Ramsey's coheirs

TL;DR

This paper employs model theoretic coheir concepts to provide concise proofs of classical and modern theorems in Ramsey Theory, including Ramsey's, Hindman's, and Hales-Jewett theorems, as well as principles leading to Carlson and Gowers partition theorems.

Contribution

It introduces a novel model theoretic approach using coheirs to simplify and unify proofs of key results in Ramsey Theory.

Findings

Short proofs of Ramsey's, Hindman's, and Hales-Jewett theorems

New Ramsey theoretic principles related to Carlson and Gowers

Unified model theoretic framework for classical Ramsey results

Abstract

We use the model theoretic notion of coheir to give short proofs of old and new theorems in Ramsey Theory. As an illustration we start from Ramsey's theorem itself. Then we prove Hindman's theorem and the Hales-Jewett theorem. Finally, we prove two Ramsey theoretic principles that have among their consequences partition theorems of Carlson and of Gowers.

Full text

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**Ramsey’s coheirs

**Eugenio Colla and Domenico ZambellaUniversità di Torino\monthname 2024

Abstract  We use the model theoretic notion of coheir to give short proofs of old and new theorems in Ramsey Theory. As an illustration we start from Ramsey’s theorem itself. Then we prove Hindman’s theorem and the Hales-Jewett theorem. Finally, we prove two Ramsey theoretic principles that have among their consequences partition theorems due to Carlson and to Gowers.

Msc: primary 05D10, secondary 03C98, 03H99.

1**  Introduction**

Ramsey theory has substantial and diverse applications to many parts of mathematics. In particular, Ramsey’s theorem has foundational applications to model theory through the Ehrenfeucth-Mostowski construction of indiscernibles and generalizations thereof. In this paper we explore the converse direction, that is, we use model theory to obtain new proofs of classical results in Ramsey Theory.

The Stone-Čech compactification, obtained via ultrafilters, is a widely employed method for proving Ramsey theoretic results. One of its first major applications is the celebrated Galvin-Glazer proof of Hindman’s theorem, see e.g. [Blass]. Our methods are related, but alternative, to the ultrafilter approach. We replace $\beta G$ (the Stone-Čech compactification of a semigroup $G$) with a large saturated elementary extension of $G$, i.e. a monster model of $\textrm{Th}(G/G)$. One immediate advantage is that we work with elements of a natural semigroup with a natural operation. In contrast, elements of $\beta G$ are ultrafilters, that is, sets of sets, and the semigroup operation among ultrafilters is far from straightforward.

This idea is not completely new: in his seminal work on the applications of topological dynamics to model theory [Newelski1, Newelski2], Newelski replaces the semigroup $\beta G$ with the space of types over $G$ with a suitably defined operation. Our approach is similar, except that, unlike Newelski, we do not pursue connections with topological dynamics, but rather offer an alternative realm of application. The investigation of alternative methods in the study of regularity phenomena has been called for by Di Nasso [Mauro2, Open problem #1]. This article contains a possible answer.

The model theoretic tools employed in this paper are relatively basic. Section 2 is meant to give an accessible overview of the necessary notions for readers whose expertise is not primarily in model theory. Our results do not require assumptions of model theoretic tameness such as stability, NIP, etc., much like those that use nonstandard analysis, for example in [Mauro]. Investigating the effect of such assumptions remains as future work.

The second author is grateful to Pierre Simon for suggesting the comparison with nonstandard analysis. Both authors would like to thank Vassilis Kanellopoulos for helpful conversation. When this paper was essentially complete, we became aware of [ACG], which is worth mentioning since it employs similar methods in a related context.

---

The paper is divided into two parts. In the first part we prove that the notion of coheir leads to short and elegant proofs of well-known results. Most proofs in this part may be considered folklore, though they have not appeared in the literature so far. They are included here to provide a self-contained, gentle introduction to the techniques that are used in the second part.

As a preliminary illustrative step, we present a proof of Ramsey’s theorem (Theorem 3.1). Then we prove a generalization of Hindman’s theorem (Theorem 5.1), which is required in the second part of the paper. We also show how to combine Ramsey’s and Hindman’s theorems in a single proposition – the Milliken–Taylor theorem (Theorem 5.3). Finally, we prove an abstract algebraic version of the Hales-Jewett theorem (Theorem 6.4) due to Sabine Koppelberg [Koppelberg].

In the second part of the paper we prove two Ramsey-theoretic properties of semigroups (Lemmas 7.1 and 8.1). As an application, we derive a generalization of Carlson’s theorem on colourings of variable words which we present in the style of Koppelberg (Theorem 7.2) and in its classical form (Corollary 7.3). Lemma 8.1 is a partition theorem that generalizes Gowers’s FINk Theorem [Gowers] in a different direction than [Lupini].

The extent of the generalizations mentioned above is limited, and they could be obtained in other ways, but our motivation here is to show the use and relevance of model theoretic methods. Numerous papers in the literature strengthen or generalize the partition theorems considered here. The comparison of the results that appear in these papers is not always straightforward – a few are compared in [AC].

---

The proofs in this paper require a modicum of familiarity with model theory. However, the results can be stated in an elementary language, and in the rest of this introduction we introduce the necessary terminology.

Throughout the paper $G$ is a semigroup and $\Sigma$ a non-empty set of endomorphisms of $G$. For ${\bar{a}}\in G^{\leq\omega}$ we write

${\rm fp}^{\Sigma}\,{\bar{a}}$

$\displaystyle\hfil=$

\displaystyle\Big{\{}\sigma_{0}\,{a_{i_{0}}}\kern-2.15277pt\cdot\dots\cdot\sigma_{k}\,{a_{i_{k}}}\ :\ i_{0}<\dots<i_{k}<|{\bar{a}}|,\ \ \bar{\sigma}\in\big{(}\Sigma\cup\{{\rm id}_{G}\}\big{)}^{k+1},\ k<|{\bar{a}}|\Big{\}}

Overlined symbols, such as $\bar{a}$ or $\bar{\sigma}$, always denote a tuple, and $a_{i}$, $\sigma_{i}$ denotes the $i$-th entry of that tuple.

When $\Sigma$ is empty, we write
${\rm fp}\,{\bar{a}}$.

1.1  Example

For future reference, we instantiate the definition above in the context of free semigroups. Let $G$ be the set of words on a finite alphabet $A\cup\{x\}$, where $x$ is a symbol not in $A$ which we call
variable.

Let $C$ be the set of words on the alphabet $A$. Words in $C$ are called
constant words

, while those in $G\smallsetminus C$ are called
variable words.

When $G$ is endowed with the operation of concatenation of words, $C$ and $G\smallsetminus C$ are subsemigroups of $G$. For $t\in G$ and $a\in A$, let $t(a)$ be the word obtained by replacing all the occurrences of $x$ in $t$ by $a$. Note that the map $\sigma_{a}:t\mapsto t(a)$ is an endomorphism of $G$. In the literature, when $G$ is as above and $\Sigma=\{\sigma_{a}:a\in A\}$, the elements of ${\rm fp}^{\Sigma}\,{\bar{s}}$ are called
extracted words.

We say that a tuple \bar{a}\in\big{(}{\rm fp}^{\Sigma}\bar{s}\big{)}^{\omega} is an
extracted sequence

if $a_{i}\in{\rm fp}^{\Sigma}s_{\restriction\,[n_{i},n_{i+1})}$ for some increasing sequence of positive integers $\langle n_{i}:i<\omega\rangle$. If, moreover, $a_{i}\notin C$ for all $i$, we say that $\bar{a}$ is an
extracted variable sequence

of $\bar{s}$.  $\Box$

The following definition will be used to express our results in the general context of semigroups.

1.2  Definition

Let
\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}

be a binary relation on $G$. We say that $G$ is
\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered

if for every finite $A\subseteq G$ there is a $c$ such that A\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}{c}. If $c$ can be found in some fixed $B\subseteq G$, we say \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered
by $B$

. We say that $G$ is
[TABLE]

-closed

if {a}\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}{b}\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}{c} implies {a}\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}{b}{\cdot}{c} for all ${a},{b},{c}\in G$. A
\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-chain

in $G$ is a tuple ${\bar{a}}\in G^{\leq\omega}$ such that {a_{i}}\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}{a_{i+1}}.

The preorder relation given by the length of the words on a free semigroup $G$ is a natural example that is both

$\kern 5.16663pt\cdot$

\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} -closed and \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered. A less straightforward relation is used in the proof of Theorem 7.2.

Finally, we recall two standard notions. Let $C\subseteq G$ be a subsemigroup. We say that $C$ is
nice

if $a\cdot b\in C$ implies $a,b\in C$. A homomorphism $\sigma:G\to C$ such that $\sigma_{\restriction C}={\rm id}_{C}$ is called
retraction

of $G$ onto $C$. Note that the set of constant words in Example 1.1 is a nice subsemigroup and that the maps $\sigma_{a}$ are retractions.

We are now ready to state Lemma 7.1.

Lemma Let $\Sigma$ be a finite set of retractions of $G$ onto a nice subsemigroup $C$. Let \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} be a relation on $G$ that makes it

$\kern 5.16663pt\cdot$

\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} -closed and \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered by $G\smallsetminus C$. Then, for every finite coloring of $G$, there is a \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-chain $\bar{a}\in(G\smallsetminus C)^{\omega}$ such that ${\rm fp}^{\Sigma}\bar{a}\smallsetminus C$ is monochromatic.   $\Box$

When $C$ and $\Sigma$ are empty and \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} holds for all pairs, the lemma reduces to Hindman’s theorem (Theorem 5.1).

The appropriate choice of $G$, $C$, $\Sigma$ and \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} yields Carlson’s partition theorem (in particular no model theoretic argument is necessary, see Theorem 7.2 and its Corollary 7.3).

In the last section we prove Lemma 8.1 which is similar to the lemma above but deals with composition of homomorphisms. This is also stated in an elementary language and a general version of a partition theorem by Gowers is derived from it.

2**  Coheirs, and coheir sequences**

We assume that the reader is familiar with undergraduate model theory and in this section we only review the few prerequisites that go beyond that. Proofs are omitted. The reader may consult any standard model theory textbook e.g. [TZ] (the intrepid reader may consult [DZ], some lecture notes which use the same notation and quirks as this paper). The notation and terminology are standard with the possible exception of Definitions 2.3 and 2.5.

A
sequence

is a function whose domain is a linear order. A
tuple

is a sequence whose domain is an ordinal. The domain of the tuple {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} is denoted by
|{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}|

and is called the
length

of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}.

2.1  Notation

Sometimes (i.e. not always) we may overline tuples as mnemonic. When a tuple {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}} is introduced, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{i}} denotes the $i$-th element of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}}. We write {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{\restriction I}}, where I\subseteq|{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}}|, for the tuple which is naturally associated to the restriction of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}} to $I$. The bar is dropped for ease of notation.  $\Box$

We denote the monster model by {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}} or, when dealing with semigroups, by {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}. We always work over a fixed set of parameters A\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}. When this set is a model, as it will often be, we denote it by $M$, or $G$ in the case of semigroups.

We say that a type p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}) is
finitely satisfied

in $A$ if every conjunction of formulas in p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}) has a solution in A^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|}. A global type that is finitely satisfiable in $A$ is invariant over $A$.

If $M$ is a model every consistent type p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x})\subseteq L(M) is finitely satisfied in $M$. For this reason in a few points in this paper it is necessary to work over a model. For simplicity, we always assume this.

The following is an easy, well-known fact.

2.2  Proposition

Every type q({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x})\subseteq L({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}) that is finitely satisfiable in $M$ has an extension to a global type finitely satisfiable in $M$.  $\Box$

If p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}) is finitely satisfied in $M$, the extensions of p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}) that are also finitely satisfied in $M$ are called

coheirs

of p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}).

In many cases it is useful to focus on elements instead of their types. We introduce the following notation to express that {\rm tp}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}/M,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}) is finitely satisfied in $M$. (The notion is standard in model-theory, it has no standard notation though.)

2.3  Definition

For every {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}z}|} we define

{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}

$\displaystyle\hfil\Leftrightarrow$

\displaystyle\varphi({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\,;{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b})\textrm{ for all }\varphi({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\,;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}z})\in L(M)\textrm{ such that }M^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|}\subseteq\varphi({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|}\,;{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b})

We call this the
coheir-heir

relation. We define the type

{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}

$\displaystyle\hfil=$

\displaystyle\Big{\{}\varphi({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\,;{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b})\ :\ \varphi({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\,;{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b})\in L(M)\textrm{ and }M^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|}\subseteq\varphi({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|}\,;{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b})\Big{\}}.

The tuples {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a} realizing this type are those such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}. We will use the symbol
{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\equiv_{A}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}

for the union of the types {\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} and {\rm tp}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}/M).  $\Box$

We imagine {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} as saying that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a} is

independent

from {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} over $M$. This is a very strong form of independence. In general it is not symmetric, that is, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} is not the same as {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a} (symmetry is equivalent to stability).

We shall use, sometimes without reference, the following easy lemma.

2.4  Lemma

The following properties hold for all small M,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}, and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}

{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\ \ \Rightarrow\ \ f{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}f{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}   for every f\in\textrm{Aut\kern 0.6458pt}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}/M) invariance

• 2.

{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\ \ \Leftrightarrow\ \ {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a_{0}}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{0}}  for all finite {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a_{0}}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{0}}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} finite character

• 3.

{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}  and  {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\ \ \Rightarrow\ \ {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} transitivity

• 4.

{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\ \ \Rightarrow\ \ there exists {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\equiv_{M,\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a} such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} coheir extension  $\Box$

Note that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\equiv_{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} is the intersection of all types in S(M,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}) that are coheirs of {\rm tp}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}/M). As there may be more than one of such coheirs, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\equiv_{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} need not be a complete over M,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}. In fact, completeness is a rather strong property.

2.5  Definition

If {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\equiv_{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} is a complete type (over M,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}) for every {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}^{<\omega}, every {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|}, and every tuple of variables {\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}, then we say that \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M} is
stationary.

We say
$n$-stationary

if the requirement above is restricted to |{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|=n.  $\Box$

Stationarity is often ensured by the following property.

2.6  Proposition

Fix a tuple of variable {\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x} of length $n$. If for every \varphi({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x})\in L({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}) there is a formula \psi({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x})\in L(M) such that \varphi(M^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|})=\psi(M^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|}) then \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M} is $n$-stationary.  $\Box$

2.7  Remark

Stationarity of \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M} over every model $M$ is equivalent to the stability of $T$. However, in unstable theories the assumption may hold for some particular model. For example, if every subset of $M^{n}$ is the trace of a definable set, then \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M} is $n$-stationary by the proposition above. This simple observation will be of help in the proof of Theorem 5.1. For natural example let $T=T_{\rm dlo}$ and let M\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}} have the order-type of $\mathds{R}$. By quantifier elimination every definable of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}} is union of finitely many intervals. By Dedekind completeness, the trace on $A$ of any interval of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}} coincides with that of an $M$-definable interval.  $\Box$

Let p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x})\in S({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}) be a global type that is finitely satisfiable in $M$. We say that the tuple {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}} is a
coheir sequence

of p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}) over $M$ if for every i<|{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}}|

\displaystyle\hfill{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{i}}

$\displaystyle\hfil\vDash$

\displaystyle p_{\restriction M,\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{\restriction i}}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}).

The following is a convenient characterization of coheir sequences.

2.8  Lemma

For {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}} a tuple of length $\omega$, the following are equivalent

{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}} is a coheir sequence over $M$;

• 2.

{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{n}}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{M}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{\restriction n}} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{n+1}}\equiv_{M,\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{\restriction n}}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{n}} for every $n<\omega$.  $\Box$

Let $I,<_{I}$ be a linear order. We call a function {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{a}}:I\to{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}^{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}|} an
$I$-sequence

, or simply a
sequence

when $I$ is clear.

If $I_{0}\subseteq I$ we call {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a_{\restriction I_{0}}}, the restriction of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{a}} to $I_{0}$, a
subsequence

of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{a}}. When $I_{0}$ is finite we identify {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a_{\restriction I_{0}}} with a tuple of length $|I_{0}|$.

2.9  Definition

Let $I,<_{I}$ be an infinite linear order and let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{a}} be an $I$-sequence. We say that $a$ is a
sequence of indiscernibles

over $A$ or, a sequence of
$A$-indiscernibles

, if {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a_{\restriction I_{0}}}\equiv_{A}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a_{\restriction I_{1}}} for every $I_{0},I_{1}\subseteq I$ of equal finite cardinality.  $\Box$

The following can be easily derived from the lemma above by induction.

2.10  Proposition

Every sequence of coheirs over $M$ is $M$-indiscernible.  $\Box$

3**  Ramsey’s theorem from coheir sequences**

We illustrate the relation between coheirs and Ramsey phenomena in the simplest possible case: Ramsey’s theorem. The subsequent sections build on this proof for more sophisticated results.

In this chapter we deal with finite partitions. The partition of a set $X$ into $k$ subsets is often represented by a map $f:X\to[k]$. The elements of $[k]=\{1,\dots,k\}$ are also called
colors

, and the partition a
coloring

, or
$k$-coloring

, of $X$. We say that $Y\subseteq X$ is
monochromatic

if $f$ is constant on $Y$.

Let $M$ be an arbitrary infinite set. Fix $n,k<\omega$ and fix a coloring $f$ of the set of all

$n$-subsets

of $M$, aleas the

complete $n$-uniform hypergraph

with vertex set $M$,

$\displaystyle\hfill f$

$\displaystyle\hfil:$

$\displaystyle{M\choose n}\ \ \to\ \ [k]$.

We say that $H\subseteq M$ is a
monochromatic subgraph

if the subgraph induced by $H$ is monochromatic. In the literature monochromatic subgraphs are also called

homogeneous sets.

The following is a very famous theorem which we prove here in an unusual way. The proof will serve as a blueprint for other constructions in this paper.

3.1  Ramsey’s Theorem

Let $M$ be an infinite set. Then for every positive integer $n$ and every finite coloring of the complete $n$-uniform hypergraph with vertex set $M$ there is an infinite monochromatic subgraph.

• Proof

Let $L$ be a language that contains $k$ relation symbols $r_{1},\dots,r_{k}$ of arity $n$. Given a $k$-coloring $f$ we define a structure with domain $M$. The interpretation of the relation symbols is

$\displaystyle\hfill r_{i}^{M}$

$\displaystyle\hfil=$

\displaystyle\Big{\{}a_{1},\dots,a_{n}\in M\ :\ f\big{(}\{a_{1},\dots,a_{n}\}\big{)}=i\Big{\}}.

We may assume that $M$ is an elementary substructure of some large saturated model {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}. Pick any type p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x})\in S({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}) finitely satisfied in $M$ but not realized in $M$ and let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}}=\langle{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{i}}:i<\omega\rangle be a coheir sequence of p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}).

There is a first-order sentence saying that the formulas r_{i}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{1}},\dots,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x_{n}}) are a coloring of ${M\choose n}$. Then by elementarity the same holds in {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{U}}. By indiscernibility, all tuples of $n$ distinct elements of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}} have the same color, say $1$. We now prove that there is a sequence $\bar{a}=\langle a_{i}:i<\omega\rangle$ in $M$ with the same property.

We construct $a_{\restriction i}$ by induction on $i$ as follows.

Assume as induction hypothesis that the subsequences of length $n$ of a_{\restriction i},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{\restriction n}} all have color $1$. Our goal is to find $a_{i}\in M$ such that the same property holds for a_{\restriction i},a_{i},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{\restriction n}}. By the indiscernibility of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}}, the property holds for a_{\restriction i},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{\restriction n}},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}{c_{n}}}. And this can be written by a formula \varphi(a_{\restriction i},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{\restriction n}},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}{c_{n}}}). As {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{c}} is a coheir sequence, by Lemma 2.8 we can find $a_{i}\in M$ such that \varphi(a_{\restriction i},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{\restriction n}},a_{i}). So, as the order is irrelevant, a_{\restriction i},a_{i},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c_{\restriction n}} satisfies the induction hypothesis.   $\Box$

4**  Idempotent orbits in semigroups**

In this and the following sections we fix a semigroup $G$ which we identify with a first-order structure. The language contains, among others, the symbol
\ \cdot\

which is interpreted as a binary associative operation on $G$. We write
{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}

for a large saturated elementary extension of $G$.

For any two sets {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}} we define

{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}

$\displaystyle\hfil=$

\displaystyle\Big{\{}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\ :\ {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}},\ {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\textrm{ and }\ {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\Big{\}}

In this and the next section we abbreviate \mathcal{O}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}/G), the orbit of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a} under \textrm{Aut\kern 0.6458pt}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}/G), with
{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}_{G}

. We write
{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}

for \mathcal{O}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}/G)\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}. Similarly for
{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}

and
{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}.

4.1  Lemma

If {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} is type definable over $G$ then so is {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} for any {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}.

• Proof

The set {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} is the union of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}\{{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\} as {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} ranges in {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}_{G}. The set {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}\{{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\} is type definable, say by the the type \exists{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y}\,p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}) where

\displaystyle\hfill p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c})

$\displaystyle\hfil=$

\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\ \ \wedge\ \ {\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\ \ \wedge\ \ {\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}

Note that, by the invariance of \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}, if f\in\textrm{Aut\kern 0.6458pt}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}/G), then \exists{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y}\,p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y},f{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}) defines {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}\{f{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\}. Therefore if q({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}z})={\rm tp}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}/G) then \exists{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}z}\,\big{[}q({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}z})\cup p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}z})\big{]} defines {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}.   $\Box$

By the invariance of \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}, for every f\in\textrm{Aut\kern 0.6458pt}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}/G) we have f[{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}]=f[{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}]\cdot_{G}f[{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}]. Therefore when {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}} are invariant over $G$, also {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}} is invariant over $G$. Below we mainly deal with invariant sets.

4.2  Proposition

For all $G$-invariant sets {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}, and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}

\displaystyle\hfill{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}\big{(}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}\big{)}

$\displaystyle\hfil\subseteq$

\displaystyle\big{(}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\big{)}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}.

• Proof

Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} be an arbitrary element of the l.h.s. where {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}. By extension (Lemma 2.4), there exists {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}} such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\equiv_{G,\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}. By transitivity (again Lemma 2.4), {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}. Therefore {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} belongs to the r.h.s. Finally, as {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\equiv_{G,\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}, also {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} belongs to the r.h.s. by invariance.   $\Box$

Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} be a non-empty set. When {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}, we say that it is

idempotent

(over $G$).

4.3  Corollary

Assume {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} are both $G$-invariant. Then if {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} is idempotent, also {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}} is idempotent.

• Proof

We check that if {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} is idempotent so is {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}

\displaystyle\hfill\big{(}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\big{)}\ \cdot_{G}\ \big{(}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\big{)}

$\displaystyle\hfil\subseteq$

\displaystyle{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\ \cdot_{G}\ \big{(}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\big{)} because {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}

$\displaystyle\hfil\subseteq$

\displaystyle\big{(}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\big{)}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}} by the lemma above

$\displaystyle\hfil\subseteq$

\displaystyle{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}   $\Box$

We show that, under the assumption of stationarity, the operation $\cdot_{G}$ is associative. The quotient map {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}\to{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}/{\equiv_{G}} is almost a homomorphism.

4.4  Proposition

Assume \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G} is $1$-stationary, see Definition 2.5. Fix {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} arbitrarily. Then {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b^{\prime}}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} for every {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b^{\prime}}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b^{\prime}}. Or, in other words,

\displaystyle\hfill({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b})_{G}

$\displaystyle\hfil=$

\displaystyle{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}.

• Proof

We prove two inclusions, only the second one requires stationarity.

$\subseteq$  As {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} holds by hypothesis, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}. The inclusion follows by invariance.

$\supseteq$  By invariance it suffices to show that the l.h.s. contains {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}\{{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\}. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}_{G} such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}. We claim that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b})_{G}. Both {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}} satisfy {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\equiv_{G}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}. By $1$-stationarity, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\equiv_{G,\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}. Hence {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}.   $\Box$

4.5  Corollary (associativity)

Assume \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G} is $1$-stationary. Then for all $G$-invariant sets {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}

\displaystyle\hfill{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}\big{(}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}\big{)}

$\displaystyle\hfil=$

\displaystyle\big{(}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\big{)}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}.

• Proof

We can assume that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}} are $G$-orbits. Say of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}, and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} respectively. We can assume that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}. By Proposition 4.4 the set on the l.h.s. equals ({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c})_{G}. By a similar argument the set on the r.h.s. equals ({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b^{\prime}}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c^{\prime}})_{G} for some elements {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b^{\prime}}, and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c^{\prime}}. Proposition 4.2 proves that inclusion $\subseteq$ holds in general. But inclusion between orbits amounts to equality.   $\Box$

The following lemma proves the existence of idempotent orbits. The proof is self-contained, i.e. it does not use Ellis’s theorem on the existence of idempotents in compact left topological semigroups (however, the argument is very similar). As a comparison, finding a proof in the setting of nonstandard analysis is listed as an open problem in [Mauro2].

4.6  Lemma

Assume \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G} is $1$-stationary. If {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} is minimal among the idempotent sets that are type-definable over $G$, then {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}_{G} for some (any) {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}.

• Proof

Fix arbitrarily some {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}. By Corollary 4.3, the set {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} is contained in {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}, idempotent and type-definable over $G$ by Lemma 4.1. Therefore by minimality {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}^{\prime}}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} be the set of those {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a} such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}_{G}. This set is non-empty because {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}. It is easy to verify that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}^{\prime}} is type-definable over G,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}. As it is clearly invariant over $G$, it is type-definable over $G$. By associativity it is idempotent. Hence, by minimality, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}^{\prime}}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}. Then {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}^{\prime}}, which implies {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}_{G}. That is, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} has idempotent orbit. Finally, by minimality, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}_{G}.   $\Box$

4.7  Corollary

Under the same assumptions of the lemma above, every idempotent set that is type-definable over $G$ contains an element with an idempotent orbit.  $\Box$

5**  Hindman’s theorem**

In this section we merge the theory of idempotents presented in Section 4 with the proof of Ramsey’s theorem to obtain Hindman’s theorem.

Let ${\bar{a}}$ be a tuple of elements of $G$ of length $\leq\omega$. In Section 1 we defined ${\rm fp}\,{\bar{a}}$ and the notions of

$\kern 5.16663pt\cdot$

\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} -closed and \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered. The relation \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} is introduced mainly for future reference. The classical Hindman’s theorem is obtained with the positive integers (as an additive semigroup) for $G$ and $<$ for \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}.

5.1  Hindman Theorem

Let \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} be a relation on $G$ that makes it

$\kern 5.16663pt\cdot$

\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} -closed and \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered. Then for every finite coloring of $G$ there is a \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-chain $\bar{a}$ such that ${\rm fp}\,\bar{a}$ is monochromatic. If there is no $g\in G$ such that G\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}g, we may further assume that the elements of the \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-chain are all distinct.

• Proof

We interpret $G$ as a structure in a language that extends the language of semigroups with a symbol for \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} and one for each subset of $G$. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}} be a saturated elementary superstucture of $G$. As observed in Remark 2.7, the language makes \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G} trivially $1$-stationary.

We write {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}^{\prime}} for the type-definable set \{{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}:G\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\}, which is non-empty because $G$ is \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered. We claim that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}^{\prime}} is idempotent. In fact, if {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}^{\prime}} then, as G\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}, we must have that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}. Therefore, from the

$\kern 5.16663pt\cdot$

\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} -closedness of $G$ we infer {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}{\cdot}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}^{\prime}}.

Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}} be an element of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}^{\prime}} with idempotent orbit as given by Corollary 4.7. We can assume that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}\notin G otherwise the sequence that is identically {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}} trivially proves the theorem. If we want the elements of the chain $\bar{a}$ to be distinct it suffices require that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}\notin G. By definition of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}, this can be directly assumed when there is no $g\in G$ such that G\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}g. Let p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x})\in S({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}) be a global coheir of {\rm tp}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}/G). Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{g}} be a coheir sequence of p({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}), that is

\displaystyle\hfill{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{i}}

$\displaystyle\hfil\vDash$

\displaystyle p_{\restriction G,\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{\restriction i}}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}).

We write {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction i}} for the tuple {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{i-1}},\dots,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}. By the idempotency of ({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}})_{G} and Proposition 4.4, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}} for all {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h}\in{\rm fp}\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction i}} and all $i$. It follows in particular that {\rm fp}\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction i}} is monochromatic, say all its elements have color $1$. Now, we use the sequence {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{g}} to define $\bar{a}\in G^{\omega}$ such that all elements of ${\rm fp}\,\bar{a}$ have color $1$.

Assume as induction hypothesis that {\rm fp}(a_{\restriction i},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}) is monochromatic of color $1$. Our goal is to find $a_{i}$ such that the same property holds for {\rm fp}(a_{\restriction i+1},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}).

First we claim that from the induction hypothesis it follows that, for all $j$, all elements of {\rm fp}(a_{\restriction i},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction j}}) have color $1$. In fact, the elements of {\rm fp}(a_{\restriction i},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction j}}) have the form b\cdot{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h} for some $b\in{\rm fp}(a_{\restriction i})$ and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h}\in{\rm fp}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction j}}). As {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}, we conclude that b\cdot{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h}\equiv_{G}b\cdot{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}, which proves the claim.

Let \varphi(a_{\restriction i},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{i+1}},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{\restriction i+1}}) say that all elements of {\rm fp}(a_{\restriction i},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction i+2}}) have color $1$. As {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{g}} is a coheir sequence we can find $a_{i}$ such that \varphi(a_{\restriction i},\,a_{i},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{\restriction i+1}}). Hence all elements of {\rm fp}(a_{\restriction i+1},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction i+1}}) have color $1$. Therefore $a_{i}$ is as required.   $\Box$

Hindman’s theorem generalizes to a proposition that subsumes Ramsey’s theorem. It is usually referred to as the Milliken–Taylor theorem [Milliken] and [Taylor]. By the following observation, we may use virtually the same proof.

5.2  Proposition

Assume \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G} is $1$-stationary. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{g}}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}^{\omega} be a coheir sequence of some global coheir of {\rm tp}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}/G) where {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g} has idempotent orbit. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{h}}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}^{\omega} be such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h_{i}}\in{\rm fp}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction I_{i}}}) for some finite non-empty $I_{i}\subseteq\omega$ such that $I_{i}<I_{i+1}$. Then {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{h}}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{g}}.

• Proof

Write $n_{i}$ for the minimum of $I_{i}$. It suffices to prove that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h_{i}}\equiv_{G,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{\restriction n_{i}}}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{n_{i}}}. Note that the type {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\equiv_{G}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{\restriction n_{i}}} is satisfied both by {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h_{i}} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{n_{i}}}, hence the claim follows by stationarity.   $\Box$

Write
${\rm fp}(\bar{a})_{n}$

for the $n$-uniform hypergraph with vertex set ${\rm fp}(\bar{a})$ and as edges those sets $\{h_{1},\dots,h_{n}\}$ such that $h_{i}\in{\rm fp}(a_{\restriction I_{i}})$ for some finite sets $I_{1}<\dots<I_{n}$.

5.3  Milliken-Taylor Theorem

Let \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} be a relation on $G$ that makes it

$\kern 5.16663pt\cdot$

\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} -closed and \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered. Then for every positive integer $n$ and every finite coloring of the complete $n$-uniform hypergraph with vertex set $G$ there is a \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-chain $\bar{a}$ such that ${\rm fp}(\bar{a})_{n}$ is monochromatic.  $\Box$

• Proof

Given a coheir sequence {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{g}} as in the proof of Theorem 5.1 we want to define $\bar{a}\in G^{\omega}$ such that ${\rm fp}(\bar{a})_{n}$ is monochromatic. By the proposition above, {\rm fp}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction i}})_{n} is monochromatic for every $i\geq n$. As in the proof of Theorem 5.1, we define by induction $\bar{a}\in G^{\omega}$ in such a way that {\rm fp}(a_{\restriction i},\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction n}})_{n} is a finite monochromatic subgraph of $G$.   $\Box$

6**  The Hales-Jewett theorem**

The Hales-Jewett theorem is a purely combinatorial statement that implies the van der Waerden theorem. The original proof by Alfred Hales and Robert Jewett is combinatorial [HJ]. An alternative proof, also combinatorial, is due by Saharon Shelah [Shelah]. Our proof is similar to the proof by Andreas Blass in [Blass] (based on ideas from [BBH]), but we use saturated models where he uses Stone-Čech compactification. We present three versions of the main theorem.

First we prove an abstract algebraic version due to Sabine Koppelberg [Koppelberg] which is easier to state and to prove (this version comes in two variants). The classical version follows easily from the algebraic one.

We work with the same notation as in Section 4. We say that an element {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} is
left-minimal

(w.r.t. {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}) if {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g} for every {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}.

6.1  Proposition

Assume \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G} is $1$-stationary. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} be idempotent and type-definable over $G$. Then {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} contains a left-minimal element {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} with idempotent orbit.

• Proof

Construct by induction a chain of type-definable idempotent sets {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} and elements {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}_{\alpha}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha} such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{0}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha+1}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}_{\alpha}. For $\alpha$ limit take the intersection. By idempotency of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}, it is straightforward to check that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha+1}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha}. The sets {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha} are type-definable and idempotent by 4.1 and 4.3. For $\alpha$ limit {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha} is non-empty by compactness, as it is intersection of a chain of closed sets.

For some $\alpha$ we cannot properly extend this construction. For this $\alpha$, for every {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha} we have {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g} . Hence every {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha} is left-minimal. As {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha} is idempotent, by Corollary 4.7 there is some {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}_{\alpha} with idempotent orbit.   $\Box$

6.2  Proposition

Assume \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G} is $1$-stationary. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} be idempotent and type-definable over $G$. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}_{G} be idempotent and such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}},\ {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}. Then

{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} contains some {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g} with idempotent orbit;

• 2.

if moreover {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} is left-minimal, then {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g} for every {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g} as in 1.

Note, parenthetically, that the set in 1 may not be type-definable, therefore Corollary 4.7 does not apply directly and we need an indirect argument.

• Proof

1.  From {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} we obtain that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} is idempotent. As it is also type-definable, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} contains a {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} with idempotent orbit by Corollary 4.7. There is an {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}_{G}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}, then {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}_{G}. From this we obtain that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b} is idempotent and contained in {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}.

2.  From {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} and the idempotency of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}_{G} we obtain {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}_{G}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}. As {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}, from the left-minimality of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}_{G} we obtain {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}. Hence {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}_{G}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}, by the idempotency of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}_{G}. Therefore {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}_{G}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}_{G}, which proves 2.   $\Box$

The following is a technical lemma that is required in many proofs below.

6.3  Proposition

Assume \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G} is $1$-stationary. Let \sigma:{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}\to{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}} be a semigroup homomorphism definable over $G$. Then for every {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a},{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}

\sigma\big{[}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}_{G}\big{]}\ =\ (\sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a})_{G}

• 2.

\sigma\big{[}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\big{]}\ =\ \sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}\sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}.

• Proof

As {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}} implies \sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\equiv_{G}\sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}, inclusion $\subseteq$ is clear. For the converse, note that the type \exists{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y}\,\big{[}\sigma\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x}\,\wedge\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\big{]} is trivially realized by \sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}. Therefore it is realized by all elements of (\sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a})_{G}. Hence all elements of (\sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a})_{G} are the image of some element in {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}_{G}.

2.  Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b^{\prime}}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}. By Proposition 4.4 we have \sigma\big{[}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\big{]}=\sigma\big{[}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\cdot{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b^{\prime}})_{G}\big{]}. Then it suffices to prove that \sigma\big{[}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\cdot{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b^{\prime}})_{G}\big{]}\subseteq\sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}\sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}, because by 1 and Proposition 4.4 both sides of the equality are orbits. As $\sigma$ preserves \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G} and orbits, we obtain that \sigma({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\cdot{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b^{\prime}}) is in \sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a}\cdot_{G}\sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}, as well as all other elements of \sigma\big{[}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}a^{\prime}}\cdot{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b^{\prime}})_{G}\big{]}.   $\Box$

6.4  Hales-Jewett Theorem (Koppelberg’s version)

Let $G$ be an infinite semigroup and let $C\subset G$ be a nice subsemigroup. Let $\Sigma$ be a finite set of retractions of $G$ onto $C$. Then, for every finite coloring of $C$, there is an $a\in G\smallsetminus C$ such that $\{\sigma\,a:\sigma\in\Sigma\}$ is monochromatic.

• Proof

Let G\preceq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}. Here {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}} is a monster model in a language that expands the natural one with a symbol for all subsets of $G$ and for every retraction in $\Sigma$. As observed in Remark 2.7, this makes \mathbin{\raise 7.74998pt\hbox to0.0pt{\kern 2.58334pt\rule{2.58334pt}{0.43057pt}\hss}\hbox to0.0pt{\kern 4.73611pt\rule{0.43057pt}{8.1805pt}\hss}\raise-1.29167pt\hbox{\smile}}_{G} trivially $1$-stationary. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}} be the definable set such that C=G\cap{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}. By elementarity, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}} is a nice subsemigroup of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}. The language contains also symbols for the retractions \sigma:{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}\to{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}.

By Proposition 6.1, there is a left-minimal {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}} with idempotent orbit.

By niceness, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}\smallsetminus{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} satisfy the assumptions of Proposition 6.2. Hence, by the first claim of that proposition, there is an idempotent {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}\smallsetminus{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}})\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}. In particular, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}\smallsetminus{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}. Now apply the second claim of Proposition 6.3, with {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}} for {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} to obtain \sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} for all $\sigma\in\Sigma$. As \sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g} is also idempotent, we apply Proposition 6.2 to conclude that \sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}. In particular the set \{\sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}:\sigma\in\Sigma\} is monochromatic.

Though the element {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g} above need not belong to $G\smallsetminus C$, by elementarity $G\smallsetminus C$ contains some $a$ with the same property and this proves the theorem.   $\Box$

Finally we show how the classical Hales-Jewett theorem follows from its abstract version.

If $C$ and $X$ are two semigroups we denote by $C*X$ their free product. That is, $C*X$ contains finite sequences of elements of $C\cup X$, below called
words,

that alternate elements in $C$ with elements in $X$. The product of two words is obtained concatenating them and, when it applies, replacing two contiguous elements of the same semigroup by their product. Note that $C$ and $X$ are nice subsemigroups of $C*X$. When $X$ is the free semigroup generated by a variable $x$, we denote $C*X$ by
$C[x]$.

If $w(x)$ is an element of $C[x]$ and $a\in C$ we denote by $w(a)$ the result of replacing $x$ by $a$ in $w(x)$.

6.5  Hales-Jewett Theorem (classical version)

Let $C$ be a semigroup generated by some finite set $A$. Let $x$ be a variable. Then for every finite coloring of $C[x]$ there is a $w(x)\in C[x]\smallsetminus C$ such that $\{w(a):a\in A\}$ is monochromatic.

• Proof

Let $G=C[x]$. For every $a\in A$ the homomorphism $\sigma_{a}:w(x)\mapsto w(a)$ is a retraction of $G$ onto $C$. Hence we can apply the theorem above.   $\Box$

We conclude with a variant of Theorem 6.4 that applies to a broader class of semigroup homomorphisms. This result is not required for the following.

For $\Sigma$ a set of maps $\sigma:G\to C$ and $c\in C$ we define

$\Sigma^{-1}[c]$

$\displaystyle\hfil=$

$\displaystyle\bigcap_{\sigma\in\Sigma}\sigma^{-1}[c]$

Clearly, when the maps in $\Sigma$ are retractions, $\Sigma^{-1}[c]$ is non-empty for all $c\in C$ because it contains at least $c$.

6.6  Hales-Jewett Theorem (yet another variant)

Let $C$ be a semigroup and let $\Sigma$ be a finite set of homomorphisms $\sigma:G\to C$ such that $\Sigma^{-1}[c]$ is non-empty for all $c\in C$. Then, for every finite coloring of $C$, there is a $g\in G$ such that the set $\{\sigma\,g\ :\ \sigma\in\Sigma\}$ is monochromatic.

• Proof

Let $G*C$ be the free product of the two semigroups. Any homomorphism $\sigma:G\to C$ extends canonically to a retraction of $G*C$ onto $C$. The elements of $G$ that occur in a word are replaced by their image under $\sigma$, finally the elements in the resulting sequence are multiplied. This extension is denoted by the same symbol $\sigma$.

Apply Theorem 6.4 to obtain some $w\in G*C$ such that $\{\sigma\,w:\sigma\in\Sigma\}$ is monochromatic. Suppose $w=c_{0}\cdot g_{0}\cdots\cdots c_{n}\cdot g_{n}$ for some $g_{i}\in G$ and $c_{i}\in C$, where one or both of $c_{0}$ or $g_{n}$ could be absent. Pick some $h_{i}\in\Sigma^{-1}[c_{i}]$ and let $g=h_{0}\cdot g_{0}\cdots\cdots h_{n}\cdot g_{n}$. Then $\{\sigma\,g:\sigma\in\Sigma\}$ is monochromatic as required to complete the proof.   $\Box$

7**  Carlson’s theorem**

This section is devoted to the following lemma and some of its consequences.

7.1  Lemma

Let $\Sigma$ be a finite set of retractions of $G$ onto a nice subsemigroup $C$. Let \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} be a relation on $G$ that makes it

$\kern 5.16663pt\cdot$

\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} -closed and \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered by $G\smallsetminus C$. Then, for every finite coloring of $G$, there is a \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-chain $\bar{a}\in(G\smallsetminus C)^{\omega}$ such that ${\rm fp}^{\Sigma}\bar{a}\smallsetminus C$ is monochromatic.

• Proof

The models {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}} are as in the proof of Theorem 6.4. The language is the same with \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} included. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}=\{{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}}\smallsetminus{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}:G\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g}\}. By Proposition 6.1 there is some left-minimal {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}} with idempotent orbit. As $G$ is \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered by $G\smallsetminus C$, the set {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}} is non-empty. As $G$ is

$\kern 5.16663pt\cdot$

\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} -closed and $C$ is nice, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} satisfy the assumptions of Proposition 6.2. Then, {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} contains some {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}} with idempotent orbit. By Proposition 6.3, we obtain that \sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} for all $\sigma\in\Sigma$. As (\sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}})_{G} is also idempotent, we apply the second claim of Proposition 6.3, with {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}} for {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} to conclude that \sigma\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} for all $\sigma\in\Sigma$. Now, let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{g}} be a coheir sequence as in Theorem 5.1, and assume the notation thereof. As {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c} then {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}c}=({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}})_{G}. Hence {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h}\equiv_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}g_{0}} for all $i$ and all {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h}\in{\rm fp}\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\reflectbox{\vec{\reflectbox{$g$}}}_{\restriction i}}\smallsetminus{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{C}}. In particular all these {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}h} have the same color, say color $1$. Now, we can use the sequence {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\bar{g}} to define $\bar{a}\in(G\smallsetminus C)^{\omega}$ such that all elements of ${\rm fp}^{\Sigma}\,\bar{a}\smallsetminus C$ have color $1$ by reasoning as in the proof of Theorem 5.1.   $\Box$

Carlson’s theorem is a result that combines the theorems of Hindman and Hales-Jewett and has a number of important consequences. We refer the reader to [DK] for a discussion of some of these consequences. The definitions in Example 1.1 will help matching the notation.

We first present a Koppelberg-style version of the theorem. It is obtained from the lemma above applying a suitable coding.

7.2  Carlson Theorem (à la Koppelberg)

Let $\Sigma$ be a finite set of retractions of $G$ onto a nice subsemigroup $C$. Let $\bar{s}\in(G\smallsetminus C)^{\omega}$. Then for every finite coloring of $G$, there is an increasing sequence of positive integers $\langle n_{i}:i<\omega\rangle$ and some $a_{i}\in{\rm fp}^{\Sigma}s_{\restriction\,[n_{i},n_{i+1})}\smallsetminus C$ such that ${\rm fp}^{\Sigma}\bar{a}\smallsetminus C$ is monochromatic.

• Proof

Let $G_{*}$ be the free semigroup generated by the alphabet

$\{\langle\sigma,g\rangle\ :\ \sigma\in\Sigma\cup\{{\rm id}_{G}\},\ g\in G\smallsetminus C\}$.

The semigroup $C_{*}$ is defined as $G_{*}$, only $\sigma$ is restricted to range over $\Sigma$. Clearly $C_{*}$ is a nice subsemigroup of $G_{*}$. We associate to each $\sigma\in\Sigma$ the endomorphism of $G_{*}$ that substitutes $\sigma$ for every occurrence of ${\rm id}_{G}$ in a word. These maps, which we denote by $\sigma_{*}$, are retractions of $G_{*}$ onto $C_{*}$.

If $g_{*}\in G_{*}$ has the form $\langle\sigma_{1},g_{1}\rangle\cdots\langle\sigma_{n},g_{n}\rangle$ we call $\sigma_{1}\,g_{1}\cdots\sigma_{n}\,g_{n}\in G$ the evaluation of $g_{*}$. We denote the evaluation by ${\rm eval}(g_{*})$. As $\tau\,\sigma=\sigma$ for every $\tau,\sigma\in\Sigma$, we have that ${\rm eval}(\sigma_{*}\,g_{*})=\sigma\,{\rm eval}(g_{*})$. The evaluation of $g_{*}\in C_{*}$ belongs to $C$ and, as $C$ is nice, the evaluation of $g_{*}\in G_{*}\smallsetminus C_{*}$ belongs to $G\smallsetminus C$.

We color each element of $G_{*}$ with the color of its evaluation.

We define the relation \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} on $G_{*}$. First, we need to define the well-formed elements of $G_{*}$. These are elements of the form $\langle\sigma_{1},s_{i_{1}}\rangle\cdots\langle\sigma_{n},s_{i_{n}}\rangle$ for some $i_{1}<\dots<i_{n}$. Now, for $h_{*},g_{*}\in G_{*}$ we define h_{*}\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}g_{*} if one of the following holds

1. $h_{*}$ is not well-formed while $g_{*}$ is;

2. the product (i.e., concatenation) $h_{*}g_{*}$ is well-formed.

It is immediate to verify that \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} is $G_{*}$ is

$\kern 5.16663pt\cdot$

\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} -closed and \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered by $G_{*}\smallsetminus C_{*}$. Therefore by Lemma 7.1 there is a \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-chain $\bar{a}_{*}\in(G_{*}\smallsetminus C_{*})^{\omega}$ such that ${\rm fp}^{\Sigma}\bar{a}_{*}\smallsetminus C_{*}$ is monochromatic. We can assume that all elements of $\bar{a}_{*}$ are well-formed (only the first element might be ill-formed, but we can drop it). Then the sequence $\langle{\rm eval}(a_{i*}):i\in\omega\rangle$ is as required by the lemma.   $\Box$

From the algebraic version of Carlson’s theorem we obtain the classical one in the same way as for the Hales-Jewett theorem (Theorem 6.5), which we refer to for the notation.

7.3  Corollary (Carlson’s theorem, classical version)

Let $C$ be a semigroup generated by some finite set $A$. Let $x$ be a variable. Let \bar{s}\in\big{(}C[x]\smallsetminus C\big{)}^{\omega}. Let $\Sigma$ contain, for every $a\in A$, the function $w(x)\mapsto w(a)$. Then, for every finite coloring of $C[x]$, there is an increasing sequence of positive integers $\langle n_{i}:i<\omega\rangle$ and some $a_{i}\in{\rm fp}^{\Sigma}s_{\restriction\,[n_{i},n_{i+1})}\smallsetminus C$ such that ${\rm fp}^{\Sigma}\bar{a}\smallsetminus C$ is monochromatic (with the terminology of Example 1.1, $\bar{a}$ is an extracted variable sequence of $\bar{s}$).  $\Box$

8**  Gowers’s partition theorem**

The following is similar to Lemma 7.1 but here $\Sigma$ contains compositions of homomorphisms.

8.1  Lemma

For $0<i<n$, let $G_{i}$ be a nice subsemigroup of $G_{i+1}$ and let $\sigma_{i}:G_{i+1}\to G_{i}$ be homomorphisms. Let \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} be a relation on $G_{n}$ that makes it

$\kern 5.16663pt\cdot$

\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} -closed and \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered by $G_{n}\smallsetminus G_{n-1}$. Finally, let \Sigma=\big{\{}\sigma_{i}\circ\dots\circ\sigma_{n-1}:0<i<n\big{\}}. Then, for every finite coloring of $G_{n}$, there is a \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-chain \bar{a}\in\big{(}G_{n}\smallsetminus G_{n-1}\big{)}^{\omega} such that ${\rm fp}^{\Sigma}\bar{a}\smallsetminus G_{n-1}$ is monochromatic.

• Proof

For convenience, we let $i$ run from [math], hence we agree that $\sigma_{0}:G_{1}\to G_{0}=G_{1}$ is the identity. Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{n}}=\{{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}_{n}}\smallsetminus{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}_{n-1}}\,:\,G_{n}\mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b}\} and {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{i}}=\sigma_{i}[{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{i+1}}]. Note that the {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{i}} are non-empty because $G_{n}$ is \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}}-covered by $G_{n}\smallsetminus G_{n-1}$. Also, as {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}_{i}} is a nice subsemigroup of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{G}_{i+1}}, we have that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{i}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{i+1}},\ {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{i+1}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{i}}\subseteq{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{i+1}}.

We claim there is some {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{n}}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{n}} with idempotent orbit such that, if we define {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i}}=\sigma_{i}\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i+1}} for $0\leq i<n$, the following holds

\displaystyle\hfill{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{n}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i}}

$\displaystyle\hfil=$

\displaystyle{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{n}}

$\displaystyle\hfil=$

\displaystyle({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{n}})_{G}.

Note that these equalities may be replaced by

\displaystyle\sharp_{i}\hfill{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i+1}}

$\displaystyle\hfil=$

\displaystyle{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i+1}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i}}

$\displaystyle\hfil=$

\displaystyle({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i+1}})_{G}.

Let {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{0}}={\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{1}} be any element of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{0}} with idempotent orbit. We assume as induction hypothesis that we have {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i}}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{i}} for $i\leq k$, with idempotent orbits, such that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i}}=\sigma_{i}\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{i+1}} and $\sharp_{i}$ hold for all $i<k$. We show how to find {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k+1}}.

We prove that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}} and the set {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{k+1}}\cap\sigma_{k}^{-1}[{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}], which below we denote by {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} for short, satisfy the assumptions of Proposition 6.2. The proof of the idempotency of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} is left to the reader. We prove that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\,\subseteq\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}, the proof of {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}\,\subseteq\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}} is similar. As {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{k+1}}\,\subseteq\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{B}_{k+1}} by nicety, it suffices to prove that {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}\cdot_{G}\sigma_{k}^{-1}[{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}] is contained in \sigma_{k}^{-1}[{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}]. This latter inclusion holds because, by the induction hypothesis,

\displaystyle\hfill\sigma_{k}\Big{[}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}\cdot_{G}\sigma_{k}^{-1}[{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}]\Big{]}

$\displaystyle\hfil=$

\displaystyle\sigma_{k}[{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}]\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}

$\displaystyle\hfil=$

\displaystyle{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k-1}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}

$\displaystyle\hfil=$

\displaystyle({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}})_{G}.

Now we apply Proposition 6.2 to find an idempotent {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k+1}}\in{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}\mathcal{A}}\cdot_{G}{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}. Therefore $\sharp_{k}$ is satisfied. Moreover \sigma_{k}\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k+1}}\in({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}})_{G} by Proposition 6.3, hence we can assume {\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k}}=\sigma_{k}\,{\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{k+1}} as claimed above.

Finally, as in the proof of Theorem 5.1, the required chain $\bar{a}$ is obtained from a coheir sequence of a global coheir of {\rm tp}({\color[rgb]{0.58984375,0.1953125,0.0390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.58984375,0.1953125,0.0390625}b_{n}}/G).   $\Box$

8.2  Remark

The lemma above continues to hold, with essentially the same proof, if for $\Sigma$ we take a set of the form

$\displaystyle\hfill\Sigma$

$\displaystyle\hfil=$

$\displaystyle\bigcup^{n-1}_{i=1}\ \Sigma_{i}\circ\dots\circ\Sigma_{n-1}$

where

$\displaystyle\hfill\Sigma_{i}\circ\dots\circ\Sigma_{n-1}$

$\displaystyle\hfil=$

\displaystyle\Big{\{}\sigma_{i}\circ\dots\circ\sigma_{n-1}\ :\ \sigma_{i}\in\Sigma_{i},\dots,\sigma_{n-1}\in\Sigma_{n-1}\Big{\}}

and where $\Sigma_{i}$ are some finite sets of homomorphisms $G_{i+1}\to G_{i}$ such that for every $g\in G_{i}$ the set $\Sigma_{i}^{-1}[g]$ is non-empty.  $\Box$

Let $G_{i}$ be the set of functions $a:\omega\to\{0,\dots,i\}$ with finite
support

that is, the set
$\mathop{\rm supp}(a)$

$=\{x\in\omega\,:\,a\,x\neq 0\}$ is finite. We introduce a semigroup operation on $G_{i}$ by defining $(a{\cdot}b)\,x=\max\{ax,bx\}$. This makes $G_{i}$ a nice subsemigroup of $G_{i+1}$.

8.3  Corollary (Gowers Partition Theorem)

With $G_{i}$ as above, let $\sigma_{i}:G_{i+1}\to G_{i}$ be homomorphisms and let $\Sigma$ be as in Lemma 8.1. Then for every finite coloring of $G_{n}$ there is an \bar{a}\in\big{(}G_{n}\smallsetminus G_{n-1}\big{)}^{\omega} such that ${\rm fp}^{\Sigma}\bar{a}\smallsetminus G_{n-1}$ is monochromatic and $\mathop{\rm supp}(a_{i})<\mathop{\rm supp}(a_{i+1})$.

The homomorphisms $\sigma_{i}$ usually considered in the literature are so-called tetris operations i.e. $(\sigma_{i}\,a)x=\max\{a\,x-1,\ 0\}$, or generalizations thereof. However the theorem is more general.

• Proof

Let \mathbin{\ooalign{\kern-1.72218pt-$\cr$<}} be the relation $\mathop{\rm supp}(a)<\mathop{\rm supp}(b)$ and apply Theorem 8.1.   $\Box$

9**  References**

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