Actions of trees
on semigroups, and
an infinitary Gowers–Hales–Jewett Ramsey theorem
Abstract.
We introduce the notion of (Ramsey) action of a tree on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales–Jewett theorems (for both located and nonlocated words), and the Farah–Hindman–McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken–Taylor theorem of Bergelson–Hindman–Williams as a particular case. We present applications of our Ramseytheoretic results to the structure of delta sets in amenable groups.
Key words and phrases:
Gowers Ramsey Theorem, Galvin–Glazer–Hindman theorem, Milliken–Taylor theorem, Hales–Jewett theorem, tetris operation, variable word, partial semigroup, ultrafilter, StoneČech compactification2000 Mathematics Subject Classification:
Primary 05D10, 54D80; Secondary 20M99, 05C05, 06A061. Introduction
The finitary Hales–Jewett theorem [14] is a fundamental combinatorial pigeonhole principle. Several years after the original proof of Hales and Jewett, two deep infinitary strengthenings of the Hales–Jewett theorem (for located and nonlocated words) have been proved in [4] using the theory of ultrafilters and algebra in the StoneČech compactification. In another direction, and around the same time, Gowers established in [12] another fundamental combinatorial pigeonhole principle, which has been since then referred to the (infinite) Gowers Ramsey theorem. Gowers’ Ramsey theorem is a farreaching generalization of Hindman’s theorem on finite unions [15]. We refer to [13, 17, 24, 25, 20] for other proofs of such a result and its finitary counterpart, where explicit bounds on the quantities involved are also obtained.
A common generalization of Gowers’ Ramsey theorem and the infinite Hales–Jewett theorems has been established by Farah–Hindman–McLeod in the setting of layered actions on adequate partial semigroups [10]. In a different direction, the infinite Gowers Ramsey theorem has been strengthened in [18] by considering multiple tetris operations. This answered a question of Bartošová and Kwiatkowska from [2], where the corresponding finitary statement is proved. A common generalization of Gowers’ Ramsey theorem for multiple tetris operations and the Milliken–Taylor theorem [19, 23] is also provided in [18].
Gowers’ Ramsey theorem for multiple tetris operation does not fit in the framework of layered actions on partial semigroups developed by Farah–Hindman–McLeod in [10]. It is therefore natural to wonder whether there exists a unifying combinatorial principle that lies at the heart of both Gowers’ Ramsey theorem for multiple tetris operations and the infinite Hales–Jewett theorems, as well as the Farah–Hindman–McLeod Ramsey theorem for layered actions on partial semigroups. The goal of the present paper is to provide such a unifying combinatorial principle within the framework, here introduced, of Ramsey actions of rooted trees on filtered semigroups. Our main result is Theorem 5.10, which provides a common generalization of all the results mentioned above. One can also obtain from such a general result more direct common generalizations of Gowers’ theorem for multiple tetris operations and the Hales–Jewett theorems (for located and nonlocated words). Such common generalizations—Theorem 4.5 and Theorem 5.12—are stated in terms of variable words with variables indexed by a finite rooted tree, and variable substitution maps that respect the tree structure. We also provide a common generalization of our main result—Theorem 5.10—and the polynomial Milliken–Taylor theorem of Bergelson–Hindman–Williams [7, Corollary 3.5]; see Theorem 6.2. All the results of this paper are infinitary, and imply by a routine compactness argument their finitary counterparts. We omit the statement of these finitary counterparts, leaving it to the interested reader. We will conclude by presenting applications of some of our Ramseytheoretic results to the structure of delta sets in amenable graphs.
The present paper consists of six sections, besides this introduction. In Section 2 we introduce and study the notion of action of an ordered set and of a rooted tree on a compact right topological semigroup. Section 3 deals with the notion of (Ramsey) action of an ordered set and of a rooted tree on a partial semigroup. General result for Ramsey actions of rooted trees on adequate partial semigroups is obtained here (Theorem 3.2 and Corollary 3.7). Section 4 explains how Gowers’ theorem for multiple tetris operations and the Hales–Jewett theorem are both subsumed by Theorem 3.2. Section 5 considers the even more general framework of (Ramsey) actions of rooted trees on filtered semigroups. It is explained here how all the previous results extend to this more general framework. This allows one to recover the infinite Hales–Jewett theorem for nonlocated words. Section 6 presents a further polynomial generalization, which subsumes this main result of the paper as well as the polynomial Milliken–Taylor theorem [7, Corollary 3.5]. Finally, Section 7 presents applications to combinatorial configurations contained in delta sets is amenable groups.
After the present paper was written, we have been informed that a general Ramsey statement subsuming Gowers’ theorem for multiple tetris operations and the infinitary Hales–Jewett theorems has been independently obtained by Solecki with different methods. We refer the reader to [22] for this alternative approach.
In the rest of this paper we denote by be the set of natural numbers including , and be the set of natural numbers different from zero. We identify an element of with the set of its predecessors. If are finite nonempty subsets of , we write if the maximum element of is smaller than the minimum element of . We also write for and if the largest element of is smaller than . Given a set we let be the set of finite subsets of . If is a set, then we denote by the space of ultrafilters on ; see [16, Chapter 3]. This is endowed with a canonical compact Hausdorff topology, having the sets for as basis of open (and closed) sets. We will use in the rest of the paper the notation of ultrafilter quantifiers; see [24, Chapter 1]. If is a formula depending on a variable ranging over , then we write as an abbreviation for . In particular, we have that is equivalent to the assertion that . By a finite coloring of the set we mean a function for some . Any such a coloring admits a canonical extension, which we still denote by , to a finite coloring of , obtained by setting if and only if , .
Acknowledgments
We would like to thank Andy Zucker for his comments on a first draft of the present paper.
2. Actions of trees on compact right topological semigroups
2.1. Compact right topological semigroups
We recall here some notions concerning compact right topological semigroups. An (additively denoted) compact right topological semigroup is a semigroup endowed with a compact topology with the property that, for every , the right translation map is continuous. In the following we assume all the compact right topological semigroup to be Hausdorff. An element of is idempotent if . A classical result of Ellis [9, Corollary 2.10]—see also [24, Lemma 2.1]—asserts that any compact right topological semigroup contains idempotent elements. One can define an order among idempotents of by setting if and only if . An idempotent element of is minimal if it is minimal with respect to such an order. The proof of [9, Corollary 2.10] also shows that for any idempotent element of there exists a minimal idempotent of such that .
A closed subsemigroup of is a nonempty closed subset of with the property that whenever . Observe that the idempotent elements of are precisely the closed subsemigroups of that contain a single element. A closed subsemigroup of is a closed bilateral ideal if and belong to whenever and . We denote by the set of closed subsemigroups of . We define an order in by setting if and only if . Clearly a subsemigroup of is a bilateral ideal if and only if . Observe that such an order extend the order on idempotents defined above, when an idempotent element of is identified with the closed subsemigroup . If is a compact right topological semigroup, we define to be the set of continuous semigroup homomorphisms . Observe that is a semigroup with respect to composition.
In the following we will regard as an ordered set endowed with such an ordering. (Here and in the following, all the ordered sets are supposed to be partially ordered.) We record here for future reference the following well known fact; see also [24, Lemma 2.3].
Lemma 2.1.
Suppose that is a compact right topological semigroup. If and , then for any idempotent there exists a minimal idempotent of such that .
Proof.
Consider a minimal idempotent element of . Observe that is an idempotent element of such that . Therefore by minimality of inside we have that and hence . Suppose now that is an idempotent element of such that . Then we have that and hence . It follows from minimality of inside that . Therefore is a minimal idempotent element of . ∎
2.2. Actions of trees on compact right topological semigroups
Suppose that is an ordered set, and is a compact right topological semigroup.
Definition 2.2.
An action of on is given by

an orderpreserving function , ,

a subsemigroup ,
such that for every there exists an function —which we call the spine of —such that maps to for every , and such that for any and such that .
Given an action of on we let be set of functions such that and for every and . When is nonempty, we endow with the product topology and the entrywise operation. This turns into a compact right topological semigroup. Observe that an idempotent in is an element of such that is an idempotent element of for every . We say that an idempotent in is orderpreserving if whenever .
Suppose now that is a rooted tree. We regard as an ordered set endowed with the canonical rooted tree order obtained by setting if and only if is a descendent of .
Definition 2.3.
A regressive homomorphism of is a function such that for every , and maps two adjacent nodes either to the same node or to adjacent nodes.
It is clear that any regressive homomorphism fixes the root, and maps every branch to itself.
Definition 2.4.
A Ramsey action of on is given by an action of on in the sense of Definition 2.2 such that is nonempty and, for every , the corresponding spine is a regressive homomorphism.
A similar proof as [18, Lemma 2.1] shows the following.
Proposition 2.5.
Suppose that is a rooted tree of height with root . If is a Ramsey action of on , then contains an orderpreserving idempotent. Furthermore, if is an idempotent element of , then contains an orderpreserving idempotent such that .
Proof.
Fix an idempotent element of . Let, for , be the function that maps every node to its th predecessor, where we convene that the th predecessor of a node of height at most is the root, and the th predecessor of every node is itself. Let be the set of nodes in of height at most . We define by recursion on idempotent elements of such that whenever , and are such that , and whenever and . Granted the construction one can then consider defined by
for any node of height . It is not difficult to verify that is an orderpreserving idempotent such that .
We proceed now with the recursive construction. We have already defined . Suppose that have been defined for some . Consider the closed subsemigroup of such that for , and for every and such that . Observe that is nonempty. Indeed, set
for any node of height . Observe that for every since is a homomorphism, is an regressive homomorphism of , and by recursive assumption. Therefore . Furthermore if , then for every and hence . Finally suppose that and are such that . We want to prove that . Suppose that the height of is . If then we have that
by the recursive assumption. Suppose now that . Then we have that, by the recursive assumption,
This concludes the proof that . Since is nonempty, it contains an idempotent element . This concludes the recursive construction. ∎
The following definition is inspired by the definition of layered action from [10, Definition 3.3].
Definition 2.6.
An action of on is a layered action if for every and one has that

is equal to either or the immediate predecessor of ;

if has an immediate predecessor , then for any minimal idempotent there exists such that for any such that .
A similar proof as [10, Theorem 3.8] gives the following.
Proposition 2.7.
Suppose that is a rooted tree of height , and is a layered action of . Then is a Ramsey action. Furthermore, contains an orderpreserving idempotent such that is a minimal idempotent in for every .
Proof.
It is clear by definition of layered action that, for every , is a regressive homomorphism. We now prove the second assertion. This will also show that is a Ramsey action.
We define minimal idempotents by recursion on the height of such that, for every and , and if . If is the root of then we let be any minimal idempotent element of . Suppose that has been defined whenever the height of is at most . Suppose now that has height and let be the immediate predecessor of .
If, for each , , one can just define using Lemma 2.1. Suppose now that for some . Let be the set of such that for every such that . By hypothesis we have that is nonempty. Observe now that . Indeed we have that for , by recursive hypothesis. Pick now a minimal idempotent of . Observe that is an idempotent such that . By minimality, and hence . Observe that if is an idempotent element such that then . This shows that is minimal in . Finally suppose that is an idempotent element such that . Then we have that, for any , is an idempotent element of such that . It follows by minimality of that , and hence . Minimality of in now shows that . This concludes the proof that is minimal in . ∎
3. Actions of trees on partial semigroups
3.1. Partial semigroups
A partial semigroup [10, Definition 1.2]—see also [4, Section 2] and [24, Section 2.2]—is a set endowed with a partially defined binary operation , with the property that for . Such an equality should be interpreted as asserting that the left hand side is defined if and only if the right hand side is defined, and in such a case they are equal. A partial semigroup is adequate [10, Definition 2.1]—or directed [24, §2.2]—if for every finite subset of the set of elements of such that is defined for every is nonempty.
A partial semigroup homomorphism [10, Definition 2.8] between partial semigroups and is a function with the property that for . Again, such an equality should be interpreted as asserting that the left hand side is defined if and only if the right hand side is defined, and in such a case they are equal. A partial semigroup homomorphism is adequate if for every finite subset of there exists a finite subset of such that the image of under is contained in . If then we say that is an adequate partial subsemigroup of is the inclusion map is an adequate partial semigroup homomorphism [10, Definition 2.10]. We say furthermore that is an adequate bilateral ideal if it is an adequate partial subsemigroup, and, for every and , belong to whenever they are defined.
If adequate partial subsemigroups of , then we let if . This should be interpreted as the assertion that, for any and , and belong to whenever they are defined. Observe that if and only if is an adequate bilateral ideal of . We denote by the space of adequate partial subsemigroups of . We regard as an ordered set with respect to the ordering just defined.
3.2. Cofinal ultrafilters on partial semigroups
Suppose that is a partial semigroup. An ultrafilter over is cofinal if , , is defined. Following [24, Chapter 2], we denote by the space of cofinal ultrafilters over . It is clear that is a closed subspace of the space of ultrafilters over . Furthermore, is a compact right topological semigroup when endowed with the operation defined by setting if and only if , , ; see [24, Corollary 2.7] and [10, Theorem 2.6]. More generally, this expression defines a function , such that, for any , the function , is continuous. In particular, for any and , the element of is well defined.
Suppose that and are partial semigroups, and is an adequate partial semigroup homomorphism. Then induces a continuous semigroup homomorphism by setting if and only if , [10, Lemma 2.14]. When is an adequate subsemigroup of and is the inclusion map, the continuous extension is onetoone. In this situation, we will identify with its image under , which is the closed subsemigroup of consisting of the cofinite ultrafilters on that contain . This defines a map , . Here denotes as in Subsection 2.1 the space of closed subsemigroups of . It is not hard to see that such a map is orderpreserving with respect to the ordering on and defined above.
3.3. Actions of ordered sets on partial semigroups
Suppose that is an ordered set, and is an adequate partial semigroup. We denote by the space of adequate partial semigroup homomorphisms . Observe that is a semigroup with respect to composition.
Definition 3.1.
An action of on is given by

an orderpreserving function , , and

a subsemigroup ,
such that such that for every there exists a function —which we call the spine of —such that maps to for every , and such that for any and such that .
Suppose that is an action of on . Then induces an action in the sense of Definition 2.2 of on the compact right topological semigroup , which we still denote by . This is obtained by setting for and considering the semigroup of continuous semigroup homomorphisms obtained as the canonical continuous extensions of elements of . Consistently with the notation introduced in Subsection 2.2, we denote by the set of functions such that and for every . An orderpreserving idempotent in is an element of such that is an idempotent element in and whenever are .
Theorem 3.2.
Suppose that is an action of a finite ordered set on the adequate partial semigroup . Suppose that is an orderpreserving idempotent. Fix a finite coloring of and consider its canonical extension to a finite coloring of . Fix a sequence of functions and a sequence of functions such that contains the range of for every and . There exists a sequence of functions such that

for every and ; and

for any , , for , and for , if is a chain in with least element , then the color of is equal to the color of .
Proof.
We now define by recursion on functions such that such that for every the following holds:

for every , , for , for such that is a chain in with least element , one has that the color of is equal to the color of , and

for every , , for , for such that is a chain in one has that the color of is equal to the color of .
Let us consider initially the case . In this case is a single point. Therefore selects a finite subset of , and selects a finite subset of . We need to find a function such that for every and such that the following holds:

for every and the color of is equal to the color of , and

for every and if is a chain in with least element , then the color of is equal to the color of .
Fix . Using the notation of ultrafilter quantifiers for the ultrafilter , we have that , , such that is a chain in with least element , one has that , the color of is equal to the color of and the color of is equal to the color of . This allows one to choose satisfying () and ().
We now consider the case . In this case selects a finite subset of , and selects a fintie subset that contains for every and . From () we deduce that

for every and , if is a chain in with least element , then the color of is equal to the color of .
Now fix . We have that , , , , one has that , the color of is equal to the color of , if is a chain in with least element then the color of is equal to the color of , if is a chain in with least element then the color of is equal to the color of , and if is a chain in with least element then the color of is equal to the color of . This allows one to choose in such a way that () and () are satisfied.
Suppose that a sequence as above has been defined up to in such a way that () and () are satisfied. From () and the fact that is an orderpreserving idempotent in , it follows that the following holds as well:

for every , , for , for such that is a chain in with least element , one has that the color of is equal to the color of .
Fix . Using (), () we see that , for every , , for , for and , if is a chain in with least element then the color of is equal to the color of , and if is a chain in with least element then the color of is equal to the color of . This allows one to choose for every in such a way that () and () are satisfied. This concludes the recursive construction. ∎
3.4. Actions of trees on partial semigroups
Suppose that is a finite rooted tree. As in Subsection 2.2, we consider as an ordered set with respect to its canonical ordering. This is defined by setting if and only if and is a descendent of .
Definition 3.3.
Suppose that is an action of a finite rooted tree on an adequate partial semigroup as in Definition 3.1. We say that is Ramsey if, for every , the corresponding spine is a regressive homomorphism, and for any finite subset of , for any finite coloring of , and any finite subset of , there exists a function such that, for any and , and the color of depends only on .
While the definition of adequate action might seem difficult to verify, it holds trivially in many examples, including the case of the action corresponding to Gowers’ theorem for multiple tetris operations.
Theorem 3.4.
Suppose that is an action of a finite rooted tree on an adequate partial semigroup given by some semigroup such that, for every , the corresponding spine is a regressive homomorphism. The following statements are equivalent:

is Ramsey;

the action induced by on is Ramsey;

for any finite coloring of , sequence of functions and sequence of functions such that contains the range of for every and , there exist functions such that

for every and ; and

for any , , for , and for , if is a chain in with least element , then the color of depends only on .

Proof.
(1)(2) Since is Ramsey, we have that for any finite subset of , for any finite coloring of , and any finite subset of , there exists a function such that, for any and , the color of depends only on . By compactness of we deduce that there exists a function such that, for any , , and any finite coloring of , the color of depends only on . This being true for any coloring of implies that for every and . Therefore .
(2)(1) Suppose that . Fix finite subsets of , of , and a finite coloring of . Consider the canonical extension of to a finite coloring of . We have that, for every and , . In particular the color of is equal to the color of . Fix . Using the notation of ultrafilter quantifiers, we have that , , and the color of is equal to the color of . Therefore we can choose an element for every such that the function witnesses that the action is Ramsey.
(3)(1) Observe that the definition of Ramsey action is the particular instance of (4) where the sequence has length .
3.5. Products of actions
We recall the notion of tensor product [16, Section 11.1]—see also [24, Section 1.2]—of ultrafilters. Suppose that and are ultrafilter on sets , respectively. Then is the ultrafilter on obtained by declaring, for any , if and only if , , . Observe that in the particular case when is the principal ultrafilter over , then if and only if . In the following we identify an element of a set with the corresponding principal ultrafilter. The following result is proved in [7, Corollary 2.8]
Theorem 3.5 (Bergelson–Hindman–Williams).
Suppose that and is a function. Suppose that for , is a semigroup and is an idempotent element. Set and suppose that is a subset of . If then for each there exist a sequence in such that
is contained in .
In the statement of Theorem 3.5 and in the following, the expression , where is a sequence in a partial semigroup , denotes the element of , where is an increasing enumeration of . Whenever we write such an expression, we will also implicitly assert that is defined in .
We will now present a generalization of Theorem 3.5 to the setting of actions of ordered sets on adequate partial semigroups. Suppose that