# Ramsey theory for layered semigroups

Jordan Mitchell Barrett
School of Mathematics and Statistics
Victoria University of Wellington
New Zealand
<EMAIL_ADDRESS>
###### Abstract

We further develop the theory of layered semigroups, as introduced by Farah,
Hindman and McLeod, providing a general framework to prove Ramsey statements
about such a semigroup $S$. By nonstandard and topological arguments, we show
Ramsey statements on $S$ are implied by the existence of “coherent” sequences
in $S$. This framework allows us to formalise and prove many results in Ramsey
theory, including Gowers’ $\mathrm{FIN}_{k}$ theorem, the Graham–Rothschild
theorem, and Hindman’s finite sums theorem. Other highlights include: a simple
nonstandard proof of the Graham–Rothschild theorem for strong variable words;
a nonstandard proof of Bergelson–Blass–Hindman’s partition theorem for located
variable words, using a result of Carlson, Hindman and Strauss; and a common
generalisation of the latter result and Gowers’ theorem, which can be proven
in our framework.

##  1 Introduction

Ramsey theory mathematically studies to what extent regular configurations
appear in disorder. A Ramsey-type result typically has the following form: for
any finite colouring of some structure $\mathcal{M}$, we can find a
monochromatic substructure $\mathcal{N}\subseteq M$ with certain properties.
The structure $\mathcal{M}$ and required properties of $\mathcal{N}$ are what
distinguish the various results. Commonly, the structure in question will be a
semigroup. An early example is van der Waerden’s theorem on monochromatic
arithmetic progressions:

######  Theorem 1 (van der Waerden).

For every $k\in\mathbb{N}$ and finite colouring of $\mathbb{N}$, there is
$a,d\in\mathbb{N}$ such that the arithmetic progression $a,\ a+d,\ a+2d,\
\ldots,\ a+(k-1)d$ is monochromatic.

Here, the structure in question is the semigroup $(\mathbb{N},+)$. A later
example, more in the style of the results of this paper, is the Hales–Jewett
theorem about the word semigroup $A^{<\omega}$ over a finite alphabet $A$. We
let $V=\big{(}A\cup\\{x\\}\big{)}^{<\omega}\setminus A^{<\omega}$ be the set
of variable words, words over $A$ which include the variable symbol $x$. Given
$u\in V$ and $a\in A$, the (nonvariable) word $u[a]$ is formed by replacing
each occurrence of $x$ in $u$ with $a$.

######  Theorem 2 (Hales–Jewett).

For every finite colouring of $A^{<\omega}$, there is a variable word $u\in V$
such that $\\{u[a]:a\in A\\}$ is monochromatic.

Infinitary Ramsey theory received a boost in the 1970s with the advent of
ultrafilter methods, as pioneered by Glazer in his proof of Hindman’s finite
sums theorem [5, Thm 10.3]. Given a semigroup $(S,+)$, we can naturally extend
$+$ to an operation $\oplus$ on the set $\beta S$ of ultrafilters on $S$.
Furthermore, $\beta S$ admits a natural topology, making it a compact right-
topological semigroup. The rich algebraic structure of $\beta S$ has powerful
applications and consequences all throughout combinatorics [18]. More
recently, nonstandard methods have also seen success in Ramsey theory [6],
particularly in studying partition regularity of Diophantine equations [9, 10,
3].

In [11], Farah, Hindman and McLeod introduced layered semigroups, as well as
shifts and layered actions thereon. The motivation was to generalise partition
results about certain spaces of variable words, such as Gowers’
$\mathrm{FIN}_{k}$ theorem, the Hales–Jewett theorem, and Bergelson, Blass and
Hindman’s theorem on located words. Layered semigroups were further explored
by Lupini [20] and Farmaki–Negrepontis [13].

This paper should be considered a “spiritual successor” to [11]. We also work
in the setting of layered semigroups, but we consider a different, much
broader class of morphisms, called regressive maps. Working in this setting,
we develop a general framework to prove partition theorems about a layered
semigroup $S$, assuming only the existence of certain “coherent” sequences in
$S$. This framework allows a general way to formulate and prove many
fundamental results of Ramsey theory.

While [11] was phrased in the language of ultrafilters, we instead formalise
our results using nonstandard analysis. We believe the nonstandard formulation
is more intuitive, but our work has an equivalent translation in the setting
of ultrafilters. §2.1 reviews the necessary concepts of nonstandard analysis,
working in an internal superstructure model. Effectively, every object $M$
under consideration is assigned a nonstandard extension ${}^{*}M$, such that
the transfer principle holds—$M$ and ${}^{*}M$ satisfy the same “elementary”
properties.

Ellis’ theory of compact semitopological semigroups (CSTSs) is also essential
to our nonstandard study of Ramsey theory, and we discuss the topological
prerequisites in §2.2. For a semigroup $S$, we define a topology on ${}^{*}S$
such that ${}^{*}S$ is “nearly” a CSTS. This gives us analogues of results in
CSTS theory—particularly the Ellis-Numakura lemma guaranteeing the existence
of idempotents, which are essential to our work.

In §3, we define layered semigroups $S$—those which can be partitioned into
countably many layers $S_{0},S_{1},\ldots$ so that $S_{0}\cup\cdots\cup S_{n}$
forms a semigroup, of which $S_{n}$ is an ideal. We see some examples which
naturally occur in Ramsey theory, some of which are in fact partial, but
adequate in a specified sense. §4 considers regressive maps on $S$—semigroup
homomorphisms $f:S\to S$ which map layers downwards, and don’t separate or
reorder them. Natural examples of maps on layered semigroups are generally
regressive, hence this notion distills the essential Ramsey-theoretic
properties of such maps.

In §3.1 and §4.1, nonstandard analysis comes in, as we consider sequences
$(\alpha_{i})$ of nonstandard elements where $\alpha_{n}\in{}^{*}S_{n}$. Such
a sequence is coherent if it is closed under all regressive maps under
consideration, and Ramsey if $\alpha_{n}$ absorbs all $\alpha_{i}$,
$i\leqslant n$ under the semigroup operation. We present the main mechanism
for proving Ramsey statements in §5. If $\mathcal{F}$ is a collection of
regressive maps on $S$, the framework is summarised diagrammatically below:

${(S,\mathcal{F})}$${\mathcal{F}\text{-coherent}}$${\mathcal{F}\text{-Ramsey}}$
Ramsey statement about $(S,\mathcal{F})$ Thm 50Thm 45

It is difficult to construct general arguments giving the implication  above,
without having to impose very strong conditions on $S$ and $\mathcal{F}$.
Therefore, the construction of an $\mathcal{F}$-coherent will usually depend
on the specific semigroup under consideration. The most general construction
we give is Lemma 76, for “complete” subsemigroups of $\mathrm{FIN}^{A}$—this
covers Gowers’ theorem and Bergelson–Blass–Hindman’s theorem on located
variable words.

However, the other two implications work much more generally, only requiring
weak, natural conditions on $S$ and $\mathcal{F}$. As a result, Ramsey
statements on $(S,\mathcal{F})$ can be reduced to the existence of coherent
sequences in $S$. Towards the end of the paper, we show how our general
framework recovers many fundamental results in Ramsey theory, including

  * •

Gowers’ $\mathrm{FIN}_{k}$ theorem, and its generalisation due to Lupini
(§6.1);

  * •

The Graham–Rothschild parameter sets theorem (§6.2);

  * •

The Galvin–Glazer theorem and Hindman’s finite sums theorem (§6.3);

  * •

An infinitary, multivariable generalisation of Bergelson, Blass and Hindman’s
partition theorem on located variable words (§7).

Collectively, these theorems imply a variety of other Ramsey-type results,
including Hindman’s finite unions theorem, the Hales–Jewett theorem, and van
der Waerden’s theorem. In each case, we give elementary nonstandard
constructions of coherent sequences, which is enough to imply the
corresponding result via our framework. We also present a common
generalisation of Gowers’ theorem and the multivariable
Bergelson–Blass–Hindman theorem (and even the Milliken–Taylor theorem) in
§7.1, which is provable using our framework. Again, an elementary nonstandard
argument constructs an $\mathcal{F}$-coherent in this case.

Throughout, we let $\mathbb{N}=\\{0,1,2,\ldots\\}$ be the set of nonnegative
integers. We may use interval notation, e.g. $[0,3]$, $[1,7)$, and this should
be interpreted in the natural numbers, i.e.
$[n,m]=\\{k\in\mathbb{N}:n\leqslant k\leqslant m\\}$. This notation will later
extend to nonstandard integers ${}^{*}\mathbb{N}$, i.e.
$[\xi,\zeta]=\\{\alpha\in{}^{*}\mathbb{N}:\xi\leqslant\alpha\leqslant\zeta\\}$.

In general, we will use uppercase Latin letters $A,B,\ldots,S,T,\ldots$ for
sets and semigroups, lowercase Latin letters $s,t,\ldots$ for elements
thereof, and lowercase Greek letters $\alpha,\beta,\ldots$ for elements of
nonstandard extensions ${}^{*}S$ of semigroups.

##  2 Prerequisites

###  2.1 Nonstandard analysis

The main results of this paper (in §5) will be proved using the tools of
nonstandard analysis. Here, we give a basic overview of the concepts
needed—for a more in-depth exposition of nonstandard methods and their
applications to Ramsey theory, see [6]. All our work can alternatively be
formulated using ultrafilter methods, as per [24, 18].

Effectively, we work inside a “universe” $\mathbb{V}$ which includes the
semigroup $(S,+)$ under consideration, as well as subsets thereof and
functions $f:S^{k}\to S^{m}$. This universe comes equipped with a star map
$*:\mathbb{V}\to\mathbb{V}$, which assigns every object $M\in\mathbb{V}$ to
its nonstandard extension ${}^{*}M$, in such a way that the following
properties hold:

######  Axioms 3 (Basic properties of the star map).

  1. (i)

For a set $A$, ${}^{*}A$ is also a set, and ${}^{\sigma}A\subseteq{}^{*}A$,
where ${}^{\sigma}A\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}\\{{}^{*}x:x\in
A\\}$. This containment is strict iff $A$ is infinite;

  2. (ii)

If $A,B$ are sets such that $A\subseteq B$, then ${}^{*}A\subseteq{}^{*}B$;

  3. (iii)

For a set $A$ and $k\in\mathbb{N}$, we have ${}^{*}(A^{k})=({}^{*}A)^{k}$;

  4. (iv)

If $f$ is a function $A\to B$, then ${}^{*}f$ is a function
${}^{*}A\to{}^{*}B$;

  5. (v)

If $f:A\to B$ and $x\in A$, then $({}^{*}f)(x)=f(x)$;

  6. (vi)

If $f:A\to B$ and $x\in A$, then $({}^{*}f)({}^{*}x)={}^{*}(f(x))$;

  7. (vii)

If $s\in S$, then ${}^{*}s=s$;

  8. (viii)

If $n\in\mathbb{N}$, then ${}^{*}n=n$.

The star map also satisfies the following key principle:

######  Axiom 4 (Transfer principle).

For any elementary111A formula is elementary if all the quantifiers are
bounded, i.e. of the form $Q\,x\in y$ for objects $x,y\in\mathbb{V}$. All
logical formulae which we consider will be elementary. formula
$\varphi(x_{1},\ldots,x_{n})$ and objects $M_{1},\ldots,M_{n}\in\mathbb{V}$,
we have

$\varphi(M_{1},\ldots,M_{n})\text{
holds}\iff\varphi({}^{*}M_{1},\ldots,{}^{*}M_{n})\text{ holds}$

For a rigorous construction satisfying Axioms 3 and 4, see [6]. Here, we will
take on faith that such a structure does exist.

The semigroup operation $+$ can be considered as a function $+:S^{2}\to S$,
thus we get a natural extension ${}^{*}+$ of this operation to ${}^{*}S$.
Abusing notation, we will use $+$ to denote both the original operation and
its nonstandard extension—this is somewhat justified by Axiom 3.(v). We will
generally do the same for functions $f:S^{k}\to S^{m}$.

A peculiarity of our approach will be that we may iterate the star map, to
obtain nonstandard extensions of nonstandard extensions, and so on. In this
way, we get objects $M,{}^{*}M,{}^{*}{}^{*}M,\ldots$. We will use ${}^{n*}M$
to denote the $n$-fold nonstandard extension of an object $M\in\mathbb{V}$.
Axioms 3 and 4 also hold when the objects under consideration are themselves
nonstandard.

######  Remark 5.

In general, the simplifying assumptions made in Axioms 3.(vii) and 3.(viii)
cannot be extended to elements of ${}^{*}S$, ${}^{*}\mathbb{N}$ or higher in
the nonstandard hierarchy. As an example, $\mathbb{N}$ is an initial segment
of ${}^{*}\mathbb{N}$ [6, Prop 2.27], so by transfer, ${}^{*}\mathbb{N}$ is an
initial segment of ${}^{*}{}^{*}\mathbb{N}$. Now, if we take
$\xi\in{}^{*}\mathbb{N}\setminus\mathbb{N}$, we have
${}^{*}\xi\in{}^{*}{}^{*}\mathbb{N}\setminus{}^{*}\mathbb{N}$ by transfer. It
follows that $\xi<{}^{*}\xi\implies\xi\neq{}^{*}\xi$.

###  2.2 $u$-semigroups

Here, we develop some further notions that prove essential in the study of
Ramsey semigroups. These are mostly based on the theory of compact
semitopological semigroups (see [24, §2]), as developed by Ellis and others.
An example is given by $\beta S$, the set of ultrafilters on $S$, whose
topological and algebraic structure is well-studied [18, 24]. $\beta S$ is
also homeomorphic to the Stone–Čech compactification of $S$ with the discrete
topology. The following map allows us to transport this structure to
${}^{*}S$.

######  Definition 6.

Elements $\alpha\in{}^{*}S$ generate ultrafilters on $S$ via the ultrafilter
map:

$\alpha\mapsto\mathcal{U}_{\alpha}=\\{A\subseteq S:\alpha\in{}^{*}A\\}$

Two elements $\alpha,\beta$ are $u$-equivalent (denoted $\alpha\sim\beta$) if
$\mathcal{U}_{\alpha}=\mathcal{U}_{\beta}$.

######  Proposition 7.

$\sim$ is an equivalence relation on ${}^{*}S$.

The relation $\sim$ on ${}^{*}S$ was first considered by Di Nasso in [7], and
has seen extensive combinatorial and Ramsey-theoretic applications in [6, 9,
10, 3]. Some of the key properties are summarised below.

######  Proposition 8 ([7, 6]).

  1. (i)

If $\alpha\in{}^{*}S$, $s\in S$, then $\alpha\sim s$ if and only if
$\alpha=s$.

  2. (ii)

For any function $f:S\to S$, if $\alpha\sim\beta\in{}^{*}S$ then
$f(\alpha)\sim f(\beta)$.

  3. (iii)

For any function $f:S\to S$, if $\alpha\in{}^{*}S$ is such that
$f(\alpha)\sim\alpha$, then $f(\alpha)=\alpha$.

  4. (iv)

For any $\alpha,\alpha^{\prime},\beta,\beta^{\prime}\in{}^{*}S$, if
$\alpha\sim\alpha^{\prime}$ and $\beta\sim\beta^{\prime}$, then
$\alpha+{}^{*}\beta\sim\alpha^{\prime}+{}^{*}\beta^{\prime}$.

  5. (v)

For any $\alpha\in{}^{*}S$, $\alpha\sim{}^{*}\alpha$.

There is a natural way to define a topology on ${}^{*}S$ as follows:

######  Definition 9.

For any semigroup $S$, equip ${}^{*}S$ with the $u$-topology—that generated by
the basic open sets ${}^{*}A$ for $A\subseteq S$. We say ${}^{*}S$ is a
compact $u$-semigroup, i.e.

  1. (i)

${}^{*}S$ is compact;

  2. (ii)

For any $\alpha,\beta\in S$, there exists $\gamma\in S$ such that
$\gamma\,\sim\,\alpha+{}^{*}\beta$;

  3. (iii)

The map $\alpha\,\mapsto\,\alpha+{}^{*}\beta$ is continuous.

######  Proposition 10.

${}^{*}S/{\sim}$ is Hausdorff, i.e. two elements $\alpha,\beta\in{}^{*}S$ are
inseparable by disjoint open sets exactly when $\alpha\sim\beta$.

######  Corollary 11.

For continuous functions $f,g:{}^{*}S\to{}^{*}S$, the set
$\\{\alpha\in{}^{*}S:f(\alpha)\sim g(\alpha)\\}$ is closed.

######  Proposition 12.

For any function $f:S\to S$, its nonstandard extension
${}^{*}f:{}^{*}S\to{}^{*}S$ is continuous with respect to the $u$-topology on
${}^{*}S$.

Idempotent ultrafilters are key to most applications of infinitary methods in
Ramsey theory—their existence follows from the Ellis–Numakura lemma.
Throughout this paper, we will use a similar notion of idempotence for
elements of ${}^{*}S$.

######  Definition 13.

Suppose $(S,+)$ is a semigroup. We say $\alpha\in{}^{*}S$ is $u$-idempotent if
$\alpha+{}^{*}\alpha\ \sim\ \alpha$.

######  Lemma 14 (Ellis–Numakura).

If $T\subseteq{}^{*}S$ is a closed $u$-subsemigroup, then $T$ contains a
$u$-idempotent element.

Often, we will need a strengthening of Lemma 14, as follows.

######  Definition 15.

Define a relation $\preccurlyeq$ on ${}^{*}S$ by

$\alpha\preccurlyeq\beta\ \iff\ \alpha+{}^{*}\beta\ \sim\ \beta+{}^{*}\alpha\
\sim\ \alpha$

$\preccurlyeq$ is a partial order (up to $u$-equivalence) on the
$u$-idempotents of ${}^{*}S$.

######  Corollary 16 ([24, Lemma 2.3]).

Any closed $u$-subsemigroup of ${}^{*}S$ contains a $\preccurlyeq$-minimal
$u$-idempotent element.

##  3 Layered semigroups

Our results concern the framework of layered semigroups, as introduced by
Farah, Hindman and McLeod in [11]. We will work with the following adaptation
of their definition:

######  Definition 17.

A layered semigroup is a (total) semigroup $S$, with a layering map
$\ell:S\to\mathbb{N}$ such that for all $s,t\in S$,
$\ell(s+t)=\max\\{\ell(s),\ell(t)\\}$.

The map $\ell$ splits $S$ into layers $S_{n}=\ell^{-1}(n)$ for each
$n\in\operatorname{range}(\ell)$. Without loss of generality, we will
suppose222By shifting down values of $\ell$ as required. that
$\operatorname{range}(\ell)$ is an initial segment of $\mathbb{N}$. We will
encounter situations where $\operatorname{range}(\ell)$ is finite (i.e. our
semigroup has finitely many layers), but also cases when
$\operatorname{range}(\ell)=\mathbb{N}$ (i.e. our semigroup has infinitely
many layers. In theory, we could allow $\operatorname{range}(\ell)=\delta$ for
ordinals $\delta>\omega$, but we will not pursue such generalisations here.

Definition 17 is equivalent to the following, which is more in the style of
Farah, Hindman and McLeod’s original definition:

######  Proposition 18.

A pair $\big{(}S,\ \ell:S\to\mathbb{N}\big{)}$ form a layered semigroup if and
only if:

  1. (i)

$S_{\leqslant
n}\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}\ell^{-1}\big{(}\\{0,\ldots,n\\}\big{)}$
is a subsemigroup of $S$;

  2. (ii)

$S_{n}=\ell^{-1}(n)$ is a two-sided ideal of $S_{\leqslant n}$.

######  Remark 19.

Farah, Hindman and McLeod’s original definition in [11] is given by only
allowing finitely many layers, and further positing that $S_{0}=\\{e\\}$,
where $e$ is a two-sided identity for any element of $S$. We have relaxed both
conditions, since we will encounter layered semigroups for which neither
holds.

To illustrate Definition 17, we present some examples of layered semigroups
which naturally arise in Ramsey theory, and which will be relevant later in
this paper.

######  Example 20.

For each $i$, let $M_{i}$ be a monoid (semigroup with identity $e_{i}$), such
that only the identity has an inverse. Let $S$ be the set of tuples
$(m_{0},m_{1},m_{2},\ldots)\in\prod_{i=0}^{\infty}M_{i}$ with finite support
(i.e. $m_{n}\neq e_{n}$ for finitely many $n$). Then, $S$ is a layered
semigroup under pointwise operations, and the layering map
$\ell\big{[}(m_{i})\big{]}=\min\\{k:\,\forall i\geqslant k\ m_{i}\neq
e_{i}\\}$.

######  Example 21.

Fix a finite alphabet $A$. For $k\in\mathbb{N}$, a $k$-parameter word is an
element of $(A\cup\\{x_{1},x_{2},\ldots,x_{k}\\})^{<\omega}$ such that all the
variables $x_{1},x_{2},\ldots,x_{k}$ appear, and their first appearances are
in increasing order. Let $W_{k}$ be the set of all $k$-parameter words.333So
$W_{0}$ is simply $A^{<\omega}$, the set of all words over $A$. Then,
$W=\bigcup_{k=0}^{\infty}W_{k}$ is a layered semigroup444The layering map
$\ell:W\to\mathbb{N}$ here maps every parameter word to the number of
variables it contains. under concatenation, called the Graham–Rothschild
semigroup.

We will also consider partial semigroups—those for which the operation is not
always defined. To exclude trivial cases, such as when the operation is never
defined, we have the following notion of adequacy for a partial semigroup.

######  Definition 22.

A partial semigroup $S$ is adequate [11, 18, Defn 1.15.6] or directed [24] if,
for any finite subset $F\subseteq S$, there exists $y$ such that $x+y$ is
defined for all $x\in F$.

We now generalise Definition 17 to the case of partial semigroups. However, it
is not enough for just $S$ to be adequate—we need each layer to be adequate
also.

######  Definition 23.

An adequate partial layered semigroup is a partial semigroup $S$, with a
layering map $\ell:S\to\mathbb{N}$ such that:

  1. (i)

For all $s,t\in S$ such that $s+t$ is defined,
$\ell(s+t)=\max\\{\ell(s),\ell(t)\\}$;

  2. (ii)

For all $s_{1},\ldots,s_{n}\in S$ with $\ell(s_{1})=\cdots=\ell(s_{n})$, there
is $t\in S$ such that $\ell(t)=\ell(s_{i})$ and $s_{i}+t$ is defined for all
$i\leqslant n$.

######  Example 24.

Let $\mathrm{FIN}$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$
with finite support, i.e.
$\operatorname{supp}(f)\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}\\{n\in\mathbb{N}:f(n)\neq
0\\}$ is finite. For $f,g\in\mathrm{FIN}$, we define $f+g$ pointwise iff the
pointwise product $f(n)g(n)=0$ for all $n\in\mathbb{N}$.555Equivalently, if
$f$ and $g$ have disjoint supports. Then, $\mathrm{FIN}$ is an adequate
partial layered semigroup under the layering map
$\ell(f)=\max(\operatorname{range}(f))$, called the Gowers semigroup.

######  Example 25.

Later, we will consider nonstandard extensions of these semigroups. We use
transfer to deduce what these extensions look like. For example,
${}^{*}\mathrm{FIN}_{k}$ will consist666Not all such functions are in
${}^{*}\mathrm{FIN}_{k}$—only those that are internal. [6, §2.5] gives a good
overview of internal/external objects. An understanding of these will not be
necessary in this paper. of functions $\varphi:{}^{*}\mathbb{N}\to[0,k]$ with
hyperfinite support (i.e. $\operatorname{supp}(\varphi)\subseteq[0,\xi]$ for
some $\xi\in{}^{*}\mathbb{N}$) and having
$\max(\operatorname{range}(\varphi))=k$. This is clear from writing the
definition of $\mathrm{FIN}_{k}$ as an elementary formula, and applying
transfer.

######  Remark 26.

In general, $\bigcup_{i<\delta}{}^{*}S_{i}\subseteq{}^{*}S$, but this
containment may be strict when $S$ has infinitely many layers. For example, in
the Graham–Rothschild semigroup $W$ of Example 21, ${}^{*}W$ consists of
parameter words having hyperfinite length777This is clear by considering a
$k$-parameter word $w\in W$ as a function $w:[1,n]\to
A\cup\\{x_{1},x_{2},\ldots,x_{k}\\}$ for some $n\in\mathbb{N}$, and applying
transfer. $\xi$ and $\zeta$-many variables for some
$\xi,\zeta\in{}^{*}\mathbb{N}$. In contrast,
$\bigcup_{i=0}^{\infty}{}^{*}W_{i}$ is the subset of ${}^{*}W$ consisting of
words with only finitely many variables. We will never need to consider all of
${}^{*}S$; just the ${}^{*}S_{i}$ will be enough for our purposes.

###  3.1 The $*$-product $\Pi_{S}$

We will consider sequences $(\alpha_{i})$ of nonstandard elements, where each
$\ell(\alpha_{i})=i$. For notational convenience, we define the following:

######  Definition 27.

Given a total layered semigroup $S$, define

$\Pi_{S}\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}\prod_{i<\delta}{}^{*}S_{i}$

where $\delta$ is the number of layers. If it is clear what layered semigroup
we are referring to, we may just use the notation $\Pi$.

We now generalise Definition 27 to adequate partial semigroups.

######  Definition 28.

For a subsemigroup $A\subseteq S$, we define

${}^{\gamma}A\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}\\{\alpha\in{}^{*}A:x+\alpha\text{
is defined for all }x\in A\\}$

######  Example 29.

Let $\mathrm{FIN}$ be the Gowers semigroup of Example 24. Then, for
$k\in\mathbb{N}$, ${}^{\gamma}\mathrm{FIN}_{k}$ consists of the cofinite
functions $\varphi\in{}^{*}\mathrm{FIN}_{k}$—those whose supports are disjoint
from $\mathbb{N}$.

######  Remark 30.

In the case of partial adequate semigroups $S$, generally the whole of $\beta
S$ is not considered, but only a special subset, which is notated $\gamma S$
[24] or $\delta S$ [18, 11]. For each $A\subseteq S$, ${}^{\gamma}A$ is the
preimage of $\gamma A\subseteq\beta A$ under the ultrafilter map (Definition
6), hence the notation.

The sets ${}^{\gamma}A$ are useful because they have the following property:

######  Proposition 31.

If $\alpha\in{}^{*}A$ and $\beta\in{}^{\gamma}A$, then $\alpha+{}^{*}\beta$ is
always defined.

###### Proof.

Since $\beta\in{}^{\gamma}A$, the elementary formula

$\varphi(A,{}^{*}A,\beta)\ =\ \forall x\in A\ \ \exists y\in{}^{*}A\ \
x+\beta=y$

holds, so by transfer,

$\varphi({}^{*}A,{}^{*}{}^{*}A,{}^{*}\beta)\ =\ \forall x\in{}^{*}A\ \ \exists
y\in{}^{*}{}^{*}A\ \ x+{}^{*}\beta=y$

also holds. Letting $x=\alpha$ gives the result. ∎

######  Proposition 32.

If $A\subseteq S$ is adequate, then ${}^{\gamma}A$ is nonempty.

###### Proof.

For each $s\in A$, let $K_{s}=\\{y\in A:s+y\text{ is defined}\\}$. Then we
have

${}^{\gamma}A=\bigcap_{s\in A}{}^{*}K_{s}$

By adequacy, each $K_{s}$ is nonempty, so each ${}^{*}K_{s}$ is nonempty, and
closed in the $u$-topology on $A$. The collection $\\{{}^{*}K_{s}:s\in A\\}$
has the finite intersection property, since for any finite $F\subseteq A$:

$\bigcap_{s\in F}{}^{*}K_{s}\ =\ {}^{*}\left[\,\bigcap_{s\in F}K_{s}\right]$

which is nonempty by adequacy of $A$. The result follows by compactness of
${}^{*}A$ (it is a closed subset of the compact space ${}^{*}S$). ∎

######  Definition 33.

Given an adequate layered semigroup $S$, define

$\Pi_{S}\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}\prod_{i<\delta}{}^{\gamma}S_{i}$

where $\delta$ is the number of layers. If it is clear what layered semigroup
we are referring to, we may just use the notation $\Pi$.

Notice that if $S$ is total, then ${}^{\gamma}A={}^{*}A$ for any $A\subseteq
S$, so Definitions 27 and 33 coincide.

######  Proposition 34.

For any layered semigroup $S$, $\Pi_{S}$ is a compact $u$-semigroup, with the
product topology and operation defined componentwise.

##  4 Regressive maps

Our Ramsey-type results will be in the context of certain functions acting on
layered semigroups. We will require our functions $f:S\to S$ to have the
following properties.

######  Definition 35.

A regressive map is a function $f:S\to S$ such that for all $s,t\in S$:

  1. (i)

$f(st)=f(s)f(t)$, i.e. $f$ is a semigroup homomorphism;

  2. (ii)

$\ell(f(s))\leqslant\ell(s)$;

  3. (iii)

$\ell(s)\leqslant\ell(t)\implies\ell(f(s))\leqslant\ell(f(t))$;

  4. (iv)

${\left\lvert\ell(f(s))-\ell(f(t))\right\rvert}\ \leqslant\
{\left\lvert\ell(s)-\ell(t)\right\rvert}$;

######  Remark 36.

Property (i) seems natural, since we are dealing with semigroups. The
“layering” of our semigroups motivates property (ii). More specifically, in
§5, we will inductively construct “coherent” sequences
$\alpha^{(0)}\in{}^{\gamma}S_{0},\ \alpha^{(1)}\in{}^{\gamma}S_{1},\ \ldots$
which are closed under all the functions $f$ under consideration. Property
(ii) makes such a construction tractable, since we can build each
$\alpha^{(k)}$ based only on the $\alpha^{(j)}$ for $j<k$.

Properties (iii) and (iv) are harder to motivate, but they are essential to
the proof of Lemma 49, which plays a big part in the proof of Theorem 50. We
will see below that many natural examples of maps on layered semigroups
satisfy these two properties.

We will consider sequences which are “well-behaved” with respect to a
collection $\mathcal{F}$ of regressive maps on $S$. Generally, $\mathcal{F}$
will be closed under composition—however, this is not required for our
arguments to work. To make effective use of the transfer principle, we do
require the following:

######  Definition 37.

Let $\mathcal{F}$ be a collection of regressive maps $f:S\to S$. $\mathcal{F}$
is locally finite (l.f.) if for all $i\in\mathbb{N}$, the set
$\mathcal{F}_{i}=\\{f|_{S_{\leqslant i}}:f\in\mathcal{F}\\}$ is finite.

######  Example 38.

Let $W$ be the Graham–Rothschild semigroup of Example 21. Now, we consider
infinite parameter words $\tilde{w}$ \- elements of
$(A\cup\\{x_{1},x_{2},\ldots\\})^{\omega}$ such that all $x_{i}$ appear, with
their initial appearances in increasing order.

Every such $\tilde{w}$ defines a function $W\to W$, $u\mapsto u[\tilde{w}]$,
called the substitution map, by replacing each occurrence of $x_{i}$ in $u$
with the $i$th character in $\tilde{w}$. Each substitution map is a regressive
map, and the collection $\mathcal{F}$ of all such maps is locally finite, and
closed under composition.

######  Example 39.

Let $\mathrm{FIN}$ be the Gowers semigroup of Example 24. Every
$F:\mathbb{N}\to\mathbb{N}$ induces a map
$\tilde{F}:\mathrm{FIN}\to\mathrm{FIN}$ by composition, i.e.
$\tilde{F}(f)=F\circ f$. When $F$ is a nondecreasing surjection, $\tilde{F}$
is a regressive map, and the collection $\mathcal{F}$ of all such maps is
locally finite, and closed under composition. These maps, first considered in
[2], are called (generalised) tetris operations.

###  4.1 Special sequences in $\Pi_{S}$

Throughout this section, fix a layered semigroup $S$ and a locally finite
collection of regressive maps $\mathcal{F}$ on $S$. Essential to the proof of
Theorem 50 is the following notion of coherence for elements of $\Pi_{S}$.

######  Definition 40.

An element $(\alpha_{i})_{i<\delta}\in\Pi_{S}$ is $\mathcal{F}$-coherent if
for all $f\in\mathcal{F}$ and $j<\delta$, we have
$f(\alpha_{j})\sim\alpha_{k}$ for some888Such a $k$ is uniquely defined.
$k\leqslant j$.

Intuitively, $(\alpha_{i})$ is $\mathcal{F}$-coherent if it is closed under
all functions $f\in\mathcal{F}$. Definition 40 doesn’t say anything about the
product of elements in $(\alpha_{i})$, so a stronger notion of coherence is
needed to draw conclusions about products.

######  Definition 41.

An $\mathcal{F}$-coherent element $(\alpha_{i})_{i<\delta}\in\Pi_{S}$ is
$\mathcal{F}$-Ramsey if, for all $i\leqslant j<\delta$,
$\alpha_{i}\succcurlyeq\alpha_{j}$.999For the case $i=j$, this implies
$\alpha_{i}$ is $u$-idempotent. We say $S$ itself is $\mathcal{F}$-Ramsey if
$\Pi_{S}$ contains an $\mathcal{F}$-Ramsey element.

######  Remark 42.

Sequences of ultrafilters satisfying the analogue of Definition 41 (as well as
being $\preccurlyeq$-minimal) are termed reductive in [19].

##  5 Main results

Throughout this section, fix a layered semigroup $S$ and a locally finite
collection of regressive maps $\mathcal{F}$ on $S$. The existence of an
$\mathcal{F}$-Ramsey in $\Pi_{S}$ is a powerful statement—it implies that
general Ramsey statements (Theorems 45 and 46) are true of our layered
semigroup $S$. When put into context, these general statements reduce to
familiar results of Ramsey theory. Before proving these statements, we need
the following concept:

######  Definition 43.

A sequence $(x_{i})\subseteq S$ is called a block sequence [14, 20] or basic
sequence [24, Thm 2.20] if, for any $n_{0}<\cdots<n_{\ell-1}$ and
$f_{0},\ldots,f_{\ell-1}\in\mathcal{F}$,
$f_{0}(x_{n_{0}})+\cdots+f_{\ell-1}(x_{n_{\ell-1}})$ is defined.

######  Remark 44.

The above definition has content only when $S$ is partial - every sequence in
a total semigroup is a block sequence.

######  Theorem 45.

Suppose $S$ is $\mathcal{F}$-Ramsey with $\delta$ layers, and $n<\delta$.
Then, for any finite colouring of $S$, there exists a block sequence
$(x_{i})_{i=1}^{\infty}\subseteq S_{n}$ such that for every $k\leqslant n$,
the following set is monochromatic:

$S_{k}\ \cap\ \big{\\{}\ f_{1}(x_{n_{1}})+\cdots+f_{\ell}(x_{n_{\ell}})\ :\
1\leqslant n_{1}<\cdots<n_{\ell},\ f_{1},\ldots,f_{\ell}\in\mathcal{F}\
\big{\\}}$

We sketch the proof. Note that for an $\mathcal{F}$-Ramsey
$(\alpha_{i})_{i<\delta}\in\Pi$, the constant sequence
$(\alpha_{n},\alpha_{n},\alpha_{n},\ldots)$ effectively satisfies the
conclusion of the theorem, except that it is in ${}^{\gamma}S_{n}$ rather than
$S_{n}$. Given the existence of $\alpha_{n}$, we repeatedly apply transfer to
deduce the existence of each $x_{i}$.

###### Proof of Theorem 45.

Fix an $\mathcal{F}$-Ramsey sequence $(\alpha_{i})_{i<\delta}\in\Pi$. For each
$i<\delta$, let $A_{i}=\big{\\{}x\in S_{i}:\ c(x)=c(\alpha_{i})\big{\\}}$, and
$A=\bigcup_{i<\delta}A_{i}$.

$\alpha_{n}$ witnesses that the sentence101010This is a finite sentence, since
$\mathcal{F}$ is locally finite by assumption.

$\exists\ \tau\in{}^{*}S_{n}:\
\bigwedge_{f_{1},f_{2}\in\mathcal{F}_{n}}\big{(}f_{1}(\tau)\in{}^{*}A\ \land\
f_{1}(\tau)+{}^{*}f_{2}(\alpha_{n})\text{ is defined and in
}{}^{*}{}^{*}A\big{)}$

is true, so by transfer, there exists $x_{1}\in S_{n}$ such that, for all
$f_{1},f_{2}\in\mathcal{F}$, $f_{1}(x_{1})\in A$, and
$f_{1}(x_{1})+f_{2}(\alpha_{n})$ is defined and in ${}^{*}A$.

We also have
$f_{1}(x_{1})+f_{2}(\alpha_{n})+{}^{*}f_{3}(\alpha_{n})\in{}^{*}{}^{*}A$ for
any choice of $f_{1},f_{2},f_{3}\in\mathcal{F}$. This is because
$f_{2}(\alpha_{n})\sim\alpha_{i}$ and $f_{3}(\alpha_{n})\sim\alpha_{j}$ for
some $i,j\leqslant n$, so either
$f_{2}(\alpha_{n})+{}^{*}f_{3}(\alpha_{n})\sim\alpha_{i}\sim
f_{2}(\alpha_{n})$, or
$f_{2}(\alpha_{n})+{}^{*}f_{3}(\alpha_{n})\sim\alpha_{j}\sim
f_{3}(\alpha_{n})$. Hence, $\alpha_{n}$ witnesses the truth of

$\displaystyle\exists\ \tau\in{}^{*}S_{n}:\
\bigwedge_{f_{1},f_{2},f_{3}\in\mathcal{F}_{n}}\big{(}\ f_{1}(\tau)\in{}^{*}A\
\land\ f_{1}($ $\displaystyle\tau)+{}^{*}f_{2}(\alpha_{n})\text{ is defined
and in }{}^{*}{}^{*}A\ $ $\displaystyle\land\ f_{1}(x_{1})+f_{2}($
$\displaystyle\tau)\text{ is defined and in }{}^{*}A\ $ $\displaystyle\land\
f_{1}(x_{1})+f_{2}($ $\displaystyle\tau)+{}^{*}f_{3}(\alpha_{n})\text{ is
defined and in }{}^{*}{}^{*}A\ \big{)}$

whence by transfer, we get $x_{2}\in S_{n}$ with similar properties.

In general, suppose we have defined $x_{1},\ldots,x_{j-1}\in S_{n}$ such that
for all $\ell<j$, all $n_{1}<\cdots<n_{\ell}<j$, and all
$f_{1},\ldots,f_{\ell},f\in\mathcal{F}$, we have

$\displaystyle f_{1}(x_{n_{1}})+\cdots+f_{\ell}(x_{n_{\ell}})\text{ is defined
and in }A$ $\displaystyle
f_{1}(x_{n_{1}})+\cdots+f_{\ell}(x_{n_{\ell}})+f(\alpha_{n})\text{ is defined
and in }{}^{*}A$

Then, by a similar argument to before, it is also true that
$f_{1}(x_{n_{1}})+\cdots+f_{\ell}(x_{n_{\ell}})+f(\alpha_{n})+{}^{*}f^{\prime}(\alpha_{n})$
is defined and in ${}^{*}{}^{*}A$ for any choice of
$f^{\prime}\in\mathcal{F}$. Thus, $\alpha_{n}$ witnesses the truth of

$\displaystyle\exists\ \tau\in{}^{*}S_{n}:\
\bigwedge_{\ell<j}\quad\bigwedge_{n_{1}<\cdots<n_{\ell}<j}\quad$
$\displaystyle\bigwedge_{f_{1},\ldots,f_{\ell},f,f^{\prime}\in\mathcal{F}}$
$\displaystyle\hskip 28.45274pt\big{(}\
f_{1}(x_{n_{1}})+\cdots+f_{\ell}(x_{n_{\ell}})+f($ $\displaystyle\tau)\text{
is defined and in }{}^{*}A$ $\displaystyle\hskip 56.9055pt\land\
f_{1}(x_{n_{1}})+\cdots+f_{\ell}(x_{n_{\ell}})+f($
$\displaystyle\tau)+{}^{*}f^{\prime}(\alpha_{n})\text{ is defined and in
}{}^{*}{}^{*}A\ \big{)}$

so by transfer, we get $x_{j}\in S_{n}$ with similar properties. Continue ad
infinitum. ∎

######  Theorem 46.

Suppose $S$ is $\mathcal{F}$-Ramsey with $\delta$ layers. Then, for any finite
colouring of $S$, there exists a block sequence
$(x_{i})_{i<\delta}\in\prod_{i<\delta}S_{i}$ such that for every $k<\delta$,
the following set is monochromatic:

$S_{k}\ \cap\ \big{\\{}\ f_{1}(x_{n_{1}})+\cdots+f_{\ell}(x_{n_{\ell}})\ :\
0\leqslant n_{1}<\cdots<n_{\ell}<\delta,\ f_{1},\ldots,f_{\ell}\in\mathcal{F}\
\big{\\}}$

###### Proof.

Identical to that of Theorem 45, but at each stage when defining $x_{i}$,
replace $\alpha_{n}$ with $\alpha_{i}$, and $S_{n}$ with $S_{i}$. Continue up
to stage $\delta$. ∎

Theorem 45 easily implies a finite version:

######  Theorem 47.

Suppose $S$ is $\mathcal{F}$-Ramsey with $\delta$ layers, and $n<\delta$,
$m\in\mathbb{N}$. Then, for any finite colouring of $S$, there exists a block
sequence $(x_{1},\ldots,x_{m})\subseteq S_{n}$ of length $m$ such that for
every $k\leqslant n$, the following set is monochromatic:

$S_{k}\ \cap\ \big{\\{}\ f_{1}(x_{n_{1}})+\cdots+f_{\ell}(x_{n_{\ell}})\ :\
n_{1}<\cdots<n_{\ell}\leqslant m,\ f_{1},\ldots,f_{\ell}\in\mathcal{F}\
\big{\\}}$

######  Corollary 48 ($m=1$).

Suppose $S$ is $\mathcal{F}$-Ramsey with $\delta$ layers, and $n<\delta$.
Then, for any finite colouring of $S$, there exists $x\in S_{n}$ such that for
every $k\leqslant n$, $S_{k}\cap\\{f(x):f\in\mathcal{F}\\}$ is monochromatic.

Incredibly, given the existence of any $\mathcal{F}$-coherent element in
$\Pi_{S}$, we can construct an $\mathcal{F}$-Ramsey, and thus show that the
general Ramsey statements above hold in $S$. The following lemma is essential
to this construction.

######  Lemma 49.

Suppose that $(\alpha_{i})_{i<\delta}\in\Pi_{S}$ is $\mathcal{F}$-coherent and
$u$-idempotent. Then, for any $f\in\mathcal{F}$ and $k<\delta$, there is
$j=j_{k}\leqslant k$ such that

$f\big{[}\alpha_{k}+{}^{*}\alpha_{k-1}+\cdots+{}^{(k-1)*}\alpha_{1}+{}^{k*}\alpha_{0}\big{]}\
\sim\
\alpha_{j}+{}^{*}\alpha_{j-1}+\cdots+{}^{(j-1)*}\alpha_{1}+{}^{j*}\alpha_{0}$

###### Proof.

By induction on $k$. Given $f\in\mathcal{F}$, the $\mathcal{F}$-coherence of
$(\alpha_{i})$ implies that $f(\alpha_{0})\sim\alpha_{0}$. Thus, the base case
$k=0$ holds with $j_{0}=0$. Now, suppose the lemma holds for $k>0$—we prove
the $k+1$ case. By $\mathcal{F}$-coherence, there is $\ell\leqslant{k+1}$ such
that $f(\alpha_{k+1})\sim\alpha_{\ell}$, and Definition 35.(iii) and 35.(iv)
imply that either $\ell=j_{k}$ or $\ell=j_{k}+1$.

Case 1:

$\ell=j_{k}$. Then,

$\displaystyle\
f\Big{[}\alpha_{k+1}+\big{(}{}^{*}\alpha_{k}+{}^{*}{}^{*}\alpha_{k-1}+\cdots+{}^{k*}\alpha_{1}+{}^{(k+1)*}\alpha_{0}\big{)}\Big{]}$
$\displaystyle\sim\ $ $\displaystyle\
f(\alpha_{k+1})+f\big{[}{}^{*}\alpha_{k}+{}^{*}{}^{*}\alpha_{k-1}+\cdots+{}^{k*}\alpha_{1}+{}^{(k+1)*}\alpha_{0}\big{]}$
$f$ homomorphism $\displaystyle\sim\ $ $\displaystyle\
\alpha_{j}+{}^{*}\alpha_{j}+{}^{*}{}^{*}\alpha_{j-1}+\cdots+{}^{j*}\alpha_{1}+{}^{(j+1)*}\alpha_{0}$
$\displaystyle\sim\ $ $\displaystyle\
\alpha_{j}+{}^{*}\big{(}{}^{*}\alpha_{j-1}+\cdots+{}^{(j-1)*}\alpha_{1}+{}^{j*}\alpha_{0}\big{)}$
$u$-idempotence of $\alpha_{j}$ $\displaystyle\sim\ $ $\displaystyle\
\alpha_{j}+{}^{*}\alpha_{j-1}+\cdots+{}^{(j-1)*}\alpha_{1}+{}^{j*}\alpha_{0}$
Proposition 8

Case 2:

$\ell=j_{k}+1$. We can prove similarly that

$f\big{[}\alpha_{k+1}+{}^{*}\alpha_{k}+\cdots+{}^{k*}\alpha_{1}+{}^{(k+1)*}\alpha_{0}\big{]}\
\sim\
\alpha_{j+1}+{}^{*}\alpha_{j}+\cdots+{}^{j*}\alpha_{1}+{}^{(j+1)*}\alpha_{0}\hfill$

In either case, $j_{k+1}\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}\ell$ is as
required. ∎

######  Theorem 50.

Suppose $S$ has an $\mathcal{F}$-coherent element
$(\alpha_{i})_{i<\delta}\in\Pi$. Then, $S$ is $\mathcal{F}$-Ramsey.

###### Proof.

The following argument is based on [20]. Recall that $\Pi$ is a compact
$u$-semigroup by Proposition 34. We show that the subset
$\Pi_{\mathcal{F}}\subseteq\Pi$ of $\mathcal{F}$-coherent elements is a closed
$u$-subsemigroup.

$\Pi_{\mathcal{F}}$ closed: apply Lemma 11, writing $\Pi_{\mathcal{F}}$ as

$\Pi_{\mathcal{F}}\ =\ \bigcap_{j<\delta}\ \bigcup_{k\leqslant j}\
\big{\\{}(\alpha_{i})\in\Pi:f(\alpha_{j})\sim\alpha_{k}\big{\\}}$

$\Pi_{\mathcal{F}}$ $u$-subsemigroup: take
$(\beta_{i})_{i=0}^{\infty},(\gamma_{i})_{i=0}^{\infty}\in\Pi_{\mathcal{F}}$.
Then, $(\beta_{i})+{}^{*}(\gamma_{i})=(\beta_{i}+{}^{*}\gamma_{i})$ is
$\mathcal{F}$-coherent, by Proposition 8 and the fact that all
$f\in\mathcal{F}$ are homomorphisms. By transfer, there exists
$(\delta_{i})\in\Pi_{\mathcal{F}}$ with
$(\delta_{i})\sim(\beta_{i}+{}^{*}\gamma_{i})$, as required.

It follows that $\Pi_{\mathcal{F}}$ is itself a compact $u$-semigroup. By
induction on $k<\delta$, we will construct a sequence of elements
$\big{(}\alpha^{(k)}_{i}\big{)}_{i<\delta}\in\Pi_{\mathcal{F}}$ with the
following properties:

  1. (i)

For all $i\leqslant k$, $\alpha^{(k)}_{i}\sim\alpha^{(k-1)}_{i}$;

  2. (ii)

For all $i\in\mathbb{N}$, $\alpha^{(k)}_{i}$ is $u$-idempotent;

  3. (iii)

For all $i\leqslant k$ and $j\geqslant i$, we have
$\alpha^{(k)}_{j}+{}^{*}\alpha^{(k)}_{i}\ \sim\ \alpha^{(k)}_{j}$.

To begin, take any $u$-idempotent
$\big{(}\alpha^{\prime}_{i}\big{)}_{i<\delta}\in\Pi_{\mathcal{F}}$ by Lemma
14. Let

$Z_{0}=\left\\{\big{(}\beta_{i}\big{)}_{i<\delta}\in\Pi_{\mathcal{F}}:\
\beta_{0}\sim\alpha^{\prime}_{0},\ \forall j\
\beta_{j}+{}^{*}\alpha^{\prime}_{0}\ \sim\,\beta_{j}\right\\}$

Then, $Z_{0}$ is a compact $u$-semigroup by Lemma 11. Furthermore, it is
nonempty, since it contains the element $\big{(}\beta_{i}\big{)}_{i<\delta}$
defined by

$\beta_{i}\ \sim\ \alpha^{\prime}_{i}+{}^{*}\alpha^{\prime}_{0}$

To see this is in $\Pi_{\mathcal{F}}$, observe that for any $f\in\mathcal{F}$,
$f\big{[}\alpha^{\prime}_{0}\big{]}\sim\alpha^{\prime}_{0}$, since $f$ is
regressive and $\big{(}\alpha^{\prime}_{i}\big{)}$ is $\mathcal{F}$-coherent.
Then we have

$f(\beta_{i})\ \sim\
f\big{[}\alpha^{\prime}_{i}+{}^{*}\alpha^{\prime}_{0}\big{]}\ \sim\
f\big{[}\alpha^{\prime}_{i}\big{]}+{}^{*}f\big{[}\alpha^{\prime}_{0}\big{]}\
\sim\ \alpha^{\prime}_{j}+{}^{*}\alpha^{\prime}_{0}\ \sim\ \beta_{j}$

for some $j\leqslant i$. It follows from the $u$-idempotence of
$\alpha^{\prime}_{0}$ that $\big{(}\beta_{i}\big{)}_{i<\delta}\in Z_{0}$.
Picking some $u$-idempotent $\big{(}\alpha^{(0)}_{i}\big{)}_{i<\delta}\in
Z_{0}$, we can verify that $\big{(}\alpha^{(0)}_{i}\big{)}$ satisfies
properties (i)–(iii) above with $k=0$.

Proceeding inductively, suppose sequences
$\big{(}\alpha^{(0)}_{i}\big{)},\big{(}\alpha^{(1)}_{i}\big{)},\ldots,\big{(}\alpha^{(k-1)}_{i}\big{)}$
have been defined as required. Let

$Z_{k}=\left\\{\big{(}\beta_{i}\big{)}_{i<\delta}\in\Pi_{\mathcal{F}}:\
\forall i\leqslant k\ \Big{(}\beta_{i}\sim\alpha^{(k-1)}_{i}\ \wedge\ \forall
j\geqslant i,\ \beta_{j}+{}^{*}\alpha^{(k-1)}_{i}\
\sim\,\beta_{j}\Big{)}\right\\}$

Then, $Z_{k}$ is a compact $u$-semigroup by Lemma 11. Furthermore, it is
nonempty, since it contains the element $\big{(}\beta_{i}\big{)}_{i<\delta}$
defined by

$\beta_{i}\ \sim\ \alpha^{(k-1)}_{i}\ +\ {}^{*}\alpha^{(k-1)}_{i-1}\ +\
{}^{*}{}^{*}\alpha^{(k-1)}_{i-2}\ +\ \cdots\ +\ {}^{i*}\alpha^{(k-1)}_{0}$

which is $\mathcal{F}$-coherent by Lemma 49. Since
$\big{(}\alpha^{(k-1)}_{i}\big{)}$ satisfies properties (i)–(iii), we have
that $\big{(}\beta_{i}\big{)}_{i<\delta}\in Z_{k}$. Picking some
$u$-idempotent $\big{(}\alpha^{(k)}_{i}\big{)}_{i<\delta}\in Z_{k}$, we can
verify that $\big{(}\alpha^{(k)}_{i}\big{)}$ satisfies properties (i)–(iii)
above.

Finally, taking

$\alpha_{i}\ \sim\ \alpha^{0}_{0}\ +\ {}^{*}\alpha^{1}_{1}\ +\
{}^{*}{}^{*}\alpha^{2}_{2}\ +\ \cdots\ +\ {}^{i*}\alpha^{i}_{i}$

for each $i\in\mathbb{N}$, we get an $\mathcal{F}$-Ramsey sequence
$\big{(}\alpha_{i}\big{)}_{i<\delta}\in\Pi_{\mathcal{F}}$. ∎

######  Remark 51.

In [11, Thm 3.8], assuming rather strong conditions on $S$ and $\mathcal{F}$,
a direct construction of an $\mathcal{F}$-coherent is given. Along with
Theorems 45 and 50, this gives sufficient conditions for a Ramsey statement on
$S$ and $\mathcal{F}$ to be true. Unfortunately, such conditions do not apply
to many examples of layered semigroups which we consider.

##  6 Applications

###  6.1 Gowers’ theorem

Let $\mathrm{FIN}$ be the Gowers semigroup of Example 24, and $\mathcal{F}$ be
the collection of generalised tetris operations on $\mathrm{FIN}$, as in
Example 39. For any $\varphi\in{}^{\gamma}\mathrm{FIN}_{1}$, the sequence
$(\alpha_{j})$ defined $\alpha_{j}(n)=j\cdot\varphi(n)$ is
$\mathcal{F}$-coherent. Thus, by Theorem 50, $\mathrm{FIN}$ is
$\mathcal{F}$-Ramsey. Applying Theorem 45 gives us the generalised Gowers’
theorem of Lupini [20, Thm 1.1]:

######  Corollary 52 (Lupini).

For any finite colouring of $\mathrm{FIN}$, there exists a block sequence
$(f_{i})_{i=1}^{\infty}\subseteq\mathrm{FIN}_{n}$ such that for every
$k\leqslant n$, the following set is monochromatic:

$\mathrm{FIN}_{k}\ \cap\ \big{\\{}\
\tilde{F}_{1}(f_{n_{1}})+\cdots+\tilde{F}_{\ell}(f_{n_{\ell}})\ :\
n_{1}<\cdots<n_{\ell},\ \tilde{F}_{1},\ldots,\tilde{F}_{\ell}\in\mathcal{F}\
\big{\\}}$

The finite version (due to Bartošová and Kwiatkowska [2, Cor 2.7]) is obtained
from Theorem 47:

######  Corollary 53 (Bartošová–Kwiatkowska).

For any $m\in\mathbb{N}$ and finite colouring of $\mathrm{FIN}$, there exists
a block sequence $(f_{1},\ldots,f_{m})\subseteq\mathrm{FIN}_{n}$ of length $m$
such that for every $k\leqslant n$, the following set is monochromatic:

$\mathrm{FIN}_{k}\ \cap\ \big{\\{}\
\tilde{F}_{1}(f_{n_{1}})+\cdots+\tilde{F}_{\ell}(f_{n_{\ell}})\ :\
n_{1}<\cdots<n_{\ell}\leqslant m,\
\tilde{F}_{1},\ldots,\tilde{F}_{\ell}\in\mathcal{F}\ \big{\\}}$

Gowers’ original theorem [14, Theorem 1] is obtained by taking the subset
$\mathcal{F}^{\prime}\subseteq\mathcal{F}$ consisting of iterates of the
tetris operation $T(n)=\max\\{n-1,0\\}$. Since any $\mathcal{F}$-coherent
sequence is also $\mathcal{F}^{\prime}$-coherent, it follows that
$\mathrm{FIN}$ is $\mathcal{F}^{\prime}$-Ramsey, so Theorem 45 gives:

######  Corollary 54 (Gowers).

For any finite colouring of $\mathrm{FIN}$, there exists a block sequence
$(f_{i})_{i=1}^{\infty}\subseteq\mathrm{FIN}_{n}$ such that for every
$k\leqslant n$, the following set is monochromatic:

$\mathrm{FIN}_{k}\ \cap\ \big{\\{}\
\tilde{T}^{(m_{1})}(f_{n_{1}})+\cdots+\tilde{T}^{(m_{\ell})}(f_{n_{\ell}})\ :\
n_{1}<\cdots<n_{\ell};\ m_{1},\ldots,m_{\ell}\in\mathbb{N}\ \big{\\}}$

where $\tilde{T}^{(m)}$ denotes the $m$th iterate of the tetris operation.

By identifying a set $A$ with its characteristic function $\mathbf{1}_{A}$,
$\mathrm{FIN}_{1}$ is identified with
$\operatorname{\mathcal{P}}_{\text{fin}}(\mathbb{N})=\\{A\subseteq\mathbb{N}:A\text{
finite}\\}$. Then, the case $k=n=1$ gives Hindman’s finite unions theorem [17,
Cor 3.3]:

######  Corollary 55 (Hindman).

For any finite colouring of
$\operatorname{\mathcal{P}}_{\text{fin}}(\mathbb{N})$, there exists a disjoint
sequence $(A_{i})_{i=1}^{\infty}$ of finite subsets of $\mathbb{N}$ such that
the set $\\{A_{n_{1}}\cup\cdots\cup A_{n_{\ell}}:\ n_{1}<\cdots<n_{\ell}\\}$
of all finite unions of elements of $(A_{i})_{i=1}^{\infty}$ is monochromatic.

Gowers’ theorem admits many variants. For $f,g\in\mathrm{FIN}$, say $f<g$ if
$\max(\operatorname{supp}(f))<\min(\operatorname{supp}(g))$. Then, we can
further restrict the operation of pointwise sum $f+g$ to only be defined when
$f<g$. The resulting semigroup is still adequate, and the
${}^{\gamma}\mathrm{FIN}_{k}$ are identical to before. Therefore, exactly the
same argument proves the strong Gowers’ theorem:

######  Corollary 56 (Strong Gowers’).

For any finite colouring of $\mathrm{FIN}$, there exists an increasing
sequence $(f_{1}<f_{2}<\cdots)\subseteq\mathrm{FIN}_{n}$ such that for every
$k\leqslant n$, the following set is monochromatic:

$\mathrm{FIN}_{k}\ \cap\ \big{\\{}\
\tilde{F}_{1}(f_{n_{1}})+\cdots+\tilde{F}_{\ell}(f_{n_{\ell}})\ :\
n_{1}<\cdots<n_{\ell},\ \tilde{F}_{1},\ldots,\tilde{F}_{\ell}\in\mathcal{F}\
\big{\\}}$

Again, $k=n=1$ gives a strengthening of Corollary 55—we can find an increasing
sequence $(A_{1}<A_{2}<\cdots)$ with the same property, where for
$A,B\subseteq\mathbb{N}$, $A<B\iff\max(A)<\min(B)$.

###  6.2 The Graham–Rothschild theorem

Let $W$ be the Graham–Rothschild semigroup of Example 21, and $\mathcal{F}$ be
the collection of substitution maps $\tilde{w}$ on $W,$ as in Example 38.

######  Theorem 57 ([4, 19]).

There is an $\mathcal{F}$-coherent $(\alpha_{i})\in\Pi_{W}$.

As we have seen, Theorem 57 is enough to deduce a Ramsey result about $W.$ In
fact, Theorems 45 and 46 give the main results from [4], infinitary versions
of the Graham–Rothschild theorem:

######  Corollary 58 (Carlson–Hindman–Strauss, Theorem 1.4).

For any finite colouring of $W,$ there exists a sequence
$(u_{i})_{i=0}^{\infty}\in\prod_{i=0}^{\infty}W_{i}$ such that for every
$k\in\mathbb{N}$, the following set is monochromatic:

$W_{k}\ \cap\ \big{\\{}\
u_{n_{1}}[\tilde{w}_{1}]{}^{\frown}\cdots{}^{\frown}u_{n_{\ell}}[\tilde{w}_{\ell}]\
:\ n_{1}<\cdots<n_{\ell},\
\tilde{w}_{1},\ldots,\tilde{w}_{\ell}\in\mathcal{F}\ \big{\\}}$

######  Corollary 59 (Carlson–Hindman–Strauss, Corollary 1.7).

For any finite colouring of $W,$ there exists a sequence
$(u_{i})_{i=0}^{\infty}\subseteq W_{n}$ such that for every $k\leqslant n$,
the following set is monochromatic:

$W_{k}\ \cap\ \big{\\{}\
u_{n_{1}}[\tilde{w}_{1}]{}^{\frown}\cdots{}^{\frown}u_{n_{\ell}}[\tilde{w}_{\ell}]\
:\ n_{1}<\cdots<n_{\ell},\
\tilde{w}_{1},\ldots,\tilde{w}_{\ell}\in\mathcal{F}\ \big{\\}}$

Corollary 48 gives Graham and Rothschild’s original theorem [15]:

######  Corollary 60 (Graham–Rothschild).

For every $m\geqslant k$ and every finite colouring of $W_{k}$, there exists
$u\in W_{m}$ such that the set
$\\{u[\tilde{w}]:\tilde{w}{\upharpoonright_{m}}\in W_{k}\\}$ is monochromatic.

The case $k=0$, $m=1$ of Corollary 60 gives the famed Hales–Jewett theorem:

######  Corollary 61 (Hales–Jewett).

For every finite colouring of $W_{0}=A^{<\omega}$, there is a variable word
$u\in W_{1}$ such that $\\{u[a]:a\in A\\}$ is monochromatic.

Corollary 61 is known to imply van der Waerden’s theorem [16, p38]:

######  Corollary 62 (van der Waerden).

For every $k\in\mathbb{N}$ and finite colouring of $\mathbb{N}$, there is
$a,d\in\mathbb{N}$ such that the arithmetic progression $\\{a+jd:0\leqslant
j<k\\}$ is monochromatic.

######  Remark 63.

A construction satisfying Theorem 57 is given by Carlson, Hindman and Strauss
[4, Thm 2.12], and by Hindman, Strauss and Zamboni [19]. In fact, in both
cases, they directly construct an $\mathcal{F}$-Ramsey, all of whose elements
are $\preccurlyeq$-minimal. Both arguments used to construct such a sequence
are extraordinarily involved, long and complicated. The author believes that
an $\mathcal{F}$-coherent could be constructed by a significantly simpler
argument, from which an $\mathcal{F}$-Ramsey could be constructed via the
general framework laid out in this paper, but has not been able to
significantly simplify their argument, nor find an alternative method.

######  Remark 64.

Nonetheless, if we restrict to the subset
$\mathcal{F}^{\prime}\subseteq\mathcal{F}$ of strong variable words, i.e.
those where all variables appear in order, a simple nonstandard argument can
construct an $\mathcal{F}^{\prime}$-coherent. We rely on the following result
from the theory of compact semitopological semigroups:

######  Lemma 65 ([24, Lemma 2.3]).

If $\alpha\in{}^{*}K$ is $u$-idempotent, and $I\subseteq{}^{*}K$ is a closed
left ideal, then there is $u$-idempotent $\beta\in I+{}^{*}\alpha$ such that
$\beta\preccurlyeq\alpha$.

######  Lemma 66 ([6, Lemma 7.9]).

There are $u$-idempotents $\nu\in{}^{*}W_{0}$,
$\overline{\omega}\in{}^{*}W_{1}$ such that $\overline{\omega}\preccurlyeq\nu$
and $\overline{\omega}[a]\sim\nu$ for every $a\in A$.

###### Proof.

Let $T\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}W_{0}\cup W_{1}$. Pick any
$\preccurlyeq$-minimal $u$-idempotent $\nu\in{}^{*}W_{0}$ by Lemma 16. Note
that ${}^{*}W_{1}$ is a closed left ideal of ${}^{*}T$, so by Lemma 65, pick
$u$-idempotent
$\overline{\omega}\in{}^{*}W_{1}{}^{\frown}{}^{*}\nu\subseteq{}^{*}W_{1}$ such
that $\overline{\omega}\preccurlyeq\nu$. Now, for any $a\in A$, the
substitution map $v\mapsto v[a]$ is a homomorphism $T\to W_{0}$, hence
$\overline{\omega}[a]\preccurlyeq\nu[a]=\nu$ since $\nu\in{}^{*}W_{0}$. Since
$\nu$ is $\preccurlyeq$-minimal, it follows that
$\overline{\omega}[a]\sim\nu$. ∎

######  Theorem 67.

There is an $\mathcal{F}^{\prime}$-coherent $(\alpha_{i})\in\Pi_{W}$.

###### Proof.

Let $\nu$, $\overline{\omega}$ be as in Lemma 66, and define $\alpha_{i}$ by
$\alpha_{0}=\nu$, $\alpha_{1}=\overline{\omega}$,
$\alpha_{i}\sim\overline{\omega}[x_{1}]{}^{\frown}{}^{*}\overline{\omega}[x_{2}]{}^{\frown}\cdots{}^{\frown}{}^{i*}\overline{\omega}[x_{i}]$
for $i\geqslant 2$. By $u$-idempotence of $\nu$, $\overline{\omega}$ and the
fact that $\overline{\omega}\preccurlyeq\nu$, $(\alpha_{i})$ is
$\mathcal{F}^{\prime}$-coherent. ∎

###  6.3 Galvin–Glazer and Hindman’s theorem

Let $S$ be an adequate partial semigroup such that $s+s$ is never defined. $S$
is trivially layered by the layering map $\ell:S\to\mathbb{N}$, $s\mapsto 0$.
Let $\mathcal{F}$ consist of only the identity map $\operatorname{id}:S\to S$,
which is a regressive map. Then, any $\alpha\in{}^{*}S$ is trivially
$\mathcal{F}$-coherent, thus $S$ is $\mathcal{F}$-Ramsey. Applying Theorem 45
with $n=0$ gives the Galvin–Glazer theorem (see [24, Thm 2.20] or [5]):

######  Corollary 68 (Galvin–Glazer).

For any finite colouring of $S$, there exists an infinite sequence
$(x_{i})_{i=0}\subseteq S$ of distinct elements such that the set

$\big{\\{}\ x_{n_{0}}+\cdots+x_{n_{\ell-1}}\ :\ n_{0}<\cdots<n_{\ell-1},\
\big{\\}}$

is $c$-monochromatic.

The special case $S=\mathbb{N}$ and $A=\\{x_{i}:i\in\mathbb{N}\\}$ gives us
Hindman’s finite sums theorem [17, Thm 3.1]:

######  Corollary 69 (Hindman).

For any finite colouring $c:\mathbb{N}\to r$, there exists an infinite set
$A\subseteq\mathbb{N}$, such that the set $\operatorname{FS}(A)$ of all
finite, nonrepeating sums from $A$ is $c$-monochromatic.

##  7 Located variable words

In this section, we give a simple nonstandard proof of Bergelson, Blass and
Hindman’s partition theorem for located variable words [1, Thm 4.1], using
Theorem 57 and the general framework of layered semigroups developed herein.
In fact, we obtain a multivariable generalisation of that result, related to a
result of Solecki [23, §4.3.1].

Fix a finite alphabet $A$. A located word over $A$ [21, §4.2] is a partial
function $w:\mathbb{N}\to A$, where $\operatorname{dom}(w)$ is finite and
nonempty. In the vein of Example 21, a located $k$-parameter word over $A$ is
a partial function $w:\mathbb{N}\to A\cup\\{x_{1},\ldots,x_{k}\\}$, where
$\operatorname{dom}(w)$ is finite, all the $x_{i}$ appear, and in increasing
order.

For each $k\in\mathbb{N}$, we let $L_{k}$ be the set of all located
$k$-parameter words over $A$ (where $L_{0}$ simply contains located words).
Then, $L=\bigcup_{i=0}^{\infty}L_{i}$ is a layered semigroup, with the
operation $+$ defined

$(w+v)(n)=\begin{cases}w(n)&\text{ if }n\in\operatorname{dom}(w)\\\
v(n)&\text{ if }n\in\operatorname{dom}(v)\\\ \text{undefined }&\text{
otherwise}\\\ \end{cases}$

only when $\operatorname{dom}(w)\cap\operatorname{dom}(v)=\varnothing$. Notice
that $L$ is an adequate partial semigroup, and the ${}^{\gamma}L_{m}$ are
exactly the cofinite located $m$-variable words $\alpha$—those whose domains
$\operatorname{dom}(\alpha)\subseteq{}^{*}\mathbb{N}$ are disjoint from
$\mathbb{N}$.

As in Example 38, substitution maps $u\mapsto u[\tilde{w}]$ are defined by
infinite (total/nonlocated) variable words $\tilde{w}$, in much the same way:

$u[\tilde{w}](n)=\begin{cases}u(n)&\text{ if
}n\in\operatorname{dom}(u),u(n)\neq x_{i}\text{ for all }i\\\
\tilde{w}_{i}&\text{ if }n\in\operatorname{dom}(u),u(n)=x_{i}\text{ for some
}i\\\ \text{undefined }&\text{ if }n\notin\operatorname{dom}(u)\\\
\end{cases}$

In fact, we can naturally identify the Graham–Rothschild semigroup $W$ as a
subset of $L$, and the above definition extends the substitution maps on $W.$
As before, all such substitution maps are regressive, and the collection
$\mathcal{F}$ of all of them is locally finite, and closed under composition.

Assuming the existence of an $\mathcal{F}$-coherent in $\Pi_{W}$, we can
construct an $\mathcal{F}$-coherent in $\Pi_{L}$ by a simple nonstandard
argument:

######  Lemma 70.

$L$ has an $\mathcal{F}$-coherent.

###### Proof.

Pick an $\mathcal{F}_{W}$-coherent sequence $(\alpha_{i})_{i=0}^{\infty}$ in
the Graham–Rothschild semigroup. Note that all substitution maps preserve the
length of words, thus each $\alpha_{i}$ must have infinite length. If any
$\alpha_{i}$ was finite, $\mathcal{F}$-coherence and Proposition 8.(i) would
imply all the $\alpha_{i}$ are finite, and of the same length
${\left\lvert\alpha_{i}\right\rvert}=n$. But then $\alpha_{n+1}$ cannot exist,
since there are no $(n+1)$-parameter words of length $n$.

Define $g:L\to L$ as the map which takes the “second half” of any located
variable word:

$g(w)(n)=\begin{cases}w(n)&n\geqslant{\left\lfloor\big{(}{\left\lvert
w\right\rvert}/2\big{)}\right\rfloor}\\\
\text{undefined}&n<{\left\lfloor\big{(}{\left\lvert
w\right\rvert}/2\big{)}\right\rfloor}\end{cases}$

Note that $g$ commutes with every substitution map, i.e.
$g\big{(}u[\tilde{w}]\big{)}=g(u)[\tilde{w}]$ for all $u\in L$,
$\tilde{w}\in\mathcal{F}$. By transfer, this is also true for nonstandard
words $\alpha\in{}^{*}L$.

Now, define $\beta_{i}=g(\alpha_{i})$ for each $i\in\mathbb{N}$. Since the
lengths
${\left\lvert\alpha_{i}\right\rvert}\in{}^{*}\mathbb{N}\setminus\mathbb{N}$
are infinite, it follows that each $\beta_{i}$ is undefined up to
${\left\lfloor\big{(}{\left\lvert\alpha_{i}\right\rvert}/2\big{)}\right\rfloor}\in{}^{*}\mathbb{N}\setminus\mathbb{N}$,
whence $\beta_{i}\in{}^{\gamma}L_{i}$. Thus,
$(\beta_{i})_{i=0}^{\infty}\in\Pi_{L}$. The $\mathcal{F}$-coherence of
$(\beta_{i})$ follows from Proposition 8.(ii)—for all $i\in\mathbb{N}$ and
$\tilde{w}\in\mathcal{F}$, there is $j\leqslant i$ such that

$\beta_{i}[\tilde{w}]\ =\ g(\alpha_{i})[\tilde{w}]\ =\
g\big{(}\alpha_{i}[\tilde{w}]\big{)}\ \sim\ g\big{(}\alpha_{j}\big{)}\ =\
\beta_{j}\qed$

It follows by Theorem 50 that $L$ is $\mathcal{F}$-Ramsey. Applying Theorems
45 gives a multivariable generalisation of the Bergelson–Blass–Hindman theorem
on located words [1, Thm 4.1]:

######  Corollary 71.

For any $n$ and finite colouring of $L$, there exists a block sequence
$(u_{i})_{i=1}^{\infty}\subseteq L_{n}$ such that for every $k\leqslant n$,
the following set is monochromatic:

$L_{k}\ \cap\ \big{\\{}\
u_{n_{1}}[\tilde{w}_{1}]+\cdots+u_{n_{\ell}}[\tilde{w}_{\ell}]\ :\
n_{1}<\cdots<n_{\ell},\ \tilde{w}_{1},\ldots,\tilde{w}_{\ell}\in\mathcal{F}\
\big{\\}}$

The original theorem of Bergelson, Blass and Hindman is the case $n=1$. If we
instead take Corollary 71 with $n=k=1$ and $A=\varnothing$, we again obtain
Hindman’s finite unions theorem (Corollary 55).

######  Remark 72.

The proof of Lemma 70 is also valid for the subcollection
$\mathcal{F}^{\prime}\subseteq\mathcal{F}$ of substitution maps corresponding
to strong variable words (see Remark 64), so along with Theorem 67, this gives
a short nonstandard proof of the weaker version of Corollary 71 for strong
variable words.

######  Remark 73.

As with Corollary 56, we can also restrict the operation $w+v$ to only be
defined when $w<v$, i.e. $\operatorname{dom}(w)<\operatorname{dom}(v)$. The
proof then goes through unchanged, and we get a stronger version of Corollary
71, where the block sequence $(u_{1}<u_{2}<\cdots)\subseteq L_{n}$ can be
taken to be increasing.

###  7.1 A common generalisation

It is notable that Gowers’ theorem and the Bergelson–Blass–Hindman theorem [1,
Thm 4.1] both generalise the finite unions theorem of Hindman [17, Cor 3.3].
Here, we present a common generalisation of both theorems [20, 21], and prove
it using the general framework of layered semigroups previously described.

Let $A$ be a finite alphabet, and $X=\\{x_{0},x_{1},x_{2},\ldots\\}$ be a
countably infinite set of variables. Note that, compared to the
Graham–Rothschild case, we have added an extra variable symbol $x_{0}$, which
we interpret to mean “undefined”. Let $\mathrm{FIN}^{A}$ be the set of (total)
$f:\mathbb{N}\to A\cup X$ which are eventually constant and equal to $x_{0}$.

For $f,g\in\mathrm{FIN}^{A}$, the sum $f+g$ is defined iff for every
$n\in\mathbb{N}$, at least one of $f(n)$, $g(n)$ is $x_{0}$. In this case,
$f+g$ is defined as

$(f+g)(n)=\begin{cases}f(n)&\text{if }f(n)\neq x_{0}\\\ g(n)&\text{if
}f(n)=x_{0}\end{cases}$

Under this operation, and the layering map $\ell(f)=\max\\{k:f(n)=x_{k}\text{
for some }n\\}$, $\mathrm{FIN}^{A}$ is a commutative, adequate, partial
layered semigroup.

Call a (total) map $F:\mathbb{N}\to A\cup X$ strong if $F(0)=x_{0}$, and all
$x_{i}$ appear as values of $F$ in increasing order (i.e. whenever
$F(n)=x_{i}$, $F(m)=x_{j}$ for $n\leqslant m$, then $i\leqslant j$). The
strong maps act on $\mathrm{FIN}^{A}$ by “composition”:

$\tilde{F}[f](n)=\begin{cases}F(k)&\text{if }f(n)=x_{k}\text{ for some
}k\geqslant 1\\\ f(n)&\text{if }f(n)\in A\cup\\{x_{0}\\}\end{cases}$

Every strong $\tilde{F}$ is a regressive map, and the collection $\mathcal{F}$
of all such strong maps is locally finite, and closed under composition.

Identifying “undefined” with the symbol $x_{0}$, the semigroup $L^{A}$ of
located words over $A$ is a subset of $\mathrm{FIN}^{A}$. As this containment
is strict, we need to strengthen Theorem 45 to ensure that the sequence can be
found inside a specified subsemigroup $T\subseteq\mathrm{FIN}^{A}$.
Henceforth, we assume $T$ is layerwise adequate, and closed under all strong
maps. We define
$\Pi_{T}\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}\prod_{i=0}^{\infty}{}^{\gamma}T_{i}$,
and say that $T$ is $\mathcal{F}$-Ramsey if $\Pi_{T}$ contains an
$\mathcal{F}$-Ramsey element. Theorem 45 admits the following strengthening:

######  Theorem 74.

Suppose $T\subseteq\mathrm{FIN}^{A}$ is $\mathcal{F}$-Ramsey, and
$n\in\mathbb{N}$. Then, for any finite colouring of $\mathrm{FIN}^{A}$, there
exists a block sequence $(f_{i})_{i=1}^{\infty}\subseteq
T_{n}\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}T\cap\mathrm{FIN}^{A}_{n}$
such that for every $k\leqslant n$, the following set is monochromatic:

$\mathrm{FIN}^{A}_{k}\ \cap\ \big{\\{}\
\tilde{F}_{1}(f_{n_{1}})+\cdots+\tilde{F}_{\ell}(f_{n_{\ell}})\ :\
n_{1}<\cdots<n_{\ell},\ \tilde{F}_{1},\ldots,\tilde{F}_{\ell}\in\mathcal{F}\
\big{\\}}$

###### Proof.

As for Theorem 45, but replace all occurrences of $S_{n}$ with $T_{n}$. ∎

The question remains—when is $T$ $\mathcal{F}$-Ramsey?

######  Definition 75.

A subsemigroup $T\subseteq\mathrm{FIN}^{A}$ is complete if for any
$\alpha\in{}^{\gamma}T_{1}$ and $k\in\mathbb{N}$, we have
$\alpha+{}^{*}\alpha[x_{2}]+\cdots+{}^{(k-1)*}\alpha[x_{k}]\in{}^{k*}T$, where
$\alpha[b]$ is the function obtained by setting $\alpha[b](n)=b$ whenever
$\alpha(n)=x_{1}$.

######  Lemma 76.

Every complete $T\subseteq\mathrm{FIN}^{A}$ is $\mathcal{F}$-Ramsey.

###### Proof.

Pick $u$-idempotent $\nu\in{}^{\gamma}T_{0}$,
$\overline{\omega}\in{}^{\gamma}T_{1}$ such that
$\overline{\omega}\preccurlyeq\nu$ and $\overline{\omega}[a]\sim\nu$ for every
$a\in A$ (see Lemma 66). Since $T$ is complete, the argument in Theorem 67
works, giving an $\mathcal{F}$-coherent in $\Pi_{T}$. Theorem 50 now shows $T$
is $\mathcal{F}$-Ramsey, by restricting to $\Pi_{T}$ rather than $\Pi_{S}$. ∎

Note that $\mathrm{FIN}^{A}$ is trivially complete, hence it is
$\mathcal{F}$-Ramsey. Applying Theorem 74 with $A=\varnothing$ and
$T=\mathrm{FIN}^{\varnothing}=\mathrm{FIN}$ gives Gowers’ theorem (Corollary
52, where we interpret each variable symbol $x_{k}$ as the number
$k\in\mathbb{N}$). $L_{A}\subseteq\mathrm{FIN}^{A}$ is also complete, and
Theorem 74 with $T=L_{A}$ gives the Bergelson–Blass–Hindman theorem for strong
variable words (Corollary 71, interpreting $x_{0}$ to mean “undefined”).

######  Remark 77.

Theorem 74 doesn’t recover the full strength of Corollary 71—we could attempt
to do so by considering a wider class $\mathcal{F}^{+}$ of regressive maps
$F:\mathbb{N}\to A\cup X$, where only the first occurrences of each $x_{i}$
must appear in increasing order. It is unclear whether $\mathrm{FIN}^{A}$ is
$\mathcal{F}^{+}$-Ramsey. If it were, we would expect that the argument to
construct an $\mathcal{F}^{+}$-coherent would be more complicated, as per
Remark 63. This would give an even more generalised version of Gowers’
theorem, where some layers of a function $f\in\mathrm{FIN}$ can be “reversed”.

In the same way as Lupini [20, §3] and Dodos–Panellopoulos [8, Thm 2.21], we
can generalise Theorem 74 to cover the Milliken–Taylor theorem. For
$m\in\mathbb{N}$, let $\mathrm{FIN}^{A[m]}$ be the collection of block
sequences $(f_{1},\ldots,f_{m})$ of elements of $\mathrm{FIN}^{A}$.

######  Theorem 78.

Suppose $T\subseteq\mathrm{FIN}^{A}$ is $\mathcal{F}$-Ramsey, and
$n\in\mathbb{N}$. Then, for any finite colouring of $\mathrm{FIN}^{A}$, there
exists a block sequence $(x_{i})_{i=1}^{\infty}\subseteq T_{n}$ such that for
every $k\leqslant n$, the following set is monochromatic:

$\displaystyle\bigg{\\{}\ \Big{(}\tilde{F}_{1}(x_{n_{1}})+$
$\displaystyle\cdots+\tilde{F}_{\ell_{1}}(x_{n_{\ell_{1}}}),\ \ldots,\
\tilde{F}_{\ell_{m-1}+1}(x_{n_{\ell_{m-1}+1}})+\cdots+\tilde{F}_{\ell_{m}}(x_{n_{\ell_{m}}})\Big{)}:$
$\displaystyle 0<\ell_{1}<\cdots<\ell_{m};\ n_{1}<\cdots<n_{\ell_{m}};\
\tilde{F}_{1},\ldots,\tilde{F}_{\ell_{m}}\in\mathcal{F}\ \bigg{\\}}\ \cap\
\mathrm{FIN}^{A[m]}_{k}$

###### Proof.

Identical to [6, Thm 8.16]. ∎

Theorem 74 is the case $m=1$, while the Milliken–Taylor theorem is the case
$A=\varnothing$, $T=\mathrm{FIN}^{\varnothing}=\mathrm{FIN}$, $n=1$.

##  8 Closing remarks

This paper outlines a general framework for proving Ramsey-type results about
layered semigroups. We have seen in §6 and §7 that many old results in Ramsey
theory can be expressed and proven via our framework. Since layered semigroups
form a rich class of structures, we also expect that one could use this
framework to prove new Ramsey-theoretic results. Unfortunately, we are not
aware of any immediate applications to known open problems.

######  Problem 79.

Use the framework of this paper to prove new Ramsey-theoretic results of
interest.

We note that there are still Ramsey statements about semigroups which we have
not been able to recover. In particular, we saw that van der Waerden’s theorem
is a corollary of the Graham–Rothschild theorem, which can be proven in our
framework, but a direct formulation remains elusive. We could attempt to
obtain it from Theorem 46 by letting $S=\mathbb{N}^{+}\times\\{0,1\\}$ and
$\ell:(n,i)\mapsto i$. Then, consider maps $f_{c}:S\to S_{0}$, where
$f_{c}:(n,0)\mapsto(n,0)$, $(m,1)\mapsto(cm,0)$ for all $c$ up to some fixed
$k$. Taking the operation $+$ on $S_{0}$ to be usual addition, Theorem 46
obtains the structure of van der Waerden’s theorem. However, it may not be
possible to extend $+$ to $S$ so that $(S,\ell)$ is layered and the $f_{c}$
are homomorphisms.

######  Question 80.

Is there a layered semigroup $S$, and regressive maps $\mathcal{F}$ on $S$,
such that Theorem 45 or Theorem 46 reduce to van der Waerden’s theorem?

Another result notably missing is Ramsey’s theorem itself. The most sensible
attempt seems to be taking
$S=\operatorname{\mathcal{P}}_{\text{fin}}(\mathbb{N})=\\{F\subseteq\mathbb{N}:F\text{
finite}\\}$, and layering it by $\ell:F\mapsto{\left\lvert F\right\rvert}$.
Then, given a sequence $(F_{i})_{i=1}^{\infty}$ of finite sets, an infinite
homogeneous set $B$ could be obtained by $B=\bigcup_{i=1}^{\infty}F_{i}$. The
challenge is how to define a semigroup operation and regressive maps that
allow us to generate any finite subset of $B$.

######  Question 81.

Is there a layered semigroup $S$, and regressive maps $\mathcal{F}$ on $S$,
such that Theorem 45 or Theorem 46 reduce to Ramsey’s theorem?

Definition 35 imposed strong conditions on the types of maps considered,
particularly conditions (iii) and (iv). While all examples considered
satisfied all these conditions, perhaps a different argument is possible which
doesn’t require these conditions. This would require a new construction in the
proof, since conditions (iii) and (iv) are necessary to Lemma 49.

######  Question 82.

Can Theorem 50 be proven when conditions (iii) and/or (iv) are weakened or
removed from Definition 35?

Our framework allows us to deduce Ramsey statements about $(S,\mathcal{F})$
from the existence of an $\mathcal{F}$-coherent in $S$. However, we have not
found a general way to construct $\mathcal{F}$-coherents in arbitrary layered
semigroups $S$, or ensure they exist. The most general construction we have
given of an $\mathcal{F}$-coherent is Lemma 76, but this still depends
crucially on the structure of $\mathrm{FIN}^{A}$. Farah–Hindman–McLeod [11,
Thm 3.8] give a fairly general construction, but it requires very strong
conditions on $\mathcal{F}$, which do not hold in many natural examples.

######  Problem 83.

Find a general way to construct an $\mathcal{F}$-coherent in a layered
semigroup $S$, making as few assumptions about $S$ and $\mathcal{F}$ as
possible.

We did not present a construction of an $\mathcal{F}$-coherent in the
Graham–Rothschild semigroup $W,$ instead deferring to a result of
Carlson–Hindman–Strauss [4, Thm 2.12]. They directly construct a
$\preccurlyeq$-minimal $\mathcal{F}$-Ramsey, and as a result, their argument
is extremely complicated. Morally, if we only need to construct an
$\mathcal{F}$-coherent, there should be a simpler argument—then our framework
would imply the Graham–Rothschild theorem.

######  Problem 84.

Find a simpler construction of an $\mathcal{F}$-coherent in the
Graham–Rothschild semigroup $W.$

Our results generalise many Ramsey-type or partition results on layered
semigroups. However, there is another thread of Ramsey theory dealing with
density results, having the form that any set $A\subseteq\mathcal{M}$ of
positive “density” contains a substructure $\mathcal{N}\subseteq\mathcal{M}$
with certain properties. Often, we take $\mathcal{M}=\mathbb{N}$ and interpret
density to mean upper density; $d(A)=\limsup_{n\to\infty}{\left\lvert
A\cap[0,n)\right\rvert}/n$ for each $A\subseteq N$. For example, Szemerédi’s
theorem is the density version of van der Waerden’s theorem:

######  Theorem 85 (Szemerédi).

For any $k\in\mathbb{N}$ and $A\subseteq\mathbb{N}$ with $d(A)>0$, there are
$a,d\in\mathbb{N}^{+}$ such that the arithmetic progression
$\\{a+cd:0\leqslant c<k\\}\subseteq A$.

By defining a suitable notion of density on $W,$ Furstenberg and Katznelson
obtained a density version of the Hales–Jewett theorem [12]. Nonstandard
methods have also been applied successfully to prove density-type results [6,
Part III]. Thus, it may be possible to develop a similar framework for proving
density theorems in layered semigroups. This would require a suitable notion
of density on layered semigroups—McLeod has already generalised some
combinatorial notions of size in $\mathbb{N}$ to this setting [22].

######  Problem 86.

Develop an analogous framework for proving density results on layered
semigroups.

Variations on the basic structure of Theorem 45 also appear in Ramsey theory.
For example, take the Graham–Rothschild semigroup $W,$ and let $LV\subseteq W$
be the subset consisting of left-variable words—those whose first character is
the variable $x_{1}$. The Hales–Jewett theorem for left-variable words states:

######  Theorem 87 ([24, Theorem 2.37]).

For any finite colouring of $W_{0}=A^{<\omega}$, there exists a word $w\in
W_{0}$ and a sequence $(u_{i})_{i=0}^{\infty}\subseteq LV\cap W_{1}$ of left-
variable words such that the following set is monochromatic:

$\big{\\{}\
w{}^{\frown}u_{n_{1}}[a_{1}]{}^{\frown}\cdots{}^{\frown}u_{n_{\ell}}[a_{\ell}]\
:\ n_{1}<\cdots<n_{\ell},\ a_{1},\ldots a_{\ell}\in A\ \big{\\}}$

Effectively, we ensure that all products of subsequences have the same
nonvariable part. $LV\subseteq W$ is a right ideal, so we could generalise
this idea to right layered semigroups $(S,\ell,R)$, with $R\subseteq S$ a
distinguished right ideal meeting every layer except $S_{0}$. Then, we instead
consider sequences in
$\Lambda_{S}\mathrel{\mathrel{\mathop{:}}\mkern-1.2mu=}{}^{*}S_{0}\times\prod_{1\leqslant
i<\delta}{}^{*}R_{i}$, and define analogous notions of right
$\mathcal{F}$-coherent and right $\mathcal{F}$-Ramsey sequences. Further
investigation is required to see if the proofs of Theorems 45 and 50 translate
to this setting. Problems may arise in translating the proof of Theorem 50, if
we require conditions such as
$\kappa+{}^{*}f(\alpha_{i})\sim\kappa+{}^{*}\alpha_{j}$, which are not
continuous in $\alpha_{i}$.

######  Problem 88.

Translate our ideas to the setting of right layered semigroups.

## Acknowledgements

Thanks go to:

  * •

Martino Lupini, for teaching me all I know about Ramsey theory and nonstandard
analysis, and for first noticing the abstract connection between Gowers’
theorem and the Graham–Rothschild theorem, eventually leading to the ideas in
this paper.

  * •

Valentino Vito, for thoroughly reading the first draft, and providing
invaluable suggestions and corrections.

  * •

The anonymous reviewer, for their thorough peer review and useful comments.

## References

  * [1] Vitaly Bergelson, Andreas Blass, and Neil Hindman. Partition theorems for spaces of variable words. Proceedings of the London Mathematical Society s3-68(3), pp. 449–476, 1994.
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