\mathcal{F}}:X^{2}\longrightarrow\omega$ as:

$\rho_{\mathcal{F}}(\alpha,\beta)=\min(\,k\in\omega\;:\;\exists
F\in\mathcal{F}_{k}(\;\\{\alpha,\beta\\}\subseteq F\;)\,).$

If $\mathcal{F}$ is clear from context, we will write $\rho_{\mathcal{F}}$
simply as $\rho.$

######  Remark 2.2.12.

Note that $\rho(\alpha,\beta)$ is well defined since $\mathcal{F}$ is cofinal
in $\text{FIN}(X)$.

######  Lemma 2.2.13.

Let $\mathcal{F}$ be a construction scheme over $X$. If $\beta\in X$ and
$k\in\omega$, then

$(\beta)_{k}=F\cap(\beta+1)$

for each $F\in\mathcal{F}_{k}$ with $\beta\in F.$

###### Proof.

By Definition 2.2.11 it is clear that $F\cap(\beta+1)\subseteq(\beta)_{k}$. It
only remains to prove the other inclusion. For this, let
$\alpha\in(\beta)_{k}$ and $G\in\mathcal{F}_{\rho(\alpha,\beta)}$ be such that
$\\{\alpha,\beta\\}\subseteq G$. By the definition of the $k$-closure we know
that $\rho(\alpha,\beta)\leqslant k$. Hence, $G\cap F\sqsubseteq G$ due to
Lemma 2.2.7. As $\beta\in G\cap F$ and $\alpha\leqslant\beta$ it must happen
that $\alpha\in G\cap F$. In particular $\alpha\in F$, so we are done. ∎

######  Proposition 2.2.14.

Let $\mathcal{F}$ be a construction scheme over $X$. Then
$\rho=\rho_{\mathcal{F}}$ is an ordinal metric.

###### Proof.

The condition (b) of Definition 2 is trivially satisfied, the condition (a) is
a consequence of the Remark 2.2.10 and the condition (d) follows from Lemma
2.2.13. Thus, we only need to prove that the condition (c) of such definition
also holds. For this, let $\alpha,\beta,\gamma\in X$ be such that
$\alpha\leqslant\beta,\gamma.$ Consider
$F\in\mathcal{F}_{\rho(\alpha,\gamma)}$ and
$G\in\mathcal{F}_{\rho(\beta,\gamma)}$ for which $\\{\alpha,\gamma\\}\subseteq
F$ and $\\{\beta,\gamma\\}\subseteq G$. We need to consider two cases.

Case 1: If $\rho(\alpha,\gamma)\leqslant\rho(\beta,\gamma)$.

###### Proof of case..

Here $F\cap G\sqsubseteq F$ by Lemma 2.2.7. As $\gamma\in F\cap G$ and
$\alpha\leqslant\gamma$ we have that $\alpha\in F\cap G$. In particular
$\alpha\in G$ so $\\{\alpha,\beta\\}\subseteq G$. Therefore
$\rho(\alpha,\beta)\leqslant\rho(\beta,\gamma)$. ∎

Case 2: If $\rho(\beta,\gamma)\leqslant\rho(\alpha,\gamma)$.

###### Proof of case..

In this case $F\cap G\sqsubseteq G$. If $\beta\leqslant\gamma$ then $\beta\in
F\cap G$ because $\gamma\in F\cap G$. Therefore $\\{\alpha,\beta\\}\subseteq
G$ which means $\rho(\alpha,\beta)\leqslant\rho(\beta,\gamma)$. On the other
hand, if $\beta\geqslant\gamma$ we can use Proposition 2.2.9 to pick
$F^{\prime}\in\mathcal{F}_{\rho(\alpha,\gamma)}$ for which $\beta\in
F^{\prime}$. Since $F^{\prime}\cap(\beta+1)=(\beta)_{\rho(\alpha,\gamma)}$ by
Lemma 2.2.13 then $\gamma\in F^{\prime}$. Again, by the same lemma,
$F^{\prime}\cap(\gamma+1)=(\gamma)_{\rho(\alpha,\gamma)}$ which means that
$\alpha\in F^{\prime}$. Consequently $\\{\alpha,\beta\\}\subseteq F^{\prime}$
so $\rho(\alpha,\beta)\leqslant\rho(\alpha,\gamma)$. ∎

∎

######  Proposition 2.2.15.

Let $\mathcal{F}$ be a construction scheme over $X$, $\rho=\rho_{\mathcal{F}}$
and let $F\in\mathcal{F}_{k}$ for some $k>0$. Consider $\langle
F_{i}\rangle_{i<n_{k}}$ the canonical decomposition of $F$ as described in
point (d) of Definition 2.2.2. If $\alpha\in F_{i}\backslash R(F)$ and
$\beta\in F_{j}\backslash R(F)$ for $i<j<n_{k}$ then $\rho(\alpha,\beta)=k.$

###### Proof.

First observe that $\rho(\alpha,\beta)\leqslant k$ as testified by $F.$ Now,
$(\beta)_{k-1}=F_{j}\cap(\beta+1)$ due to the Lemma 2.2.13. In this way,
$\alpha\not\in(\beta)_{k-1}$, which means that $\rho(\alpha,\beta)>k-1$. This
finishes the proof. ∎

######  Lemma 2.2.16.

Let $\mathcal{F}$ be a construction scheme over $X$ and
$\rho=\rho_{\mathcal{F}}$. For every $k\in\omega$ and $F\in\mathcal{F}_{k}$,
we have that $\rho^{F}=k.$

###### Proof.

For $k=0$ the result follows from Remark 2.2.10, so we will only consider the
case where $k>0$. Let $F\in\mathcal{F}_{k}$. By Definition 2.2.11 it should be
clear that $\rho^{F}\leqslant k$. To prove the other inequality let $\langle
F_{i}\rangle_{i<n_{k}}$ be the canonical decomposition of $F$ as described in
point (d) of Definition 2.2.2. Now, consider $\alpha\in F_{0}\backslash R(F)$
and $\beta\in F_{1}\backslash R(F)$. By Proposition 2.2.15,
$\rho(\alpha,\beta)=k$. Hence $k\leqslant\rho(\alpha,\beta)\leqslant\rho^{F}$.
∎

######  Lemma 2.2.17.

Let $\mathcal{F}$ be a construction scheme over $X$ and
$\rho=\rho_{\mathcal{F}}$. Each element of $\mathcal{F}$ is maximally closed
with respect to $\rho$.

###### Proof.

Let $k\in\omega$ and $F\in\mathcal{F}_{k}$. Consider $\alpha=\max(F)$. By
Lemma 2.2.13, $F=F\cap(\alpha+1)=(\alpha)_{k}$. In this way $F$ is closed
thanks to the point (4) of Proposition 2.1.7. To see that $F$ is maximally
closed let $G\in\text{FIN}(X)$ be a closed set with $\rho^{G}=k$ and
$F\subseteq G$. Consider $\beta=\max(G)$. Through the use of Proposition 2.2.9
we can take $H\in\mathcal{F}_{k}$ for which $\beta\in H$. Then
$(\beta)_{k}=H\cap(\beta+1)$ due to Lemma 2.2.13. Furthermore, by the point
(1) of Proposition 2.1.7 we know that $\rho(\delta,\beta)\leqslant k$ for each
$\delta\in G$. In particular this means that $F\subseteq G\subseteq H$. Since
both $F$ and $H$ have size $m_{k}$, we conclude that $F=H=G$. This finishes
the proof. ∎

######  Proposition 2.2.18.

Let $\mathcal{F}$ be a construction scheme over $X$ and
$\rho=\rho_{\mathcal{F}}$. Then

$\mathcal{F}_{k}=\\{\,H\in\text{FIN}(X)\,:\,H\textit{ is maximally closed and
}\rho^{H}=k\,\\}$

for each $k\in\omega$.

###### Proof.

The inclusion from left to right is a direct consequence of Lemmas 2.2.16 and
2.2.17. To prove the one from right to left, let $H\in\text{FIN}(X)$ be a
maximally closed set of diameter $k$. Consider $\alpha=\max(H)$ and take
$F\in\mathcal{F}_{k}$ for which $\alpha\in F.$ Then
$(\alpha)_{k}=F\cap(\alpha+1)=F$. This means that $H\subseteq F$ due to the
point (1) of Proposition 2.1.7. Since $\rho^{F}=k$ by Lemma 2.2.16 and $H$ is
maximally closed, it follows that $H=F$. Thus, the proof is over. ∎

######  Theorem 2.2.19.

Let $\mathcal{F}$ be a construction scheme over $X$. Then
$\rho=\rho_{\mathcal{F}}$ is locally finite, regular and homogeneous.

###### Proof.

The function $\rho$ is locally finite because each closed set $H$ is contained
in $F\cap(\max(H)+1)$ where $F\in\mathcal{F}_{\rho^{H}}$ is such that
$\max(H)\in F$. In particular, this means that $|H|$ is bounded by
$m_{\rho^{H}}$. Now, $\rho$ is regular due to Proposition 2.2.18 and the point
(d) of Definition 2.2.2. The only thing left to do is to prove that $\rho$ is
homogeneous. This proof is carried by induction over the diameter $k$ of the
maximally closed sets involved. The case where $k=0$ is trivial as maximally
closed sets of diameter $0$ are singletons. So suppose that $k\in\omega$, and
we have proved homogeneity for all maximally closed sets of diameter less than
$k$. Let $F,G$ be two maximally closed sets with $\rho^{F}=\rho^{G}=k$. By
Proposition 2.2.18, $F,G\in\mathcal{F}_{k}$. Consequently $|F|=|G|=m_{k}$.
Thus, we can let $h:F\longrightarrow G$ be the only increasing bijection. We
claim that $\rho(\alpha,\beta)=\rho(h(\alpha),h(\beta))$ for each
$\alpha,\beta\in F$. For this, take $\langle F_{i}\rangle_{i<n_{k}}$ and
$\langle G_{i}\rangle_{i<n_{k}}$ the canonical decompositions of $F$ and $G$
respectively. It is easy to see that $h[F_{i}]=G_{i}$ for each $i<n_{k}$ and
that $h[R(F)]=R(G)$. Consequently, if $\alpha,\beta\in F_{i}$ for some
$i<n_{k}$ we can use the induction hypothesis over $F_{i}$ and $G_{i}$ to
conclude that $\rho(\alpha,\beta)=\rho(h(\alpha),h(\beta))$. On the other
hand, if there is no $i$ for which both $\alpha$ and $\beta$ belong to
$F_{i}$, there are distinct $i,j<n_{k}$ for which $\alpha\in F_{i}\backslash
R(F)$ and $\beta\in F_{j}\backslash R(F)$. Observe that $h(\alpha)\in
G_{i}\backslash R(G)$ and $h(\beta)\in G_{j}\backslash R(F)$. Therefore, by
Proposition 2.2.15 we have that
$\rho(\alpha,\beta)=k=\rho(h(\alpha),h(\beta))$. This finishes the proof. ∎

The following corollary follows from the proof of the previous theorem as well
as of Proposition 2.2.4.

######  Corollary 2.2.20.

If $\mathcal{F}$ is a construction scheme over $X$ and
$\rho:X^{2}\longrightarrow\omega$ is a locally finite, homogeneous and regular
ordinal metric, then:

  * $\blacksquare$

$\mathcal{F}^{\rho_{\mathcal{F}}}=\mathcal{F}$,

  * $\blacksquare$

$\rho_{\mathcal{F}^{\rho}}=\rho.$

###  2.3 Two more canonical functions

Apart from the ordinal metric associated to a construction scheme, there are
two important functions that we need to analyze before we enter the world of
applications. The first one being the $\Delta$ function and the second one
being the $\Xi$ function.

For the rest of this section, fix a construction scheme $\mathcal{F}$ over $X$
and let $\rho=\rho_{\mathcal{F}}$ be its associated ordinal metric.

The $\Delta$ function just measures the least moment in which the
$k$-cardinality of two ordinals differs.

######  Definition 2.3.1 (The $\Delta$ function).

We define $\Delta:X^{2}\longrightarrow\omega+1$ as:

$\Delta(\alpha,\beta)=\begin{cases}\min(\,k\in\omega\,:\,\lVert\alpha\rVert_{k}\not=\lVert\beta\rVert_{k}\,)&\textit{
if }\alpha\not=\beta\\\ \omega&\textit{ if }\alpha=\beta\end{cases}$

######  Remark 2.3.2.

$\Delta$ is well defined since
$\lVert\alpha\rVert_{\rho(\alpha,\beta)}\not=\lVert\beta\rVert_{\rho(\alpha,\beta)}$
whenever $\alpha\not=\beta.$ Moreover,
$\Delta(\alpha,\beta)\leqslant\rho(\alpha,\beta)$.

######  Lemma 2.3.3.

Let $\alpha,\beta\in X$ be distinct ordinals and $k\in\omega$. If
$\lVert\alpha\rVert_{k}=\lVert\beta\rVert_{k}$ then $k<\Delta(\alpha,\beta)$.

###### Proof.

By Corollary 2.1.27 $(\alpha)_{k}$ is $\rho$-isomorphic to $(\beta)_{k}$. Let
$h:(\alpha)_{k}\longrightarrow(\beta)_{k}$ be the only increasing bijection.
Note that $h(\alpha)=\beta$. Now consider an arbitrary $i\leqslant k$. Then
$(\alpha)_{i}\subseteq(\alpha)_{k}$. In this way,
$h[(\alpha)_{i}]=(h(\alpha))_{i}=(\beta)_{i}$. This means that
$\lVert\alpha\rVert_{i}=\lVert\beta\rVert_{i}$. Therefore
$k<\Delta(\alpha,\beta)$. ∎

######  Lemma 2.3.4.

Let $\alpha,\beta,\delta\in X$ be distinct ordinals such that
$\Delta(\alpha,\beta)<\Delta(\beta,\delta)$. Then
$\Delta(\alpha,\delta)=\Delta(\alpha,\beta)$.

###### Proof.

$\Delta(\alpha,\delta)\leqslant\Delta(\alpha,\beta)$ since
$\lVert\alpha\rVert_{\Delta(\alpha,\beta)}\not=\lVert\beta\rVert_{\Delta(\alpha,\beta)}=\lVert\delta\rVert_{\Delta(\alpha,\beta)}$.
Now, to see that $\Delta(\alpha,\delta)\geqslant\Delta(\alpha,\beta)$ take an
arbitrary $i<\Delta(\alpha,\beta)$. Then
$\lVert\alpha\rVert_{i}=\lVert\beta\rVert_{i}=\lVert\delta\rVert_{i}$. This
finishes the proof. ∎

######  Lemma 2.3.5.

Let $\alpha,\beta\in X$ be distinct ordinals, $k<\Delta(\alpha,\beta)$ and
$h:(\alpha)_{k}\longrightarrow(\beta)_{k}$ be the only increasing bijection.

If $\delta\in(\alpha)_{k}$ then:

  1. $(a)$

$\Delta(\delta,h(\delta))\geqslant\Delta(\alpha,\beta)$,

  2. $(b)$

$\rho(\alpha,\beta)\geqslant\rho(\delta,h(\delta))$.

In other words, if $i<\lVert\alpha\rVert_{k}$ then:

  1. $(a)$

$\Delta((\alpha)_{k}(i),(\beta)_{k}(i))\geqslant\Delta(\alpha,\beta)$,

  2. $(b)$

$\rho(\alpha,\beta)\geqslant\rho((\alpha)_{k}(i),(\beta)_{k}(i))$.

###### Proof.

###### Proof of $(a)$.

Let $l=\Delta(\alpha,\beta)-1$ and
$h^{\prime}:(\alpha)_{l}\longrightarrow(\beta)_{l}$ be the only increasing
bijection. As $k<\Delta(\alpha,\beta)$ then
$(\alpha)_{k}\subseteq(\alpha)_{l}$ and
$h^{\prime}[(\alpha)_{k}]=(\beta)_{k}$. Hence $h^{\prime}|_{(\alpha)_{k}}=h$
so $h^{\prime}(\delta)=h(\delta)$. Therefore
$h^{\prime}[(\delta)_{l}]=(h(\delta))_{l}$. In this way,
$\lVert\delta\rVert_{l}=\lVert h(\delta)\rVert_{l}$. Thus, by Lemma 2.3.3 we
conclude that $\Delta(\alpha,\beta)\leqslant\Delta(\delta,\phi(\delta))$. ∎

###### Proof of $(b)$.

Let $F\in\mathcal{F}_{\rho(\alpha,\beta)}$ be such that $\alpha,\beta\in F$.
Since $k<\Delta(\alpha,\beta)\leqslant\rho(\alpha,\beta)$ then
$(\alpha)_{k}\cup(\beta)_{k}\subseteq F$. In particular $\delta,h(\delta)\in
F$ so $\rho(\alpha,\beta)\geqslant\rho(\delta,h(\delta)).$ ∎

∎

######  Remark 2.3.6.

Recall that if $\alpha$ and $\beta$ are so that
$\lVert\alpha\rVert_{k}=\lVert\beta\rVert_{k}$ and
$h:(\alpha)_{k}\longrightarrow(\beta)_{k}$ is the only increasing bijection,
then the following happens:

  * $\blacksquare$

$h(\alpha)=h(\beta),$

  * $\blacksquare$

$(\alpha)_{k}\cap(\beta)_{k}\sqsubseteq(\alpha)_{k},(\beta)_{k},$

  * $\blacksquare$

$h|_{(\alpha)_{k}\cap(\beta)_{k}}$ is the identity function,

  * $\blacksquare$

$h[(\alpha)_{k}\backslash(\beta)_{k}]=(\beta)_{k}\backslash(\alpha)_{k}.$

######  Lemma 2.3.7.

Let $\alpha,\beta\in X$ be distinct ordinals and $k<\Delta(\alpha,\beta)$. If
$h:(\alpha)_{k}\longrightarrow(\beta)_{k}$ is the only increasing bijection
and $\delta,\gamma\in(\alpha)_{k}$ are such that $\delta\leqslant\gamma$ and
$h(\delta)\not=\delta$ then the following happens:

  1. $(a)$

$h(\gamma)\not=\gamma,$

  2. $(b)$

$\rho(\alpha,\beta)\geqslant\rho(\gamma,h(\gamma))\geqslant\rho(\delta,h(\delta))\geqslant\Delta(\delta,h(\delta))\geqslant\Delta(\gamma,h(\gamma))\geqslant\Delta(\alpha,\beta).$

###### Proof.

Remember that $C=(\alpha)_{k}\cap(\beta)_{k}$ is an initial segment of both
$(\alpha)_{k}$ and $(\beta)_{k}$. In this way, it is easy to see that
$h(\xi)=\xi$ if and only if $\xi\in C.$ Note that since $\delta\not\in C$ and
$\gamma>\delta$, then $\gamma\notin C$. Therefore $h(\gamma)\not=\gamma$. This
proves the point (a).

The inequalities from (b) follow directly from Lemma 2.3.5 and the fact that
$h|_{(\gamma)_{k}}$ is the only increasing bijection from $(\gamma)_{k}$ to
$(h(\gamma))_{k}$.

∎

######  Definition 2.3.8 (The $\Xi$ function).

Let $\alpha\in X$. $\Xi_{\alpha}:\omega\longrightarrow\omega\cup\\{-1\\}$ is
the function defined as:

$\Xi_{\alpha}(k)=\begin{cases}0&\textit{if }k=0\\\ -1&\textit{if }k>0\textit{
and }\lVert\alpha\rVert_{k}\leqslant r_{k}\\\
\frac{\lVert\alpha\rVert_{k}-\lVert\alpha\rVert_{k-1}}{m_{k-1}-r_{k}}&\textit{otherwise}\end{cases}$

It is not hard to check that if $k\in\omega\backslash 1$ and
$F\in\mathcal{F}_{k}$ is such that $\alpha\in F$, then:

$\Xi_{\alpha}(k)=\begin{cases}-1&\textit{ if }\alpha\in R(F)\\\ i&\textit{ if
}\alpha\in F_{i}\backslash R(F)\par\end{cases}$

The main reason for defining the $\Xi$ function is that without appealing to
any extra axioms it is really hard to give useful properties regarding the
behavior of the $k$-cardinality of pairs of ordinals. The $\Xi$ function
reveals two important and natural critical points leading the dance of the
$k$-cardinality of ordinals $\alpha$ and $\beta$ as $k$ grows larger. As the
next lemma shows, below $\Delta(\alpha,\beta)$ and above $\rho(\alpha,\beta)$,
this dance is smooth and pleasant whereas between $\Delta(\alpha,\beta)$ and
$\rho(\alpha,\beta)$ it seems that almost anything is possible.

In here, $F$ represents a member of $F_{k}$ with $n_{k}=3$.

######  Lemma 2.3.9.

Let $\alpha<\beta\in X$ and $k\in\omega\backslash 1$. Then:

  1. $(a)$

If $k<\Delta(\alpha,\beta)$, then $\Xi_{\alpha}(k)=\Xi_{\beta}(k).$

  2. $(b)$

If $k=\rho(\alpha,\beta)$, then $0\leqslant\Xi_{\alpha}(k)<\Xi_{\beta}(k).$

  3. $(c)$

If $k>\rho(\alpha,\beta)$, then either $\Xi_{\alpha}(k)=-1$ or
$\Xi_{\alpha}(k)=\Xi_{\beta}(k).$

  4. $(d)$

If $k=\Delta(\alpha,\beta)$ then
$0\leqslant\Xi_{\alpha}(k)\not=\Xi_{\beta}(k)\geqslant 0.$

###### Proof.

(a) is direct from the definition of $\Delta(\alpha,\beta)$ so we will only
prove the remaining points.

###### Proof of $(b)$..

Fix $F\in\mathcal{F}_{\rho(\alpha,\beta)}$ for which $\alpha,\beta\in F$. We
know there are $i,j<n_{k}$ such that $\alpha\in F_{i}$ and $\beta\in F_{j}$.
Since $F_{i},F_{j}\in\mathcal{F}_{\rho(\alpha,\beta)-1}$, minimality of $\rho$
implies that $\alpha\not\in F_{j}$ and $\beta\not\in F_{i}$. In particular,
this means $i\not=j$ and $\alpha,\beta\not\in R(F)$. Thus, by Definition 2.3.8
we have that $\Xi_{\alpha}(k)=i$ and $\Xi_{\beta}(k)=j$. Moreover, since
$\alpha<\beta$ then $i<j$. ∎

###### Proof of $(c)$..

For this, suppose that $\Xi_{\alpha}(k)\not=-1$ and let $F\in\mathcal{F}_{k}$
be such that $\alpha,\beta\in F$. Since $\alpha<\beta$, we also have
$\Xi_{\beta}(k)\not=-1$. Thus, there is $j<n_{k}$ for which $\beta\in
F_{j}\backslash R(F)$. Recall that $F_{j}\in\mathcal{F}_{k-1}$. In this way,
$\alpha\in(\beta)_{k-1}=(\beta+1)\cap F_{j}$. As $\alpha\not\in R(F)$, this
means $\Xi_{\alpha}(k)=j=\Xi_{\beta}(k)$. So we are done. ∎

###### Proof of $(d)$..

Let $F,G\in\mathcal{F}_{k}$ be such that $\alpha\in F$ and $\beta\in G$. Also,
let $h:F\longrightarrow G$ be the increasing bijection. Then
$h[(\alpha)_{i}]=(h(\alpha))_{i}$ for any $i\leqslant k$. In this way,

$k=\Delta(\alpha,\beta)=\Delta(h(\alpha),\beta)\leqslant\rho(h(\alpha),\beta)\leqslant
k.$

That is, $k=\rho(h(\alpha),\beta)$. So by the part (b) of this lemma we
conclude that $\Xi_{h(\alpha)}(k)$ and $\Xi_{\beta}(k)$ are both distinct and
greater or equal to $0$. To finish just note that
$\Xi_{\alpha}(k)=\Xi_{h(\alpha)}(k)$. ∎

∎

As an easy consequence of the previous lemma we have the following corollary.

######  Corollary 2.3.10.

Suppose that $X=\omega_{1}$ and let $\alpha\in X$. Then:

  1. $(1)$

$\Xi_{\alpha}(k)=-1$ for infinitely many $k\in\omega$.

  2. $(2)$

If $\alpha$ is infinite, then $\Xi_{\alpha}(k)\geqslant 1$ for infinitely many
$k\in\omega$.

  3. $(3)$

If $\tau$ (the type of $\mathcal{F}$) is a good type, then
$\Xi_{\alpha}(k)\geqslant 0$ for infinitely many $k\in\omega$.

###### Proof.

In order to prove (3) just observe that since $\tau$ is a good type then
$r_{k}=0$ for infinitely many $k$’s. For any such $k$ it is necessarily true
that $\Xi_{\alpha}(k)\geqslant 0$. We now prove the remaining points.

###### Proof of $(1)$.

Let $l\in\omega$. We will find $k>l$ for which $\Xi_{\alpha}(k)=-1$. Since
$X=\omega_{1}$ there is an uncountable $S\subseteq\omega_{1}\backslash\alpha$
and $s\in\omega$ such that $\rho(\alpha,\beta)=\rho(\alpha,\delta)=s$ for any
$\beta,\delta\in S$. In virtue of Lemma 2.1.13 there are $\beta<\delta\in S$
for which $\rho(\beta,\delta)>\max(l,s)$. Let $k=\rho(\beta,\delta)$. By the
point (b) in Lemma 2.3.9 we know that
$0\leqslant\Xi_{\beta}(k)<\Xi_{\delta}(k)$. If $\Xi_{\alpha}(k)\not=-1$ then
we would have that $\Xi_{\beta}(k)=\Xi_{\alpha}(k)=\Xi_{\delta}(k)$ due to
part (c) of that same lemma, which is impossible. Hence, we are done. ∎

###### Proof of $(2)$.

Fix $l\in\omega$. We shall find $k>l$ for which $\Xi_{\alpha}(k)\geqslant 1$.
Since $(\alpha)_{l}$ is finite and $\alpha$ is infinite there is
$\xi\in\alpha\backslash(\alpha)_{l}$. By the definition of the $l$-closure we
have that $k=\rho(\xi,\alpha)>l$. Then $\Xi_{\alpha}(k)\geqslant 1$ do to the
point (b) of Lemma 2.3.9. ∎

∎

######  Lemma 2.3.11.

Let $\alpha,\beta\in X$ be distinct ordinals, $k<\Delta(\alpha,\beta)$ and
$h:(\alpha)_{k}\longrightarrow(\beta)_{k}$ be the only increasing bijection.
If $\gamma\in(\alpha)_{k}$ is such that $\gamma\not=h(\gamma)$ then the
following are equivalent:

  1. $(a)$

$\Delta(\gamma,h(\gamma))>\Delta(\alpha,\beta)$.

  2. $(b)$

$\Xi_{\gamma}(\Delta(\alpha,\beta))=-1$.

Furthermore, if $\Xi_{\gamma}(\Delta(\alpha,\beta)\geqslant 0$ then
$\Xi_{\gamma}(\Delta(\alpha,\beta))=\Xi_{\alpha}(\Delta(\alpha,\beta))$ and
$\Xi_{h(\gamma)}(\Delta(\alpha,\beta))=\Xi_{\beta}(\Delta(\alpha,\beta)).$

###### Proof.

###### Proof of $(a)\Rightarrow(b)$..

We argue by contradiction. Suppose that
$\Xi_{\gamma}(\Delta(\alpha,\beta))\geqslant 0$. Since
$\rho(\alpha,\gamma)\leqslant k<\Delta(\alpha,\beta)$ then
$\Xi_{\gamma}(\Delta(\alpha,\beta))=\Xi_{\alpha}(\Delta(\alpha,\beta))\geqslant
0$ by the part (c) of Lemma 2.3.9. Now, since we are assuming that
$\Delta(\gamma,h(\gamma))>\Delta(\alpha,\beta)$, then
$0\leqslant\Xi_{\gamma}(\Delta(\alpha,\beta))=\Xi_{h(\gamma)}(\Delta(\alpha,\beta))$
due to the point (a) of Lemma 2.3.9. Thus, we can argue in the same way as
before to conclude that
$\Xi_{h(\gamma)}(\Delta(\alpha,\beta))=\Xi_{\beta}(\Delta(\alpha,\beta))$.
Therefore, according the part (d) of the same lemma,

$\Xi_{\gamma}(\Delta(\alpha,\beta))=\Xi_{\alpha}(\Delta(\alpha,\beta))\not=\Xi_{\beta}(\Delta(\alpha,\beta))=\Xi_{h(\gamma)}(\Delta(\alpha,\beta)).$

We conclude using the part (a) of Lemma 2.3.9 that
$\Delta(\gamma,h(\gamma))=\Delta(\alpha,\beta)$. ∎

###### Proof of $(b)\Rightarrow(a)$..

Suppose that $\Xi_{\gamma}(\Delta(\alpha,\beta))=-1$. Then the conclusion of
the part (d) of Lemma 2.3.9 can not hold when applied to $\gamma$, $h(\gamma)$
and $\Delta(\alpha,\beta).$ In virtue of this, $\Delta(\alpha,\beta)$ must be
distinct from $\Delta(\gamma,h(\gamma)).$ ∎

∎

######  Corollary 2.3.12.

Let $\xi<\alpha<\beta\in X$. If $\rho(\xi,\beta)<\rho(\alpha,\beta)$ then
$\rho(\xi,\alpha)<\rho(\alpha,\beta)$.

###### Proof.

Since $\rho$ is an ordinal metric,
$\rho(\xi,\alpha)\leqslant\max(\rho(\xi,\beta),\rho(\alpha,\beta))=\rho(\alpha,\beta)$.
Suppose towards a contradiction that $\rho(\alpha,\beta)=\rho(\xi,\alpha)$.
Then by the part (b) of Lemma 2.3.9 we have that

$0\leqslant\Xi_{\xi}(\rho(\alpha,\beta))<\Xi_{\alpha}(\rho(\alpha,\beta))<\Xi_{\beta}(\rho(\alpha,\beta)).$

On the other hand, since $\rho(\alpha,\beta)>\rho(\beta,\xi)$ then
$\Xi_{\xi}(\rho(\alpha,\beta))=-1$ or
$\Xi_{\xi}(\rho(\alpha,\beta))=\Xi_{\beta}(\rho(\alpha,\beta))$ due to the
part (c) of Lemma 2.3.9. Both cases contradict the previous inequality so we
are done. ∎

The definition of the $\Xi$ function can be extended (with some restrictions)
to arbitrary elements of $\text{FIN}(X)$. More precisely, by virtue of
Proposition 2.1.20 we know that:

######  Lemma 2.3.13.

Let $A\in\text{FIN}(X)$, $k>\rho^{A}$ and $F\in\mathcal{F}_{k}$ such that
$A\subseteq F_{i}$. Then there is $i<n_{k}$ for which $A\subseteq F$.
Furthermore, it is easy to see that if $A\not\subseteq R(F)$ then this $i$ is
unique and does not depend on the choice of $F$.

This leads to the following definition.

######  Definition 2.3.14 (set-valued $\Xi$ function).

Let $A\in\text{FIN}(X)$ for each $k>\rho^{A}$ we define

$\Xi_{A}(k)=\begin{cases}-1&\textit{ if }A\subseteq R(F)\\\ i&\textit{ if
}A\not\subseteq R(F)\textit{ and }A\subseteq F_{i}\end{cases}$

Here, $F\in\mathcal{F}_{k}$ is such that $A\subseteq F.$

######  Remark 2.3.15.

It is not hard to check that actually $\Xi_{A}(k)=\Xi_{\max(A)}(k)$ for each
$k>\rho^{A}$.

The following lemma is easy to prove and it is left to the reader as it is
just a variation of Lemma 2.3.9.

######  Lemma 2.3.16.

For any distinct $A,B\in\text{FIN}(X)$ and $k\geqslant\rho^{A\cup B}$ the
following happens:

  1. $(a)$

If $0\leqslant\Xi_{A}(k)\not=\Xi_{B}(k)\geqslant 0$ then $k=\rho^{A\cup B}.$

  2. $(b)$

If $k>\rho^{A\cup B}$ then $\Xi_{A}(k)=\Xi_{B}(k)$ or
$\min(\Xi_{A}(k),\Xi_{B}(k))=-1$.

###  2.4 Capturing Axioms

Suppose that $A$ and $B$ are two different elements of $\text{FIN}(X)$ and $k$
is a natural number bigger (or equal) than the diameter of both $A$ and $B$.
Then $(A)_{k}$ and $(B)_{k}$ are closed sets by the part (4) of Proposition
2.1.7. By Lemma 2.1.18 and Theorem 2.1.26 we know that these two sets have the
same size if and only if they are $\rho$-isomorphic. However, if this
situation occurs, nothing assures that if we take
$h:(A)_{k}\longrightarrow(B)_{k}$ to be the increasing bijection then
$h[A]=B$. In particular, if $k\geqslant\rho^{A\cup B}$, then either
$|(A)_{k}|\not=|(B)_{k}|$ (so these closures are not even $\rho$-isomorphic)
or $(A)_{k}=(B)_{k}$666Since $k\geqslant\rho^{A\cup
B}\geqslant\max(\rho^{A},\rho^{B})$, then $(A)_{k}=(\max(A))_{k}$ and
$(B)_{k}=(\max(B))_{k}$.. Note that in the latter case, the increasing
bijection $h$ between $(A)_{k}$ and $(B)_{k}$ is the identity function which
means that $h[A]\not=B$. In virtue of this observation, the biggest $k$ for
which we can expect $(A)_{k}$ and $(B)_{k}$ to satisfy this stronger
isomorphism condition is $\rho^{A\cup B}-1$.

######  Definition 2.4.1.

Let $A,B\in FIN(X)$ be such that $\rho^{A},\rho^{B}<\rho^{A\cup B}$. We say
that $A$ and $B$ are strongly $\rho$-isomorphic if for $l=\rho^{A\cup B}$ it
happens that:

  * $\blacksquare$

$|(A)_{l-1}|=|(B)_{l-1}|$. That is, $(A)_{l-1}$ is $\rho$-isomorphic to
$(B)_{l-1}$,

  * $\blacksquare$

$h[A]=B$ where $h$ is the increasing bijection from $(A)_{l-1}$ to
$(B)_{l-1}$.

The proof of the following lemma is easy.

######  Lemma 2.4.2.

If $A$ and $B$ are strongly $\rho$-isomorphic, then $\rho^{A}=\rho^{B}$ and
$(A)_{k}$ is isomorphic to $(B)_{k}$ for each $k<\rho^{A\cup B}.$

The following proposition gives us a better picture of how two strongly
$\rho$-isomorphic sets look inside the closure of their union.

######  Proposition 2.4.3.

Let $A,B\in\text{FIN}(X)$ be such that $\max(\rho^{A},\rho^{B})<\rho^{A\cup
B}=l$. Then $A$ and $B$ are strongly $\rho$-isomorphic if and only if for any
$F\in\mathcal{F}_{l}$ with $A\cup B\subseteq F$ the following happens:

  * $\blacksquare$

$\Xi_{A}(l),\Xi_{B}(l)\not=-1$.

  * $\blacksquare$

$h[A]=B$ where $h$ is the increasing bijection from $F_{\Xi_{A}(l)}$ to
$F_{\Xi_{A}(l)}$.

###### Proof.

###### Proof of $\Leftarrow$..

Let $F\in\mathcal{F}_{l}$ be such that $A\cup B\subseteq F$. Note that
$(A)_{l-1}\sqsubseteq F_{\Xi_{A}(l)}$ and $(B)_{l-1}\sqsubseteq
F_{\Xi_{B}(l)}$. By this and since $|(A)_{l-1}|=|(B)_{l-1}|$ we have that
$h[(A)_{l-1}]=(B)_{l-1}$ where $h:F_{\Xi_{A}(l)}\longrightarrow
F_{\Xi_{B}(l)}$ is the increasing bijection. Hence, the restriction of
$h|_{(A)_{l-1}}$ is the increasing bijection from $(A)_{l-1}$ to $(B)_{l-1}$.
In this way we conclude that $h[A]=B$. Now suppose towards a contradiction
that $\min(\Xi_{A}(l),\Xi_{B}(l))=-1$. Without loss of generality we can
assume that $\Xi_{A}(l)=-1$. That is, $A\subseteq R(F)$. Since $h|_{R(F)}$ is
the identity function this means that $A=B$. This is a contradiction to the
fact that $\max(\rho^{A},\rho^{B})<\rho^{A\cup B}$. We conclude that
$\min(\Xi_{A}(l),\Xi_{B}(l))\geqslant 0$ so we are done. ∎

###### Proof of $\Rightarrow$..

For this, let $F\in\mathcal{F}_{l}$ be such that $A\cup B\subseteq F$ and
$h^{\prime}:F_{\Xi_{A}(l)}\longrightarrow F_{\Xi_{B}(l)}$ be the increasing
bijection. Since $h^{\prime}[A]=B$ it is easy to see that
$h^{\prime}[(A)_{l-1}]=(B)_{l-1}$. In this way, $|(A)_{l-1}|=|(B)_{l-1}|$. To
finish just note that $h=h^{\prime}|_{(A)_{l-1}}$ is the increasing bijection
to $(B)_{l-1}$ so $h[A]=B$. ∎

∎

######  Remark 2.4.4.

The two conditions imposed to $F$ in the previous proposition are equivalent
to the existence of distinct $i,j<n_{l}$ (namely, $\Xi_{A}(l)$ and $\Xi_{B}l$)
and a unique $S\subseteq m_{l-1}$ such that $F_{i}[S]=A$ and $F_{j}[S]=B$
(such $S$ being $F^{-1}_{\Xi_{A}(l)}[A]$).

######  Lemma 2.4.5.

Let $A,B\in\text{FIN}(X)$ be strongly $\rho$-isomorphic and let $l=\rho^{A\cup
B}$. If $F\in\mathcal{F}_{l}$ is such that $A\cup B\subseteq F$, then $A\cap
B=A\cap R(F)$. In particular, $A\cap B\sqsubseteq A$ and $\Xi_{A\cap
B}(l)=-1$.

###### Proof.

Let $h:F_{\Xi_{A}(l)}\longrightarrow F_{\Xi_{B}(l)}$ be the increasing
bijection. The inclusion from left to right is clear since $A\cap
B=A\cap(A\cap B)\subseteq A\cap(F_{\Xi_{A}(l)}\cap F_{\Xi_{B}(l)})=A\cap
R(F)$. To prove the other one just note that $h|_{R(F)}$ is the identity
function. Hence, if $\alpha\in A\cap R(F)$ then $\alpha=h(\alpha)\in B$ due to
the second point of Proposition 2.4.3.

∎

As a corollary we have:

######  Corollary 2.4.6.

Let $A,B\in\text{FIN}(X)$ be strongly $\rho$-isomorphic and let $l=\rho^{A\cup
B}$. If $F\in\mathcal{F}_{l}$ is such that $A\cup B\subseteq F$ and $A\cap
B=\emptyset$ then $A\subseteq F_{\Xi_{A}(l)}\backslash R(F)$ and $B\subseteq
F_{\Xi_{A}(l)}\backslash R(F)$. In particular, $\Xi_{\alpha}(l)=\Xi_{A}(l)$
and $\Xi_{\beta}(l)=\Xi_{B}(l)$ for any $\alpha\in A$ and $\beta\in B$.

######  Remark 2.4.7.

In virtue of Proposition 2.4.3, if $A=\\{\alpha\\}$ and $B=\\{\beta\\}$ for
$\alpha$ and $\beta$ distinct ordinals, then $A$ and $B$ are strongly
$\rho$-isomorphic if and only if $\Delta(\alpha,\beta)=\rho(\alpha,\beta)$.

######  Definition 2.4.8 (Captured families).

Let $\mathcal{D}$ be a finite subset of $\text{FIN}(X)$ and $l\in\omega$. We
say that that $\mathcal{D}$ is captured at level $l$ if
$2\leqslant|\mathcal{D}|\leqslant n_{l}$ and:

  1. $(1)$

$l=\rho^{\cup\mathcal{D}}>\rho^{D}$ for each $D\in\mathcal{D}$,

  2. $(2)$

$\\{\Xi_{D}(l)\,:\,D\in\mathcal{D}\\}=\\{0,\dots,|\mathcal{D}|-1\\}=|\mathcal{D}|$.777by
Lemma 2.3.16, this condition implies that $\rho^{E\cup D}=l$ for any distinct
$D,E\in\mathcal{D}$.

  3. $(3)$

For any distinct $D,E\in\mathcal{D}$, $D$ is strongly $\rho$-isomorphic to
$E$.

Equivalently, for any $F\in\mathcal{F}_{l}$ with $\bigcup\mathcal{D}\subseteq
F$ there is $S\subseteq m_{l-1}$ such that

$\mathcal{D}=\\{F_{i}[S]\,:\,i<|\mathcal{D}|\\}.$

Additionally, if $|\mathcal{D}|=n_{l}$ we say that $\mathcal{D}$ is fully
captured at level $l$. Whenever $D\in FIN(X)$, we say that $D$ is captured
(resp. fully captured) in case $\\{\\{\alpha\\}\,:\,\alpha\in D\\}$ is
captured (resp. fully captured). Finally, if $\mathcal{D}$ is captured at
level $l$ and we write $\mathcal{D}$ as a list of elements, say
$\mathcal{D}=\\{D_{0},D_{1},\dots,D_{n-1}\\}$, we always assume that
$\Xi_{D_{i}}(l)=i$ for each $i<n$.

In here, the set $\mathcal{D}=\\{A,B,C\\}$ is being captured as testified by
$F$ and $S$. Note that $\mathcal{D}$ forms a $\Delta$-system whose root is
contained in $R(F)$.

We will frequently use the following lemma without any explicit mention to it.
Its proof is left to the reader.

######  Lemma 2.4.9.

Let $\mathcal{D}$ be a finite subset of $\text{FIN}(X)$ which is captured at
some level $l\in\omega$. If $n$ is the common cardinality of the members of
$\mathcal{D}$ and $S$ is a non-empty finite subset of $n$, then

$\\{D[S]\,:\,D\in\mathcal{D}\\}$

is also captured at level $l$ as long as this set has at least two elements.

The following proposition is a direct consequence of the Remark 2.4.7.

######  Proposition 2.4.10.

Let $l\in\omega$ and $D\in\text{FIN}(X)$ be such that $|D|\geqslant 2$. Then
$D$ is captured at level $l$ if and only if:

  1. $(1)$

For any distinct $\alpha,\beta\in D$,
$\Delta(\alpha,\beta)=l=\rho(\alpha,\beta)$,

  2. $(2)$

For each $i<|D|$, $\Xi_{D(i)}(l)=i$ . In other words,
$\\{\,\Xi_{\alpha}(l)\,:\,\alpha\in D\,\\}=|D|$.

######  Lemma 2.4.11.

Let $\mathcal{D}$ be a finite subset of $\text{FIN}(X)$ and $l\in\omega$. For
any $A\in\mathcal{D}$, let $\alpha_{A}=\max(A)$. Suppose that there are
$j\in\omega$, $a<m_{j}$ and $C\subseteq a+1$ so that the following conditions
hold for any $A\in\mathcal{D}$:

  1. $(a)$

$\rho^{A}\leqslant j,$

  2. $(b)$

$\lVert\alpha_{A}\rVert_{j}=a$,

  3. $(c)$

$(\alpha_{A})_{j}[C]=A.$

If $D=\\{\alpha_{A}\,:\,A\in\mathcal{D}\\}$ is captured at some level $l$,
then $\mathcal{D}$ is also captured at level $l$.

###### Proof.

First observe that that $l>j$. In order to prove that $\mathcal{D}$ is
captured at level $l$, we will prove that the three points in Definition 2.4.8
are satisfied.

Claim 1: $l=\rho^{\cup\mathcal{D}}$ and $l>\rho^{A}$ for each
$A\in\mathcal{D}$.

###### Proof of claim.

We know that $l>\rho^{A}$ for each $A\in\mathcal{D}$ due to the condition (a)
written above. $l\leqslant\rho^{\cup\mathcal{D}}$ because if
$A,B\in\mathcal{D}$ are distinct then
$\alpha_{A},\alpha_{B}\in\bigcup\mathcal{D}$ and
$\rho(\alpha_{A},\alpha_{B})=l$. Lastly, if $F\in\mathcal{F}_{l}$ is such that
$D\subseteq F$ then
$A\subseteq(\alpha_{A})_{j}\subseteq(\alpha_{A})_{l}\subseteq F$ for any
$A\in\mathcal{D}$. In this way, $\bigcup\mathcal{D}\subseteq F$. Thus,
$\rho^{\mathcal{D}}\leqslant l$. ∎

Claim 2: $\\{\Xi_{A}(l)\,:\,A\in\mathcal{D}\,\\}=n_{l}$.

###### Proof of claim.

As $l>\rho^{A}$ for each $A\in\mathcal{D}$, then
$\Xi_{A}(l)=\Xi_{\alpha_{A}}(l)$ for any $A\in\mathcal{D}$ due to the Remark
2.3.15. The claim follows from this fact. ∎

Claim 3: For any distinct $A,B\in\mathcal{D}$, $A$ is strongly
$\rho$-isomorphic to $B$.

###### Proof of claim.

Let $h:(A)_{l-1}\Longrightarrow(B)_{l-1}$ be the increasing bijection. As $D$
is fully captured, we know that $\\{\alpha_{A}\\}$ is strongly isomorphic to
$\\{\alpha_{B}\\}$. In this way, $h(\alpha_{A})=\alpha_{B}$. Since $k\leqslant
l-1$, we conclude that $h[(\alpha_{A})_{k}]=(\alpha_{B})_{k}$. Thus,

$h[A]=h[\,(\alpha_{A})_{k}[C]\,]=(\alpha_{B})_{k}[C]=B.$

This finishes the proof. ∎

∎

######  Definition 2.4.12 ($n$-capturing schemes).

Let $\mathcal{P}$ be a partition of $\omega$ and $n\in\omega$. We say that
$\mathcal{F}$ is $n$-$\mathcal{P}$-capturing if for each uncountable
$\mathcal{S}\subseteq\text{FIN}(X)$ and $P\in\mathcal{P}$ there are infinitely
many $l\in P$ with some $\mathcal{D}\in[\mathcal{S}]^{n}$ which is captured at
level $l$. Whenever $\mathcal{P}=\\{\omega\\}$, we simply say that
$\mathcal{F}$ is $n$-capturing.

######  Definition 2.4.13 (capturing schemes).

Let $\mathcal{P}$ be a partition of $\omega$. We say that $\mathcal{F}$ is
$\mathcal{P}$-capturing (resp. capturing) if $\mathcal{F}$ is
$n$-$\mathcal{P}$-capturing (resp. $n$-capturing) for each $n\in\omega$.

######  Definition 2.4.14 (fully capturing schemes).

Let $\mathcal{P}$ be a partition of $\omega$. We say that $\mathcal{F}$ is
$\mathcal{P}$-fully capturing if for each uncountable
$\mathcal{S}\subseteq\text{FIN}(X)$ and $P\in\mathcal{P}$ there are infinitely
many $l\in P$ with some $\mathcal{D}\in\text{FIN}(\mathcal{S})$ which is fully
captured at level $l$. Whenever $\mathcal{P}=\\{\omega\\}$, we simply say that
$\mathcal{F}$ is fully capturing.

The following Lemma was first proved in [112] (Lemma 7.1) and it presents
useful equivalences of the previous definitions.

######  Lemma 2.4.15.

Let $\mathcal{F}$ be a construction scheme and $\mathcal{P}$ be a partition of
$\omega$ compatible with $\tau$. Then:

  * $\blacksquare$

For each $n\in\omega$, $\mathcal{F}$ is $n$-$\mathcal{P}$-capturing if and
only if for each $S\in[X]^{\omega_{1}}$ and $P\in\mathcal{P}$ there is
$D\in[S]^{n}$ which is captured at some level $l\in P$.

  * $\blacksquare$

$\mathcal{F}$ is $\mathcal{P}$-fully capturing if and only if for each
$S\in[X]^{\omega_{1}}$ and $P\in\mathcal{P}$ there is $D\in\text{FIN}(S)$
which is fully captured at some level $l\in P$.

###### Proof.

We will only prove the second point as the first one is proved in a similar
way. In order to prove the nontrivial direction of this statement, let
$\mathcal{S}$ be an uncountable subset of $\text{FIN}(X)$ and fix
$k\in\omega$. For any $A\in S$, let $\alpha_{A}=\max(A)$. Due to the
pigeonhole principle, we can find an uncountable
$\mathcal{S}^{\prime}\subseteq S$, $k<j\in\omega$, $a<m_{j}$ and $C\subseteq
a$ so that the following conditions hold for any $A\in\mathcal{S}^{\prime}$:

  1. $(a)$

$\rho^{A}\leqslant j.$

  2. $(b)$

$\lVert\alpha_{A}\rVert_{j}=a$,

  3. $(c)$

$(\alpha_{A})_{j}[C]=A.$

By the point (b), it follows that $\rho(\alpha_{A},\alpha_{B})>j$ for any two
distinct $A,B\in\mathcal{S}^{\prime}$. According to our assumptions, there is
$\mathcal{D}\in\text{FIN}(\mathcal{S})$ for which
$D=\\{\alpha_{A}\,:\,A\in\mathcal{D}\\}$ is fully captured at some level $l$.
Then $\mathcal{D}$ is captured in level $l$ by virtue of the Lemma 2.4.11.
Thus, the proof is over.

∎

We are now ready to state the Capturing axioms. The axiom $FCA(part)$ was
introduced in [112]. The other axioms were later studied in [47], [58], [68]
and [69].

Fully Capturing Axiom [FCA]: There is a fully capturing construction scheme
over $\omega_{1}$ of every possible good type.

Fully Capturing Axiom with Partitions [FCA(part)]: There is a
$\mathcal{P}$-fully capturing construction scheme over $\omega_{1}$ for every
good type $\tau$ and each partition $\mathcal{P}$ compatible with $\tau$.

$n$-Capturing Axiom [CAn]: There is an $n$-capturing construction scheme over
$\omega_{1}$ of every possible good type satisfying that $n\leqslant n_{k}$
for each $k\in\omega\backslash 1$.

$n$-Capturing Axiom with Partitions [CAn(part)]: There is a
$\mathcal{P}$-$n$-capturing construction scheme over $\omega_{1}$ for every
good type $\tau$ satisfying that $n\leqslant n_{k}$ for each
$k\in\omega\backslash 1$ and each partition $\mathcal{P}$ compatible with
$\tau$.

Capturing Axiom [CA]: $CA_{n}$ holds for each $n\in\omega$ and there is a
capturing construction scheme over $\omega_{1}$ for every good type satisfying
that the sequence $\langle n_{k+1}\rangle_{k\in\omega}$ is non-decreasing and
unbounded.

Capturing Axiom with partitions [CA(part)]: $CA_{n}(part)$ holds for each
$n\in\omega$ and there is a capturing construction scheme over $\omega_{1}$
for every good type $\tau$ satisfying that the sequence $\langle
n_{k+1}\rangle_{k\in\omega}$ is non-decreasing and unbounded and each
partition $\mathcal{P}$-compatible with $\tau$.

Note that the following relations hold between the previously defined
capturing axioms where the arrows denote the implication relation.

${CA_{2}(part)}$${CA_{3}(part)}$${\dots}$${CA(part)}$${FCA(part)}$${CA_{2}}$${CA_{3}}$${\dots}$${CA}$${FCA}$

In [58], Damian Kalajdzievski and Fulgencio Lopez proved that none of the
arrows going from $CA_{n+1}$ to $CA_{n}$ (resp. from $CA_{n+1}(part)$ to
$CA_{n}(part)$) can be reversed. They also proved the following theorem.

######  Theorem 2.4.16.

Let $\kappa$ be an uncountable cardinal. Then
$\mathbb{1}_{\mathbb{C}_{\kappa}}\vDash\text{\say{ $\text{FCA}(part)$ }}$.

For the sake of completeness, we will prove such results later on.
Furthermore, we will show that it is consistent that there is an $n$-capturing
construction scheme which is not $\mathcal{P}$-$n$-capturing for any partition
of $\omega$ in at least two infinite pieces. For this, we need to introduce a
new cardinal invariant.

######  Definition 2.4.17 (Parametrized Martin’s numbers).

Let $\mathcal{F}$ be a construction scheme and $2\leqslant n\in\omega$. We
define $\mathfrak{m}^{n}_{\mathcal{F}}$ as follows:

$\mathfrak{m}^{n}_{\mathcal{F}}=\begin{cases}\omega&\textit{if
}\mathcal{F}\textit{ is not }n\textit{-capturing}\\\
\min(\mathfrak{m}(\mathbb{P})\,:\,\mathbb{P}\textit{ is }ccc\textit{ and
}\mathbb{P}\Vdash\text{\say{$\mathcal{F}\textit{ is
}n-capturing$}})&\textit{if }\mathcal{F}\textit{ is
}n\textit{-capturing}\end{cases}$

$\mathfrak{m}^{2}_{\mathcal{F}}$ is denoted simply as
$\mathfrak{m}_{\mathcal{F}}.$

Of course, if $\mathcal{F}$ is $n$-capturing then
$\mathfrak{m}^{n}_{\mathcal{F}}\geqslant\omega_{1}.$ The following theorem
will be proved in Chapter 5.

######  Theorem 2.4.18.

For any $n$-capturing construction scheme $\mathcal{F}$ there is a
$ccc$-forcing $\mathbb{P}$ for which

$\mathbb{P}\Vdash\text{\say{ $\mathfrak{m}_{\mathcal{F}}^{n}>\omega_{1}$ }}.$

We end this chapter by announcing the main theorem of the thesis.

######  Theorem 2.4.19.

The $\Diamond$-principle implies $\text{FCA}(part)$.

##  Chapter 3 Applications

###  3.1 Gaps, towers and almost disjoint families

This section is dedicated to the study of gaps and towers. In a general
framework, one of the main reasons for studying gaps in Boolean algebras is
because of their close relation with the existence of extensions of both
homomorphisms and embeddings from one Boolean algebra to another one. More
concretely, in [97] Roman Sikorski implicitly gave a criterion for the
extension of homomorphisms in Boolean algebras. Such criterion tells us that
the objects the we now know as gaps are the only thing which can prevent an
homomorphism to be extended. As we will center our attention solely on gaps
over $\mathscr{P}(\omega)/\text{FIN}$, all the definitions included here are
stated only for such algebra.

The reader interested in knowing more about towers and gaps is referred to
[13], [14], [37], [80], [91], [102], [110], [113], [121], [122] and [132].

We start by recalling the main definitions regarding this chapter.

######  Definition 3.1.1 (Towers).

Let $X$ be a countable set and $\mathcal{T}\subseteq[X]^{\omega}$. We say that
$\mathcal{T}$ is a tower111It is also common in the literature to call these
objects pre-towers, and towers are pre-towers which are maximal with respect
to the end-extension. As we won’t be dealing with maximal pre-towers, we’ve
opted to simply refer to them as ”towers” to simplify the notation. if it is
well-ordered with respect to $\subseteq^{*}$. Furthermore, for an ordinal
$\kappa$, we say that $\mathcal{T}$ is a $\kappa$-tower whenever it is a tower
and $ot(\mathcal{T})=\kappa.$

######  Definition 3.1.2 (Gaps and Pregaps).

Let $X$ be a countable set and $\mathcal{L},\mathcal{R}\subseteq[X]^{\omega}$.
We say that $(\mathcal{L},\mathcal{R})$ is pregap, and write it as
$\mathcal{L}\perp\mathcal{R}$, if $L\cap R=^{*}\emptyset$ for all
$L\in\mathcal{L}$ and $R\in\mathcal{R}$. An element $C\in[X]^{\omega}$ is said
to separate $(\mathcal{L},\mathcal{R})$ if $L\subseteq^{*}C$ and $C\cap
R=^{*}\emptyset$ for each $L\in\mathcal{L}$ and $R\in\mathcal{R}$. Finally, we
say that $(\mathcal{L},\mathcal{R})$ is a gap if it is a pregap and there is
no $C\in[X]^{\omega}$ separating it.

######  Remark 3.1.3.

It is well known that there are no countable gaps. That is, there is no gap
$(\mathcal{L},\mathcal{R})$ with both $\mathcal{L}$ and $\mathcal{R}$
countable.

######  Remark 3.1.4.

Whenever the domain of a pregap is not specified (the set $X$ of the previous
definition) we assume that such domain is $\omega$.

######  Definition 3.1.5 (Type of a gap).

Let $(X,\leqslant_{X})$ and $(Y,\leqslant_{Y})$ be two partial orders. We say
that a pregap $(\mathcal{L},\mathcal{R})$ is an $(X,Y)$-pregap if
$(\mathcal{L},\subseteq^{*})$ and $(\mathcal{R},\subseteq^{*})$ are isomorphic
to $(X,\leqslant_{X})$ and $(Y,\leqslant_{Y})$ respectively. Furthermore, if
$(\mathcal{L},\mathcal{R})$ is a gap we will refer to it as an $(X,Y)$-gap.
Whenever both sides of a pregap $(\mathcal{L},\mathcal{R})$ are index by some
set $I$, say $\mathcal{L}=\langle A_{i}\rangle_{i\in I}$ and
$\mathcal{R}=\langle B_{i}\rangle_{i\in I}$ we may refer to
$(\mathcal{L},\mathcal{R})$ as $(A_{i},B_{i})_{i\in I}$. Of course, if $I$ is
a partial order and we specify that the pregap is an $(I,I)$-gap it is
understood such an indexing is order preserving.

It is easy to construct an $(\omega_{1},\omega_{1})$-gap using $CH$. However,
in 1909, Felix Hausdorff gave a clever recursive construction of an
$(\omega_{1},\omega_{1})$-gap without appealing to any extra axioms. This is
quite surprising, as it is required to overcome $\mathfrak{c}$ obstacles in
only $\omega_{1}$-many steps. In Theorem 3.1.21, we will prove an analogous
result but for large collection of partial orders of size $\omega_{1}$.