# Introduction to Ramsey Spaces (AM-174)

Stevo Todorcevic
Pages: 296
https://www.jstor.org/stable/j.ctt7sxqn

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Introduction
(pp. 1-2)

This book is intended to be an introduction to a rich and elegant area of Ramsey theory that concerns itself with coloring infinite sequences of objects and which is for this reason sometimes called infinite-dimensional Ramsey theory. Transferring basic pigeon hole principles to their higher dimensional versions to increase their applicability is thus the subject matter of this theory. In fact, this tendency in Ramsey theory could be traced back to the invention of the original Ramsey theorem, which is nothing other than a higher dimensional version of the principle that says that a finite coloring of an infinite set...

4. Chapter One Ramsey Theory: Preliminaries
(pp. 3-26)

Recall the notion of acoidealon some index setS, a collection${\cal H}$of subsets ofSwith the following properties:

(1)$\not 0 \notin {\cal H}$but$S\; \in {\cal H}$,

(2)$M\; \subseteq \;N$and$M\; \in {\cal H}$imply$N\; \in {\cal H}$,

(3)$M\; = \;{N_0} \cup \;{N_1}$and$M\; \in {\cal H}$imply${N_i} \in {\cal H}$for some$i\; = \;0,\;1$.

We shall be concerned here only with infinite index sets S and we shall always make the implicit assumption that the coideal${\cal H}$isnonprincipalwhich means that we shall consider coideals with the first condition strengthened as follows:

(1’)$S \in {\cal H}$, but$\{ x\} \; \notin \;{\cal H}$for all$x\; \in \;S$.

Thus, coideals are notions of largeness for subsets of various setsSthat...

5. Chapter Two Semigroup Colorings
(pp. 27-48)

Acompact semigroup Sis a nonempty semigroup with a compact Hausdorff topology for which

$x\; \mapsto \;xs$

is a continuous map for all$s\; \in \;S$. The reader should be warned that this terminology is nonstandard, since usually “compact semigroup” means that the semigroup operation is jointly continuous in both factors. We chose this terminology only to avoid the somewhat awkward “compact semitopological semigroup” that corresponds better to the notion we study here.

Example 2.1.1If X is a compact Hausdorff space, then the Tychonov cube${X^X}$is a compact semigroup with the composition operation, since for each g the map$f \mapsto f\; \circ g$is...

6. Chapter Three Trees and Products
(pp. 49-62)

In this section by atreewe mean a rooted finitely branching tree of height$\omega$with no terminal nodes. Given a treeT, and$n\; \in \;\omega$, let$T(n)$denote thenth level ofT. AsubtreeofTis a subset ofTwith an induced tree-ordering. Note that in general for a subtreeSofTthenth level$S(n)$may not be alevel set, i.e., included in some level$T(m)$ofTalthough we typically work with subsetsSofTfor which all levels are level subsets ofT. One such subtree is the subtree

$T(A)\; = \;\bigcup\limits_{n \in A} {T(n)}$...

7. Chapter Four Abstract Ramsey Theory
(pp. 63-92)

An infinite-dimensional Ramsey theoretic result is usually given under some restriction on the colorings. It turns out that an appropriate variation of the classical topological notion of sets with the Baire property leads us to a restriction that seems optimal. The purpose of this section is to present a variation that is used in the rest of the book.

Definition 4.1Let X be a given set and let${\cal P}$be a collection of nonempty subsets of X that we call basic sets. For$P\; \in \;{\cal P}$, let

$\mathcal{P}\; \upharpoonright P\; = \;\{ Q\; \in \;\mathcal{P}:\;Q\; \subseteq \;P\}$.

We say that a subset Y of X is$\mathcal{P}-Baire$if for every...

8. Chapter Five Topological Ramsey Theory
(pp. 93-134)

The special case of the Abstract Ramsey Theorem when$\mathcal{R} = \mathcal{S}$, when$\leqslant \, = \, \leqslant ^\circ$, and when$r = s$is of independent interest since in this case the basic sets

$[a,\;B] = \{ A\; \in \;\mathcal{R}:A \leqslant B\;\& \;(\exists n)\;{r_n}(A)\; = \;a\}$

for$a\; \in \;\mathcal{A}\mathcal{R}$and$B\; \in \mathcal{R} = \mathcal{S}$form a base for a topology on$\mathcal{R}$that we call thenatural topologyof$\mathcal{R}$and which extends the usual metrizable topology on$\mathcal{R}$when we consider it a subspace of the Tychonov cube$\mathcal{A}{\mathcal{R}^\mathbb{N}}$. The axioms$A.1, A.2, A.3,$and$A.4$from the previous chapter reduce to the following set of axioms (still denoted the same way) about a triple

$(\mathcal{R},\; \leqslant ,\;r)$

of objects, where$\mathcal{R}$is a nonempty set,...

9. Chapter Six Spaces of Trees
(pp. 135-178)

In this section, unless otherwise specified, by atreewe always mean a rooted finitely branching tree of some height$\leqslant \omega$. Given a treeTand$n\; \in \;\omega$, let$T(n)$denote thenthlevelofT. Thus, theheightofTis simply the minimal$n \leqslant \omega$such that$T(n)\; = \;\not 0$. For a set$A\; \subseteq \omega$, let$T(A)\; = \;\bigcup\nolimits_{n \in A} {T(n)}$. Throughout most of this chapter, we letUbe afixedrooted finitely branching tree of height$\omega$with no terminal nodes and we study its subtrees. Recall that asubtreeofUis simply a subsetTofUthat, with the induced ordering, is...

10. Chapter Seven Local Ramsey Theory
(pp. 179-218)

The local Ellentuck theory in its most fundamental form deals with Ramsey spaces of the form

$({\mathbb{N}^{[\infty ]}},\;\mathcal{H},\; \subseteq ,\;r)$, (7.1)

where${\mathbb{N}^{[\infty ]}}$is the collection of all infinite subsets of$\mathbb{N},\;\not 0\; \ne \;\mathcal{H} \subseteq \;{\mathbb{N}^{[\infty ]}}$and where$r = ({r_n})$is the standard sequence of restriction maps:

${r_n}(A) = {\text{the}}\;{\text{first}}\;n\;{\text{members}}\;{\text{of}}\;A$. (7.2)

Thus, the extremal case$\mathcal{H} = {\mathbb{N}^{[\infty ]}}$is the original Ellentuck space, but as we shall soon see there are many other interesting choices for$\mathcal{H}$. The need to develop such a theory comes from the fact that sometimes when considering colorings of${\mathbb{N}^{[\infty ]}}$one needs a monochromatic cube${M^{[\infty ]}}$of a special form, not just an arbitrary cube given to...

11. Chapter Eight Infinite Products of Finite Sets
(pp. 219-236)

The Ramsey theory of infinite products of finite sets has several aspects. The first aspect is in describing a field$\mathcal{M}$of subsets¹ of${\mathbb{N}^\infty }$with the following property: For every sequence$({m_i})$of positive integers, there is a sequence$({n_i})$of positive integers such that for every$\mathcal{M}$-measurable coloring

$c:\prod\limits_{i = 0}^\infty {{H_i}} \to \{ 0,\;1\}$(8.1)

such that${H_i} \subseteq \mathbb{N}$and$|{H_i}|\; = {n_i}$for alli, there exist${J_i} \subseteq {H_i}$with$|{J_i}|\; = {m_i}$for allisuch thatcis constant on the subproduct$\prod\nolimits_{i = 0}^\infty {{J_i}}$.We use the symbol

$\left( {\begin{array}{*{20}{c}} {{n_0}} \\ {{n_1}} \\ {{n_2}} \\ \vdots \\ \end{array} } \right)\;{ \to _\mathcal{M}}\;\left( {\begin{array}{*{20}{c}} {{m_0}} \\ {{m_1}} \\ {{m_2}} \\ \vdots \\ \end{array} } \right)$(8.2)

as a shorthand for this statement. Another aspect of the theory is in finding out how fast the...

12. Chapter Nine Parametrized Ramsey Theory
(pp. 237-258)

The purpose of this and the next few sections is to show that the Ellentuck space$({\mathbb{N}^{[\infty ]}},\; \subseteq ,\;r)$can be parametrized by the products of finite sets. The parametrized theory is built in steps starting from the following basic pigeon hole principle proved above in Section 3.3.

Lemma 9.1There is an$R:\;\mathbb{N}_ + ^{ < \infty } \to {\mathbb{N}_ + }$such that for every infinite sequence$({m_i})$of positive integers and for every coloring

$c:\bigcup\nolimits_{k \in \mathbb{N}} {\prod\nolimits_{i < k} {R({m_0},\; \ldots ,\;{m_i})\; \to \;\{ 0,\;1\} } }$(9.1)

there exist${H_i} \subseteq \;R({m_0},\; \ldots ,\;{m_i})$, with$|{H_i}|\; = {m_i}$for all i and an infinite set$A \subseteq \mathbb{N}$such that c is monochromatic on

$\bigcup\nolimits_{k \in A} {\prod\nolimits_{i < k} {{H_i}} }$. (9.2)

From now on, we fix$R:\;\mathbb{N}_ + ^{ < \infty } \to {\mathbb{N}_ + }$satisfying Lemma 9.1 and, modifying...

13. Appendix
(pp. 259-270)
14. Bibliography
(pp. 271-278)
15. Subject Index
(pp. 279-284)
16. Index of Notation
(pp. 285-287)