aa r X i v : . [ m a t h . C O ] M a r GOWERS’ RAMSEY THEOREM FOR GENERALIZED TETRISOPERATIONS
MARTINO LUPINI
Abstract.
We prove a generalization of Gowers’ theorem for FIN k where, instead of thesingle tetris operation T : FIN k → FIN k − , one considers all maps from FIN k to FIN j for0 ≤ j ≤ k arising from nondecreasing surjections f : { , , . . . , k + 1 } → { , , . . . , j + 1 } .This answers a question of Bartoˇsov´a and Kwiatkowska. We also prove a common general-ization of such a result and the Galvin–Glazer–Hindman theorem on finite products, in thesetting of layered partial semigroups introduced by Farah, Hindman, and McLeod. Introduction
Gower’s theorem on FIN k is a generalization of Hindman’s theorem on finite unions whereone considers, rather than finite nonempty subsets of ω , the space FIN k of all finitely sup-ported functions from ω to { , , . . . , k } with maximum value k . Such a space is endowedwith a natural operation of pointwise sum, which is defined for pairs of functions with disjointsupport. Gowers considered also the tetris operation T : FIN k → FIN k − defined by letting( T b ) ( n ) = max { b ( n ) − , } for b ∈ FIN k . Gowers’ theorem can be stated, shortly, by sayingthat for any finite coloring of FIN k there exists an infinite sequence ( b n ) which is a blocksequence —in the sense that every element of the support of b n precedes every element of thesupport of b n +1 —with the property that the intersection of FIN k with the smallest subsetof FIN ∪ · · · ∪ FIN k that contains the b n ’s and it is closed under pointwise sum of disjointlysupported functions and under the tetris operation, is monochromatic [4]. Gowers then usedsuch a result—or more precisely its symmetrized version where one considers functions from ω to {− k, . . . , k } —to prove an oscillation stability result for the sphere of the Banach space c . Other proof of Gowers’ theorem can be found in [5, 7, 11].Gowers’ theorem of FIN k as stated above implies through a standard compactness ar-gument its corresponding finitary version . Explicit combinatorial proofs of such a finitaryversion have been recently given, independently, by Tyros [12] and Ojeda-Aristizabal [9].Particularly, the argument from [12] yields a primitive recursive bound on the associated Gowers numbers .A broad generalization of Gowers’ theorem has been proved by Farah, Hindman, andMcLeod in [3, Theorem 3.13] in the framework, developed therein, of layered partial semi-groups and layered actions . Such a result provides, in particular, a common generalization ofGowers’ theorem and the Hales–Jewett theorem; see [3, Theorem 3.15]. As general as [3, The-orem 3.13] is, it nonetheless does not cover the case where one considers FIN k endowed withthe multiple tetris operations described below, since these do not form a layered action inthe sense of [3, Definition 3.3].In [1], Bartoˇsov´a and Kwiatkowska considered a generalization of Gowers’ theorem, where multiple tetris operations are allowed. Precisely, they defined for 1 ≤ i ≤ k the tetris Date : April 1, 2016.2000
Mathematics Subject Classification.
Primary 05D10; Secondary 54D80.
Key words and phrases.
Gowers Ramsey Theorem, Hindman theorem, Milliken-Taylor theorem, idempo-tent ultrafilter, Stone- ˇCech compactification, partial semigroup. operation T i : FIN k → FIN k − by T i ( b ) : n (cid:26) b ( n ) − b ( n ) ≥ i , and b ( n ) otherwise.Adapting methods from [12], Bartoˇsov´a and Kwiatkowska proved in [1] the strengtheningof the finitary version of Gowers’ theorem where multiple tetris operations are considered.The authors then provided in [1] applications of such a result to the dynamics of the Lelekfan.Question 8.3 of [1] asks whether the infinitary version of Gowers’ theorem on FIN k holdswhen one considers multiple tetris operations. In this paper, we show that this is the case,via an adaptation of Gowers’ original argument using idempotent ultrafilters. In order toprecisely state our result, we introduce some terminology, to be used in the rest of the paper.We denote by ω the set of nonnegative integers, and by N the set of nonzero elementsof ω . We identify an element k of ω with the set { , , . . . , k − } of its predecessors. Asmentioned above, FIN k denotes the set of functions from ω to k + 1 with maximum value k and that vanish for all but finitely many elements of ω . We also let FIN ≤ k be the union ofFIN j for j = 1 , , . . . , k . The support Supp ( b ) of an element b of FIN k is the set of elementsof ω where b does not vanish. For finite nonempty subsets F, F ′ of ω , we write F < F ′ if themaximum element of F is smaller than the minimum element of F ′ .Suppose that 0 ≤ j ≤ k and f : k + 1 → j + 1 is a nondecreasing surjection. We alsodenote by f the generalized tetris operation f : FIN k → FIN j defined by f ( b ) = f ◦ b . It isclear that the class of generalized tetris operations is precisely the set of mappings that canbe obtained as composition of the multiple tetris operations T i for i = 1 , , . . . , k .We say that ( b n ) is a block sequence in FIN k if b n ∈ FIN k and Supp ( b n ) < Supp ( b n +1 )for every n ∈ ω . If j ∈ N , then we define the tetris subspace TS j ( b n ) of FIN j generated by( b n ) to be the set of elements of FIN j of the form f ◦ b + · · · + f n ◦ b n for some n ∈ ω , j , . . . , j n ∈ j + 1 such that max { j , . . . , j n } = j , and nondecreasingsurjections f i : k + 1 → j i + 1 for i ∈ n . A block sequence ( b ′ n ) in FIN k is a block subsequence of ( b n ) if ( b ′ n ) is contained in TS k ( b n ).In the following we will use some standard terminology concerning colorings. An r - coloring (or coloring with r colors) of a set X is a function c : X → r , and a finite coloring is an r -coloring for some r ∈ ω . A subset A of X is monochromatic (for the given coloring c ) if c is constant on A . Using this terminology, we can state our infinitary Gowers’ theorem forgeneralized tetris operations as follows. Theorem 1.1.
Suppose that k ∈ N . For any finite coloring of FIN ≤ k , there exists an infiniteblock sequence ( b n ) in FIN k such that TS j ( b n ) is monochromatic for every j = 1 , , . . . , k . Theorem 3.1 implies via a standard compactness argument its corresponding finitary ver-sion. If k, n ∈ ω , then we denote by FIN k ( n ) the set of functions f : n → k + 1 withmaximum value k , and by FIN ≤ k ( n ) the union of FIN j ( n ) for j = 1 , , . . . , k . The notion ofblock sequence ( b , . . . , b m − ) and tetris subspace TS j ( b , . . . , b m − ) of TS j ( n ) generated by( b , . . . , b m − ) are defined similarly as their infinite counterparts. Corollary 1.2.
Given k, r, ℓ ∈ N there exists n ∈ N such that for any r -coloring ofFIN ≤ k ( n ), there exists a block sequence ( b , b , . . . , b ℓ − ) in FIN k ( n ) of length ℓ such thatTS j ( b , . . . , b ℓ − ) is monochromatic for any j = 1 , , . . . , k .We will also prove below a more general statement than Theorem 1.1, where one considerscolorings of the space FIN [ m ] k of block sequences of FIN k of a fixed length m . We will alsoprovide a common generalization of such a result and the Galvin–Glazer–Hindman theorem We thank S lawomir Solecki for pointing this out.
OWERS’ RAMSEY THEOREM FOR GENERALIZED TETRIS OPERATIONS 3 on finite products, in the setting of layered partial semigroups introduced by Farah, Hindman,and McLeod in [3].As mentioned above, the original Gowers theorem from [4] was used to prove the following oscillation-stability result for the positive part of the sphere of c . Recall that c denotesthe real Banach space of vanishing sequences of real numbers endowed with the supremumnorm. Let PS( c ) be the positive part of the sphere of c , which is the set of elements of c of norm 1 with nonnegative coordinates. The support Supp ( f ) of an element f of c isthe set n ∈ ω such that f ( n ) = 0. A normalized positive block basis is a sequence ( f n ) offinitely-supported elements of PS( c ) such that Supp ( f n ) < Supp ( f n +1 ) for every n ∈ ω .Gower’s oscillation-stability result asserts that for any Lipschitz map F : PS( c ) → R and ε > f n ) such that the oscillation of F on the positive part ofthe sphere of the subspace of c spanned by ( f n ) is at most ε [4, Theorem 6]. Such a resultis proved by considering a suitable discretization of PS( c ) that can naturally be identifiedwith FIN k ; see the proof of [4, Theorem 6] and also [11, Corollary 2.26]. Under such anidentification, the tetris operation on FIN k corresponds to multiplication by positive scalarsin c .Similarly, one can observe that the multiple tetris operations T i for i = 1 , , . . . , k describedabove correspond to the following nonlinear operators on c . Fix λ, t ∈ [0 ,
1] and considerthe operator S t,λ on c mapping f to the function n (cid:26) λf ( n ) if | f ( n ) | ≥ t , f ( n ) otherwise.Given a normalized positive block basis ( f n ), one can consider the smallest subspace of c that contains ( f n ) and it is invariant under S t,λ for every t, λ ∈ [0 , Theorem 1.3.
Suppose that F : PS( c ) → R is a Lipschitz map, and ε > . There existsa positive normalized block sequence ( f n ) such that the oscillation of F on the positive partof the sphere of the smallest subspace of c containing ( f n ) and invariant under S t,λ for t, λ ∈ [0 , is at most ε . The rest of this paper consists of three sections. In Section 2 we present a proof ofTheorem 1.1. In Section 3 we explain how the proof of Theorem 1.1 can be modified toprove its multidimensional generalization. Finally in Section 4 we recall the theory of layeredpartial semigroups developed in [3], and present in this setting a common generalization ofthe (multidimensional version of) Theorem 1.1 and the Galvin–Glazer–Hindman theorem onfinite products.
Acknowledgments.
We are grateful to David Fernandez, Aleksandra Kwiatkowska, S lawomirSolecki, and Kostas Tyros for their comment and suggestions. We are also thank AleksandraKwiatkowska for pointing out a mistake in an earlier version of this paper, and Ilijas Farahfor referring us to [3] and to the theory of layered partial semigroups.2.
Gowers’ theorem for generalized tetris operations
Our proof of Theorem 1.1 uses the tool of idempotent ultrafilters, similarly as Gowers’original proof from [4]. In the following we will frequently use the notation of ultrafilterquantifiers [11, § U is an ultrafilter on FIN k and ϕ ( x )is a first-order formula, then ( U b ) ϕ ( b ) means that the set of b ∈ FIN k such that ϕ ( b ) holdsbelongs to U . A similar notation applies to ultrafilters on an arbitrary set.Adopting the terminology of [11, § U on FIN k is cofinite if ( ∀ n ∈ ω ) ( U b ), b ( n ) = 0. The set γ FIN k of cofinite ultrafilters on FIN k is endowed witha canonical semigroup operation , defined by setting A ∈ U + V if and only if ( U b ) ( V b ′ ), MARTINO LUPINI
Supp ( b ) < Supp ( b ′ ) and b + b ′ ∈ A . Furthermore γ FIN k is endowed with a canonicalcompact Hausdorff topology. Such a topology has a clopen basis consisting of sets of theform b A = {U ∈ γ FIN k : A ∈ U } for A ⊂ FIN k . Endowed with such a topology and semigroupoperation, γ FIN k is a compact right topological semigroup [11, § U 7→ U + V is continuous for any V ∈ γ FIN k . Any generalized tetrisoperation f : FIN k → FIN j admits a canonical extension to a continuous homomorphism f : γ FIN k → FIN j , defined by letting A ∈ f ( U ) iff ( U b ) f ( b ) ∈ A . In the following wewill repeatedly use the well know result, due to Ellis and Namakura, that any compact righttopological semigroup contains an idempotent element [11, Lemma 2.1].The following lemma can be seen as a refinement of [4, Lemma 3], and it is the core ofthe proof of Theorem 1.1. Lemma 2.1.
There exists a sequence ( U k ) of cofinite ultrafilters U k on FIN k such that forany 0 < j ≤ k and for any nondecreasing surjection f : k + 1 → j + 1, U k + U j = U j + U k = U k and f ( U k ) = U j . Proof.
We define by recursion on k sequences ( p ( k ) j ) of idempotent ultrafilters p ( k ) j ∈ γ FIN j such that, for every k, i, j ∈ N and nondecreasing surjection f : j + 1 → i + 1,(1) f ( p ( k ) j ) = p ( k ) i ,(2) p ( k +1) j = p ( k ) j for j ≤ k , and(3) p ( k ) j + p ( k ) j − = p ( k ) j for 2 ≤ j ≤ k .Granted the construction of the sequences ( p ( k ) j ), we can set U k := p ( k )1 + p ( k )2 + · · · + p ( k ) k .Observe that (3) implies that p ( k ) k + p ( k ) j = p ( k ) k for j ≤ k . Hence U k is idempotent, and U k + U k +1 = U k +1 + U k = U k +1 for every k ∈ N by (2). Furthermore it follows from (1) and(3) that f ( U k ) = U j for any nondecreasing surjection f : k + 1 → j + 1. Indeed supposethat f : k + 1 → j + 1 is a nondecreasing surjection. Then f | i +1 : i + 1 → f ( i ) + 1 is anondecreasing surjection for every i ∈ k + 1. Hence we can conclude by (1),(3), and the factthat the p ( k ) i ’s are idempotent that f ( U k ) = f ( p ( k )1 ) + · · · + f ( p ( k ) k ) = p ( k ) f (1) + p ( k ) f (2) + · · · + p ( k ) f ( k ) = p ( j )1 + · · · + p ( j ) j = U j .We now show how to construct the sequences ( p ( k ) j ). In the following we will convene that p ( k )0 is the function on ω constantly equal to 0, which can be seen as the unique element ofFIN . We let Π be the product of γ FIN j for j ∈ N . Observe that Π has a natural compactright topological semigroup structure, where the topology is the product topology and theoperation is the entrywise sum.For k = 1, consider the compact semigroup Σ ⊂ Π of sequences ( q j ) satisfying f ( q j ) = q i for any i, j ∈ N and nondecreasing surjection f : j + 1 → i + 1. We observe that Σ isnonempty. Define for j ∈ N the set M j := { b ∈ FIN j : ∀ n ∈ ω , b ( n ) ∈ { , j }} . Fix for any j ≥ f j : j + 1 → j . Observe that f j maps M j bijectively onto M j − . Furthermore, for any 0 < i ≤ j and nondecreasing surjection f : j + 1 → i + 1, onehas that f | M j = ( f i +1 ◦ f i ◦ · · · ◦ f j ) | M j . We denote by (cid:0) f j | M j (cid:1) − : M j − → M j the inversemap of f j . Let p be any element of γ FIN . One can define a sequence ( q j ) that belongs toΣ and such that M j ∈ q j by recursion on j ∈ N , by letting q j +1 = (cid:0) f j +1 | M j +1 (cid:1) − ( q j ) . Thisconcludes the proof that Σ is nonempty. We can then let ( p (1) j ) be an idempotent elementof Σ . This concludes the construction for k = 1.Suppose that the sequences ( p ( ℓ ) j ) have been defined for ℓ = 1 , , . . . , k − p ( k ) j ). Consider the compact semigroup Σ k ⊂ Πof sequences ( q j ) such that, for any i, j ∈ N and nondecreasing surjection f : j + 1 → i + 1, OWERS’ RAMSEY THEOREM FOR GENERALIZED TETRIS OPERATIONS 5 f ( q j ) = q i , q j = p ( k − j for j ∈ k , and q j + p ( k − i = q j for j ∈ N and i ≤ min { k − , j } . Weneed to show that Σ k is nonempty. Set q j := p ( k − j + p ( k − j − + · · · + p ( k − for every j ∈ N . Observe that for j ≤ k − q j = p ( k − j in view of (3). If j ≥ k then q j = p ( k − j + p ( k − j − + · · · + p ( k − k − ,again by (3). Since p ( k − i is idempotent for any i ∈ N , it follows from (1) that f ( q j ) = q i forany nondecreasing surjection f : j + 1 → i + 1. Furthermore for j ∈ N and i ≤ min { k − , j } one has that q j + p ( k − i = q j in view of (2). This shows that the sequence ( q j ) belongs toΣ k . One can then let ( p ( k ) j ) be any idempotent element of Σ k . This concludes the recursiveconstruction. (cid:3) Theorem 1.1 can now be deduced from Lemma 2.1 through a standard argument. Wepresent a sketch of the proof, for convenience of the reader.
Proof of Theorem 1.1.
Suppose that U , U , . . . , U k are the cofinite ultrafilters constructedin Lemma 2.1. Fix a finite coloring c of FIN j for 1 ≤ j ≤ k , and let A j be an elementof U j such that c is constant on A j for j = 1 , , . . . , k . We define, by recursion on n ∈ ω , b n ∈ FIN k with Supp ( b i ) < Supp ( b j ) for i < j , such that the following conditions hold: forany j , . . . , j n +2 ∈ k + 1 and nondecreasing surjections f i : k + 1 → j i + 1 for i ∈ n + 3,(1) f ◦ b + · · · + f n ◦ b n belongs to A max { j ,...,j n } ,(2) ( U k y ), f ◦ b + · · · + f n ◦ b n + f n +1 ◦ y belongs to A max { j ,...,j n +1 } , and(3) ( U k y ) ( U k z ), f ◦ b + · · · + f n ◦ b n + f n +1 ◦ y + f n +2 ◦ z belongs to A max { j ,...,j n +2 } .Suppose that such a sequence has been defined up to n . From (2) and (3), we can concludethat there exists b n +1 ∈ FIN k such that Supp ( b n +1 ) > Supp ( b n ) satisfying (1) and (2). Then(3) follows from (2), the properties of ultrafilter quantifiers, and the facts that, for 1 ≤ j ≤ k , U j + U k = U k + U j = U k and f ( U k ) = U j for any nondecreasing surjection f : k + 1 → j + 1.This concludes the recursive construction. In view of (1), the sequence ( b n ) obtained throughthis construction has the property that TS j ( b n ) is contained in A j , and hence c is constanton TS j ( b n ) for j = 1 , . . . , k . (cid:3) A multidimensional generalization
Gowers’ theorem on FIN k can be seen as a generalization of Hindman’s theorem for setsof finite unions [6]. Such a theorem asserts that for any finite coloring of FIN , there existsa block sequence ( b n ) in FIN such that TS ( b n ) is monochromatic. Observe that one canidentify FIN with the set of nonempty finite subsets of ω . Then TS ( b n ) is just the collectionof all finite unions of the elements of the given sequence. Hindman’s theorem on finite unionsis the particular instance of Gowers’ theorem for k = 1.In another direction, Hindman’s theorem on finite unions was generalized, independently,by Milliken and Taylor [8, 10]; see also [2]. Fix m ∈ N and consider the set FIN [ m ]1 of blocksequences in FIN of length m . The Milliken-Taylor theorem on finite unions asserts that,for any finite coloring of FIN [ m ]1 , there exists an infinite block sequence ( b n ) in FIN suchthat the set TS ( b n ) [ m ] of m -tuples of the form( b n + · · · + b n ℓ − , b n ℓ + · · · + b n ℓ − , . . . , b n ℓm − + · · · + b n ℓm − )for 0 < ℓ < ℓ < · · · < ℓ m and 0 ≤ n < n < · · · < n ℓ m − , is monochromatic.The multidimensional analog of Gowers’ theorem for a single tetris operation is provedin [11, Corollary 5.26]. The corresponding finite version is considered in [12]. In a similarspirit, one can consider a multidimensional generalization of Theorem 3.1. Let FIN [ m ] k be MARTINO LUPINI the space of block sequences in FIN k of length m , and FIN [ m ] ≤ k be the union of FIN [ m ] j for j = 1 , , . . . , k . If ( b n ) is a block sequence in FIN k and 1 ≤ j ≤ k , then we define the tetrissubspace TS j ( b n ) [ m ] of FIN [ m ] k generated by ( b n ) to be the set of elements of FIN [ m ] j of theform ( a , . . . , a m − ), where a d for d ∈ m is equal to f n d ◦ b n d + · · · + f n d +1 − ◦ b n d +1 − for some n = 0 < n < n < · · · < n m , 0 ≤ j i ≤ k and nondecreasing surjections f i : k + 1 → j i + 1 for i ∈ n m such that max (cid:8) j n d , . . . , j n d +1 − (cid:9) = j . We can then state themultidimensional generalization of Theorem 1.1 as follows: Theorem 3.1.
Suppose that m, k ∈ N . For any finite coloring of FIN [ m ] ≤ k , there existsan infinite block sequence ( b n ) in FIN k such that TS j ( b n ) [ m ] is monochromatic for every j = 1 , , . . . , k . In order to prove the Milliken-Taylor theorem, one can consider an idempotent cofiniteultrafilter U on FIN , and then the Fubini power V := U ⊗ m . This is defined as the cofiniteultrafilter on FIN [ m ]1 such that A ∈ V if and only if ( U b ) · · · ( U m b m ), ( b , . . . , b m ) ∈ A ;see [11, § V witnesses that the Milliken-Taylor theorem holds.A similar approach works for Theorem 3.1. Indeed, consider the cofinite ultrafilter U k onFIN k given by Lemma 2.1 and its Fubini power V k := U ⊗ mk on FIN [ m ] k . Then any elementof V k witness that Theorem 3.1 holds. The proof of such a fact is analogous to the proof ofTheorem 1.1, and only notationally heavier. The details are left to the interested reader.As usual, it follows by compactness from Theorem 3.1 the corresponding finite version,which recovers Corollary 2.3 of [1] Corollary 3.2.
Suppose that m, k, ℓ, r ∈ N . There exists n ∈ N such that for any r -coloringof FIN j ( n ) [ m ] , there exists a block sequence ( b , . . . , b ℓ − ) in FIN k of length ℓ such thatTS j ( b , . . . , b ℓ − ) [ m ] is monochromatic for j = 1 , , . . . , k .4. A generalization for layered partial semigroups
Recall that a partial semigroup [3, Definition 1.2] is a set S endowed with a partiallydefined binary operation ( x, y ) xy satisfying ( xy ) z = x ( yz ). This equation should beinterpreted as asserting that the left hand side is defined if and only if the right hand sideis defined, and in such a case the equality holds. Suppose that x is an element of a partialsemigroup S . Following [3, Definition 2.1], we let ϕ S ( x ) be the set of elements y of S suchthat xy is defined. More generally, for a subset A of S , we let ϕ S ( A ) be the set of elements y of S such that xy is defined for every x ∈ A . As in [3, Definition 2.1], we say that a partialsemigroup S is adequate —also called directed in [11, § ϕ S ( A ) is nonempty for everyfinite subset A of S .When S is a directed partial semigroup, the set γS of ultrafilters U over S with theproperty that ( ∀ x ∈ S ) ( U y ) xy is defined, is a closed nonempty subset of the space βS ofultrafilters over S . One can define a compact right topological semigroup operation on γS by setting A ∈ U V if and only if ( U y ) ( V z ) yz ∈ A [11, Corollary 2.7]. More generally, itis shown in [3, Theorem 2.2] that the operation on S extends to a continuous map from βS × γS to βS .Suppose that S, T are partial semigroups, and σ : S → T is a function. We say that σ is a partial semigroup homomorphism if for any x, y ∈ S , σ ( x ) σ ( y ) is defined whenever xy is defined, and in such a case σ ( xy ) = σ ( x ) σ ( y ) [3, Definition 2.8]. We say that σ : S → T is an adequate partial semigroup homomorphism if it is a partial semigroup homomorphismwith the property that for any finite subset A of S there exists a finite subset B of T suchthat ϕ T ( B ) is contained in the image under σ of ϕ S ( A ). OWERS’ RAMSEY THEOREM FOR GENERALIZED TETRIS OPERATIONS 7 If T is a partial semigroup and S ⊂ T , then S is an adequate partial subsemigroup if theinclusion map S ֒ → T is an adequate partial semigroup homomorphism [3, Definition 2.10].We say that a subset S of a partial semigroup T is an adequate ideal if it is an adequatepartial subsemigroup, and for any x ∈ S and y ∈ T one has that xy and yx belong to S whenever they are defined [3, Definition 2.15]. Lemma 2.14 and Lemma 2.16 of [3] showthat, if S ⊂ T is an adequate partial subsemigroup, then γS can be canonically identifiedwith a subsemigroup of γT . If furthermore S is an adequate ideal of T , then γS is an idealof γT . We now recall the definition of layered partial semigroup from [3, § e of a partial semigroup is an identity element if ex and xe are defined and equal to x for any x ∈ S . Definition 4.1. A layered partial semigroup with k layers is a partial semigroup S endowedwith a partition { S , . . . , S k } such that S = { e } for some identity element e for S , and forevery n = 1 , , . . . , k , letting S ≤ n = S ∪ · · · ∪ S k , one has that S ≤ n is an adequate partialsemigroup, S n is an adequate partial subsemigroup of S , and an adequate ideal of S ≤ n .In the following we will assume that S is a layered partial semigroup with k layers aswitnessed by the partition { S , . . . , S k } , and set S ≤ n = S ∪ · · · ∪ S n . Observe that itfollows from the definition of layered partial semigroup that γS n is an ideal of γS ≤ n , and asubsemigroup of γS for n = 1 , , . . . , k . Definition 4.2.
Suppose that A = ( F , M , F , M , . . . , F k , M k ) is a tuple such that forevery n = 1 , , . . . , k , F n is a nonempty finite collection of partial semigroup homomorphismsfrom S ≤ n to S ≤ n − , and M n is an adequate subsemigroup of S n for n = 1 , , . . . , k . We saythat A is a tetris action on S if and only if it satisfies for any n = 2 , , . . . , k , and σ ∈ F n the following conditions:(1) the image of M n under σ is an adequate partial subsemigroup of M n − ;(2) the image of S n under σ is an adequate partial subsemigroup of S n − ;(3) the restriction of σ to S ≤ n − either belongs to F n − , or it is the identity map of S ≤ n − , and(4) for any σ , σ ∈ F n one has that σ | M n = σ | M n .From now on we assume that ( F , M , F , M , . . . , F k , M k ) is a tetris action on S as inDefinition 4.2. It follows from [3, Lemma 2.4] that for any n = 2 , , . . . , n , any element σ of F n admits a continuous extension σ : βS ≤ n → βS ≤ n − such that: • if p ∈ βS n , q ∈ γS ≤ n − , and σ ( q ) ∈ γS ≤ n − , then σ ( pq ) = σ ( p ) σ ( q ); • if p ∈ βS ≤ n − , q ∈ γS n , and σ ( q ) ∈ γS n , then σ ( pq ) = σ ( p ) σ ( q ); • σ maps γS n to γS n − and γM n to γM n − .In particular, σ induces continuous semigroup homomorphism σ : γS n → γS n − mappingthe subsemigroup γM n to γM n − . The same proof as Lemma 2.1 shows the following: Lemma 4.3.
There exist idempotent elements U n ∈ γS n for n = 1 , , . . . , k such that σ ( U n ) = U n − and U n U n − = U n − U n = U n for every n = 2 , . . . , k and σ ∈ F n .Given a tetris action, one can define as in [3, Definition 3.9] the collection G n of mapsfrom S k to S n of the form σ n +1 ◦ σ n +2 ◦ · · · ◦ σ k , where σ j ∈ F j for j = n + 1 , . . . , k . We alsolet G be the union of G n for n = 1 , , . . . , k . Definition 4.4. A block sequence in S k is a sequence ( b n ) such that f ( b ) · · · f n ( b n ) isdefined for any n ∈ ω and f , . . . , f n ∈ G .The notion of block sequence in S n for some n ≤ k is defined similarly. We let S [ m ] n be theset of block sequences in S n of length m , and S [ m ] ≤ n be the union of S [ m ] j for j = 1 , , . . . , n . If( b n ) is a block sequence in S k , then we define the tetris subspace TS j ( b n ) ⊂ S [ n ] j of the j -thlayer generated by ( b n ) to be the set of elements of S [ n ] j of the form ( a , . . . , a m − ) where MARTINO LUPINI for some 0 = n < n < · · · < n m ∈ ω , j i ∈ k + 1, and f i ∈ G j i for i ∈ n m one has thatfor every d ∈ m , max { j n d , . . . , j n d +1 − } = j and a d = f n d ( b n d ) · · · f n d +1 − ( b n d +1 − ). Thenusing Lemma 4.3 one can prove as in Theorem 3.1 the following result, which is a commongeneralization of the Galvin-Glazer theorem and Theorem 3.1. Theorem 4.5.
Suppose that S is a layered partial semigroup endowed with a tetris action asabove. Fix m ∈ N and a finite coloring of S [ m ] ≤ k . Then there exists an infinite block sequence ( b n ) in S k such TS j ( b n ) [ m ] is monochromatic for every j = 1 , , . . . , k . It is clear that Theorem 4.5 has the Galvin-Glazer theorem [11, Theorem 2.20] as a par-ticular case. Set now S j := FIN j for j = 0 , , . . . , k , and S := S ∪ · · · ∪ S n . Define a partialsemigroup operation on S by ( b, b ′ ) b + b ′ whenever Supp ( b ) < Supp ( b ′ ), where b + b ′ isthe pointwise sum. Then S = S ∪ · · · ∪ S k is a layered partial semigroup in the sense ofDefinition 4.1. Denote by F n for n = 1 , , . . . , k the collection of multiple tetris operations T , . . . , T n : FIN n → FIN n − defined in the introduction. Let also M n ⊂ S n be the set of b ∈ S n such that b ( i ) ∈ { , n } for every i ∈ ω . It is then easy to see that ( F n , M n ) kn =1 isa tetris action on S in the sense of Definition 4.2. Furthermore the conclusions of Theorem4.5 in the particular case of such a tetris action yields Theorem 3.1. References
1. Dana Bartoˇsov´a and Aleksandra Kwiatkowska,
Gowers’ Ramsey Theorem with multiple operations anddynamics of the homeomorphism group of the Lelek fan , arXiv:1411.5134 (2014).2. Vitaly Bergelson, Neil Hindman, and Kendall Williams,
Polynomial extensions of the Milliken-TaylorTheorem , Transactions of the American Mathematical Society (2014), no. 11, 5727–5748.3. Ilijas Farah, Neil Hindman, and Jillian McLeod,
Partition theorems for layered partial semigroups , Journalof Combinatorial Theory. Series A (2002), no. 2, 268–311.4. W. Timothy Gowers, Lipschitz functions on classical spaces , European Journal of Combinatorics (1992), no. 3, 141–151.5. , Ramsey methods in Banach spaces , Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1071–1097.6. Neil Hindman,
Finite sums from sequences within cells of a partition of N , Journal of CombinatorialTheory. Series A (1974), 1–11.7. Vassilis Kanellopoulos, A proof of W. T. Gowers’ c theorem , Proceedings of the American MathematicalSociety (2004), no. 11, 3231–3242.8. Keith R. Milliken, Ramsey’s theorem with sums or unions , Journal of Combinatorial Theory. Series A (1975), 276–290.9. Diana Ojeda-Aristizabal, Finite forms of Gowers theorem on the oscillation stability of C , Combinatorica(2015), 1–13.10. Alan D. Taylor, A canonical partition relation for finite subsets of ω , Journal of Combinatorial Theory.Series A (1976), no. 2, 137–146.11. Stevo Todorcevic, Introduction to Ramsey spaces , Annals of Mathematics Studies, vol. 174, PrincetonUniversity Press, Princeton, NJ, 2010.12. Konstantinos Tyros,
Primitive recursive bounds for the finite version of Gowers’ c theorem , Mathematika (2015), no. 3, 501–522. Mathematics Department, California Institute of Technology, 1200 E. California Blvd, MC253-37, Pasadena, CA 91125
E-mail address : [email protected] URL ::