Abstract approach to finite Ramsey theory and a self-dual Ramsey theorem
aa r X i v : . [ m a t h . C O ] S e p ABSTRACT APPROACH TO FINITE RAMSEY THEORYAND A SELF-DUAL RAMSEY THEOREM
S LAWOMIR SOLECKI
Abstract.
We give an abstract approach to finite Ramsey theory andprove a general Ramsey-type theorem. We deduce from it a self-dualRamsey theorem, which is a new result naturally generalizing both theclassical Ramsey theorem and the dual Ramsey theorem of Graham andRothschild. In fact, we recover the pure finite Ramsey theory from ourgeneral Ramsey-type result in the sense that the classical Ramsey the-orem, the Hales–Jewett theorem (with Shelah’s bounds), the Graham–Rothschild theorem, the versions of these results for partial rigid sur-jections due to Voigt, and the new self-dual Ramsey theorem are allobtained as iterative applications of the general result. Introduction
Abstract approach to Ramsey theory and its applications.
Wegive an abstract approach to pure (unstructured) finite Ramsey theory. Thespirit of the undertaking is similar to Todorcevic’s approach to infinite Ram-sey theory [20, Chapters 4 and 5], even though, on the technical level, thetwo approaches are different. There are three main points to the presentpaper. The first one is the existence of a single, relatively simple type ofalgebraic structure, called Ramsey domain over a composition space (or anormed composition space), that underlies Ramsey theorems. The secondpoint is the existence of a single Ramsey theorem, which is a result aboutthe algebraic structures just mentioned. Particular Ramsey theorems areinstances, or iterative instances, of this general result for particular Ramseydomains, much like theorems about, say, modules have particular instancesfor concrete modules. The latter point opens up a possibility of classify-ing concrete Ramsey situations, at least in limited contexts; see Section 10.Finally, the third main point is a new concrete Ramsey theorem obtainedusing the abstract approach to Ramsey theory. We call this theorem theself-dual Ramsey theorem. It is a common generalization of the classical
Mathematics Subject Classification.
Key words and phrases.
Ramsey theory, Hales–Jewett theorem, Graham-Rothschildtheorem, self-dual Ramsey theorem.Research supported by NSF grant DMS-1001623.
Ramsey theorem and the dual Ramsey theorem. As far as proofs of theknown results are concerned, one advantage of the approach given here isits uniformity. Our approach also provides a hierarchy of the Ramsey resultsaccording to the number of times the abstract Ramsey theorem is appliedin their proofs. For example, the classical Ramsey theorem requires onesuch application, the Hales–Jewett theorem requires two applications, theGraham–Rothschild theorem three, and the self-dual Ramsey theorem four.The following vague observation is at the starting point of the abstractapproach to Ramsey theory. (This observation may not be apparent at thispoint, but it will become clear with the reading of the paper.) Roughlyspeaking, a Ramsey-type theorem is a statement of the following form. Weare given a set S chosen arbitrarily from some fixed family S and a numberof colors d . We find a set F from another fixed family F with a “scrambling”function, usually a type of composition, defined on F × S : F × S ∋ ( f, x ) → f . x ∈ F . S.
The arrangement is such that for each d -coloring of F . S there is f ∈ F with f . S monochromatic. Our undertaking consists of finding a general,algebraic framework in which we isolate an abstract pigeonhole principle andprove that it implies a precise version of the above abstract Ramsey-typestatement.The paper is structured as follows.Section 1: Later in this introduction, we recall the central theorems ofthe finite unstructured Ramsey theory and place our new concrete Ramseyresult—the self-dual Ramsey theorem—in this context.Section 2: We fix the notions which are used to state the concrete Ram-sey results in this paper. These notions are formulated in the language ofinjections and surjections. We show how to translate Ramsey statements asin Section 1 to statements in this language.Section 3: We define the new algebraic notions needed to phrase and provethe abstract Ramsey theorem. The progression of notions is as follows:— actoid , a most basic notion of action;— set actoid over an actoid, a lift of the operations on an actoid tosubsets;— composition spaces , actoids with an added operator;— Ramsey domains , set actoids over composition spaces fulfilling addi-tional conditions.Section 4: Using the algebraic notions introduced in the previous section,we phrase the main Ramsey theoretic conditions. So the first half of thissection contains formulations of:
AMSEY THEORY 3 — the
Ramsey property for Ramsey domains;— the pigeonhole principle for Ramsey domains.In the second half of the section, we prove in Theorem 4.1 and Corollary 4.3that, under mild assumptions, the pigeonhole principle implies the Ramseyproperty for Ramsey domains.Section 5: We prove that, in many situations, to get a Ramsey theorem itsuffices to check only a localized, and therefore easier, version of the abstractpigeonhole principle. We start this section by formulating:— the local pigeonhole principle for Ramsey domains,and follow it by defining— normed composition spaces , composition spaces with a norm to apartial ordering.After that we prove Theorem 5.3 and Corollary 5.4 that show that the localpigeonhole principle, under mild conditions, implies the Ramsey propertyfor Ramsey domains over normed composition spaces.Section 6: We show two results allowing us to propagate the pigeonholeprinciple. In the first one, Proposition 6.2, we get the pigeonhole principlefor naturally defined products assuming it holds for the factors. The secondresult, Proposition 6.4, involves a notion of interpretability and establishespreservation of the pigeonhole principle under interpretability.Section 7: We give examples of composition spaces and Ramsey domains.More examples can be found in papers [17] and [18]. (Note that the termi-nology in [17] differs somewhat from the one in the present paper.)Section 8: This section contains applications of the abstract Ramsey ap-proach to concrete situations. As a consequence of the general theory weobtain a new self-dual Ramsey theorem. We give its statement and explainits relationship with other results in Subsection 1.2 below. Here, let us onlymention one interesting feature of the proof of this theorem: the role of thepigeonhole principle is played by the Graham–Rothschild theorem. We alsogive other applications of the general theory to concrete examples. We showhow to derive as iterative applications of the abstract Ramsey result the clas-sical Ramsey theorem, the Hales–Jewett theorem, the Graham–Rothschildtheorem as well as the versions of these results for partial rigid surjectionsdue to Voigt. We note that in the proof of the Hales–Jewett theorem thebounds we obtain on the parameters involved in it turn out to be primitiverecursive and are essentially the same as Shelah’s bounds from [14]. Wewill, however, leave it to the reader to check the details of this estimate.More applications of the abstract approach to Ramsey theory involving fi-nite trees can be found in [17] and [18]. (Note again that the terminologyin the present paper differs from that in [17].)
S LAWOMIR SOLECKI
Section 9: In Theorem 9.3, we give an interesting example for whichRamsey theorem fails. The objects that are being colored can be viewed asLipschitz surjections with Lipschitz constant 1 between initial segments ofthe set of natural numbers. This example is motivated by considerations intopological dynamics.Section 10: We make concluding remarks and state a problem on classi-fying Ramsey theorems in a natural, but limited, set-up.It may be worthwhile to point out that, on the conceptual level, theelegant approach of Graham, Leeb and Rothschild [2] and of Spencer [19] tofinite Ramsey theorems for spaces is very much different from the approachpresented here. The differences on the technical level are equally large. Onemain such difference is that, unlike here, the setting of [2] and [19] has aconcrete pigeonhole principle built into it, which in that approach is theHales–Jewett theorem.The pure Ramsey theory, which is the subject matter of this paper, is afoundation on which the Ramsey theory for finite structures is built, butis not a part of it. Consequently, the methods of the present paper havenothing directly to say about the structural Ramsey theory as developed forrelational structures by Neˇsetˇril and R¨odl in [8], [9], [10], and by Pr¨omel in[12] and, more recently, for structures that incorporate both relations andfunctions by the author in [15], [16].1.2.
Self-dual Ramsey theorem and finite Ramsey theory.
We con-sider 0 to be a natural number. As is usual, we adopt the convention thatfor a natural number n , [ n ] = { , . . . , n } . In particular, [0] = ∅ .The aim of this subsection is to survey the fundamental results of finiteRamsey theory, in particular, we recall the classical Ramsey theorem andthe dual Ramsey theorem of Graham and Rothschild. In their context, westate the new self-dual Ramsey theorem. Later we recall further Ramseytheoretic results that will be relevant in the sequel.The classical Ramsey theorem can be stated as follows. Ramsey’s Theorem.
Given the number of colors d and natural numbers k and l , there exists a natural number m such that for each d -coloring of allsubsets of [ m ] of size k there exists a subset B of [ m ] of size l such that { A | A ⊆ B and A has size k } is monochromatic. AMSEY THEORY 5
The dual Ramsey theorem, due to Graham and Rothschild [3], concernspartitions and can be stated as follows. Recall that a partition P is coarserthan a partition Q if each set in P is a union of some sets in Q . Dual Ramsey Theorem.
Given the number of colors d and natural num-bers k, l > , there exists a natural number m such that for each d -coloringof all partitions of [ m ] with k pieces there exists a partition Q of [ m ] with l pieces such that {P | P a partition with k pieces coarser than Q} is monochromatic. It is natural to ask if a “self-dual” Ramsey theorem exists that combinesthe two statements above. We formulate now such a self-dual theorem. Wewill be coloring pairs consisting of a partition and a set interacting witheach other in a certain way. Let R be a partition of [ n ] and let C bea subset of [ n ]. Let m ∈ N . We say that ( R , C ) is an m - connection if R and C have m elements each and, upon listing R as R , . . . , R m withmin R i < min R i +1 and C as c , . . . , c m with c i < c i +1 , we have c i ∈ R i for i ≤ m and c i < min R i +1 for i < m . We say that an l -connection ( Q , B )is an l - subconnection of an m -connection ( R , C ) if Q is a coarser partitionthan R and B ⊆ C .Here is the self-dual Ramsey theorem. Its reformulation in terms of sur-jections and injections is Theorem 2.1. It is proved in Subsection 8.3. Self-dual Ramsey Theorem.
Let d > . For each k, l ∈ N , k, l > , thereexists m ∈ N such that for each d -coloring of all k -subconnections of an m -connection ( R , C ) there exists an l -subconnection ( Q , B ) of ( R , C ) suchthat all k -subconnections of ( Q , B ) get the same color. Ramsey’s Theorem is just the theorem above for colorings that do notdepend on the first coordinate; Dual Ramsey Theorem theorem is the abovetheorem for colorings that do not depend on the second coordinate.We recall now some other results of finite Ramsey theory, partly to remindthe reader of the main results of the theory, and partly because we will needthem in our proof of the self-dual Ramsey theory. We will illustrate the ab-stract approach to Ramsey theory developed in this paper by proving theseresults. This exercise is to support our assertion that all the unstructuredRamsey theoretic results, including the self-dual Ramsey theorem, can beobtained as iterative instances of the abstract Ramsey theorem.We state these results here using the language of parameter sets as isdone in Neˇsetˇril’s survey [7]. Later, in Section 2, we will restate them in thelanguage of injections and surjections. Let
A, l, n ∈ N with A , l not both S LAWOMIR SOLECKI equal to 0. By an l -dimensional A -parameter set on n we understand a pairof the form(1.1) V = ( g, G ) , where G consists of l non-empty, pairwise disjoint subsets of [ n ] and g : [ n ] \ S G → [ A ]. Note that if V is of dimension 0, then G = ∅ , and V = ( g, ∅ )can, and will, be identified with the function g : [ n ] → A . The set of all suchfunctions is denoted by A n . Note also that if A = 0, then there is only onechoice for g —the empty function—and V can be identified with G , whichin this case is a partition of [ n ] into l pieces. Thus, 0-parameter sets areidentified with partitions.A k -dimensional A -parameter set on nU = ( f, F )is an A -parameter subset of V as in (1.1) if each set in F is the union ofsome sets in G , f extends g , and f restricted to each set in G included in itsdomain is constant. In the particular case, when U is 0-dimensional and isidentified with f : [ n ] → A , we say that f is a function in V . This translatesto f being an extension of g to [ n ] that is constant on each set in G . In theparticular case, when V is a 0-parameter set and, therefore, so is U , andthey are both identified with the partitions G and F , respectively, we havethat F is coarser than G .The first theorem is from [4]. Hales–Jewett Theorem.
Fix A ∈ N , A > , and d > . For each l ∈ N there exists m ∈ N such that for each d -coloring of A m there exists an l -dimensional A -parameter set U such that all functions in U get the samecolor. The second theorem is from [3].
Graham–Rothschild Theorem.
Fix A ∈ N and d > . Given k, l ∈ N ,with k, l > if A = 0 , there exists m ∈ N such that for each d -coloringof all k -dimensional A -parameter subsets of an m -dimensional A -parameterset V there exists an l -dimensional A -parameter subset U of V such that all k -dimensional A -parameter subsets of U get the same color. Note that the Dual Ramsey Theorem is an instance of the Graham–Rothschild Theorem for A = 0 and k, l > k -dimensional A -parameter set on mU = ( f, F ) AMSEY THEORY 7 is a partial A -parameter subset of V as in (1.1) if m = n or m + 1 = min a for some a ∈ G , and U is an A -parameter subset of the A -parameter set (cid:16) g ↾ ([ m ] \ [ G ) , { b ∩ [ m ] | b ∈ G , b ∩ [ m ] = ∅} (cid:17) . Note that each A -parameter subset of V is a partial A -parameter subset of V . When U is 0-dimensional and is identified with the function f : [ m ] → A ,we say that f is a partial function in V as in (1.1). The notion of partialfunction in V can be rephrased by saying that m = n or m + 1 = min a forsome a ∈ G , f extends g ↾ ([ m ] \ S G ) to [ m ], and f is constant on each b ∩ [ m ] for b ∈ G .Let A ≤ m stand for the set of all functions f : [ m ′ ] → A with m ′ ≤ m . Hales–Jewett Theorem, Voigt’s version.
Fix A ∈ N and d > . Foreach l ∈ N there exists m ∈ N such that for each d -coloring of A ≤ m thereexists an l -dimensional A -parameter set U on some m ′ ≤ m such that allpartial functions in U get the same color. Graham–Rothschild Theorem, Voigt’s version.
Fix A ∈ N and d > .For each k, l ∈ N there exists m ∈ N such that for each d -coloring of all k -dimensional partial A -parameter subsets of an m -dimensional A -parameterset V there exists an l -dimensional partial A -parameter subset U of V suchthat all k -dimensional partial A -parameter subsets of U get the same color. Walks.
Uspenskij in [21] asked for an identification of the universalminimal flow of the homeomorphism group of the generic compact connectedmetric space called the pseudo-arc. In view of the papers by Irwin and theauthor [5] and by Kechris, Pestov, and Todorcevic [6], it became apparentto the author that such an identification can be accomplished if a certainRamsey statement were true. However, in Section 9, we prove that thisRamsey statement is false. We state this theorem below after we havedefined the objects involved in it. The theorem gives an interesting andnatural class of rigid surjections (for a definition see Section 2) for whichRamsey theorem fails. The coloring that make it fail was found as a by-product of an analysis of the Ramsey statement with the abstract approach.A walk is a function s : [ n ] → [ m ] that is surjective and such that s (1) = 1and | s ( i + 1) − s ( i ) | ≤ i ∈ [ n − Coloring of Walks Theorem.
Let m ≥ . There exists a coloring withtwo colors of all walks from [ m ] to [3] such that for each walk t : [ m ] → [6] the set { s ◦ t | s : [6] → [3] a walk } is not monochromatic. S LAWOMIR SOLECKI The language of injections and surjections; reformulationsof Ramsey results
In the paper, we consistently use the language of rigid surjections andincreasing injections rather than that of partitions and sets. (This languagewas proposed in [13].) In our opinion, this choice is more satisfying fromthe theoretical point of view and it easily accommodates objects comingfrom topology such as walks in Section 9. Note, however, that the abstractapproach is also applicable to the partition-and-set formalism. A canonicalway of translating statements in one language into the other is explained inSubsection 2.4.Recall that for N ∈ N , we let[ N ] = { , . . . , N } . In the sequel, we use letters
K, L, M, N, P, Q , possibly with subscripts, tostand for natural numbers.2.1.
Classes of injections and surjections.
We fix here some notationand some notions needed in the sequel.By an increasing injection we understand a strictly increasing functionfrom [ K ] to N for some K ∈ N . Let(2.1) II = { i | i is an increasing injection } . For K ≤ L , let (cid:18) LK (cid:19) = { i ∈ II | i : [ K ] → [ L ] } . Since an increasing injection from [ K ] to [ L ] is determined by, and of courseitself determines, its image, that is, a K element subset of [ L ], the set (cid:0) LK (cid:1) defined above can be thought of as the set of all K element subsets of [ L ].Let S = { v | ∃ K, L ( K ≤ L and v : [ L ] → [ K ] is a surjection) } . We adopt the convention that for v ∈ S writing v : [ L ] → [ K ] signifies that v is onto [ K ].A rigid surjection is a function s : [ L ] → [ K ] that is surjective and suchthat for each y ∈ [ L ] there is x ∈ [ K ] with s ([ y ]) = [ x ]. In other words, foreach y ∈ [ L ], we have s ( y ) ≤ ≤ x K, L be natural numbers. We call a pair ( s, i ) a connection between L and K if s : [ L ] → [ K ], i : [ K ] → [ L ] and for each x ∈ [ K ] s ( i ( x )) = x and ∀ y < i ( x ) s ( y ) ≤ x. In other words, i is a left inverse of s with the additional property that ateach x ∈ [ K ] the value i ( x ) is picked only from among those elements of s − ( x ) that are “visible from x ,” that is, from those y ′ ∈ s − ( x ) for which s ↾ ( { y | y < y ′ } ) ≤ x. We write ( s, i ) : [ L ] ↔ [ K ] . It is easy to see that if ( s, i ) is a connection, then i is an increasing injectionand s is a rigid surjection. Also for each rigid surjection s there is anincreasing injection i (usually many such injections) for which ( s, i ) is aconnection, and for each increasing injection i there is a rigid surjection s (again, usually many such surjections) with ( s, i ) a connection.Finally, for technical reasons, we need the notion of augmented surjection,which are ordered pairs whose elements are a rigid surjection and an increas-ing surjection with these elements appropriately interacting with each other.Let AS = { ( s, p ) | ∃ K, L ∈ N ( s, p : [ L ] → [ K ] , p ∈ IS , s ≤ p, ∀ x ∈ [ K ] s (max p − ( x )) = x ) } . It is easy to see that ( s, p ) ∈ AS implies that s is a rigid surjection. Elementsof AS are called augmented surjections . The canonical composition of a rigid surjection and a function. In the sequel in various situations, we will require to compose functions withrigid surjections. It will always be done in a particular way. This canonicalcomposition will be a restriction of the usual composition to a certain initialsegment. Here is a precise definition. Let v : [ L ] → [ K ] be a function andlet s : [ N ] → [ M ] be a rigid surjection. The canonical composition of v and s , which we denote by v ◦ s , is defined if and only if L ≤ M . In this case, let N ≤ N be the largest number such that s ( y ) ≤ L for all y ≤ N . Define(2.3) v ◦ s to be the usual composition of v with s ↾ [ N ]. It is easy to see that v ◦ s isthe restriction of the usual composition of v and s , which is a partial functionon [ N ], to the largest initial segment of [ N ] on which this composition isdefined. If M = K , then v ◦ s is the usual composition of v and s .Note that if v is a surjection, then v ◦ s : [ N ] → [ K ] is a surjection. If v is a rigid surjection, then v ◦ s is a rigid surjection. If v and s are walks,then so is v ◦ s . If v and s are increasing surjections, then so is v ◦ s .It is easy to verify that if v is a function, s , t are rigid surjections, and( v ◦ s ) ◦ t and v ◦ ( s ◦ t ) are both defined, then( v ◦ s ) ◦ t = v ◦ ( s ◦ t ) . This observation will be frequently used in the sequel.2.3. The self-dual Ramsey theorem and other Ramsey-type theo-rems in terms of injections and surjections. We present here reformu-lations in the language of surjections and injections of the Ramsey theorems,including the self-dual Ramsey theorem, presented in Subsection 1.2. Weexplain how the translation works in Subsection 2.4. .Given connections ( s, i ) : [ L ] ↔ [ K ] and ( t, j ) : [ M ] ↔ [ L ], define( t, j ) · ( s, i ) : [ M ] ↔ [ K ]as ( s ◦ t, j ◦ i ) . Note that the orders of the compositions in the two coordinates are differentfrom each other. One sees easily that the composition of two connections isa connection.The following theorem is a reformulation of the self-dual Ramsey theoremfrom Subsection 1.2. We think of it as the official statement of the theorem.Its proof is given in Subsection 8.3. AMSEY THEORY 11 Theorem 2.1. Let d > be a natural number. Let K and L be naturalnumbers. There exists a natural number M such that for each d -coloring ofall connections between M and K there is ( t , j ) : [ M ] ↔ [ L ] such that { ( t , j ) · ( s, i ) | ( s, i ) : [ L ] ↔ [ K ] } is monochromatic. Below we enclose a list of Ramsey-type theorems stated in Subsection 1.2formulated here in the language of increasing injections and rigid surjections.These statements of the theorems will be proved and used in this paper. Ramsey’s Theorem. Given d > and natural numbers K and L , thereexists a natural number M such that for each d -coloring of all increasinginjections from [ K ] to [ M ] there exists an increasing injection j : [ L ] → [ M ] such that { j ◦ i | i : [ K ] → [ L ] an increasing injection } is monochromatic. Dual Ramsey Theorem. Given d > and natural numbers K and L ,there exists a natural number M such that for each d -coloring of all rigidsurjections from [ M ] to [ K ] there exists a rigid surjection t : [ M ] → [ L ] such that { s ◦ t | s : [ L ] → [ K ] a rigid surjection } is monochromatic. Hales–Jewett Theorem. Given d > and < A ≤ L , there exists M ≥ A with the following property. For each d -coloring of the set { v : [ M ] → [ A ] | v ↾ [ A ] = id [ A ] } there exists a rigid surjection s : [ M ] → [ L ] such that s ↾ [ A ] = id [ A ] and { v ◦ s | v : [ L ] → [ A ] , v ↾ [ A ] = id [ A ] } is monochromatic. Hales–Jewett Theorem, Voigt’s version. Given d > , < A ≤ L ,there exists M ≥ A with the following property. For each d -coloring of theset { v : [ M ′ ] → [ A ] | A ≤ M ′ ≤ M and v ↾ [ A ] = id [ A ] } there exists a rigid surjection s : [ M ′ ] → [ L ] for some M ′ ≤ M such that s ↾ [ A ] = id [ A ] and { v ◦ s | v : [ L ′ ] → [ A ] , A ≤ L ′ ≤ L, and v ↾ [ A ] = id [ A ] } is monochromatic. Graham–Rothschild Theorem. Given d > , A ≤ K ,and A ≤ L , thereexists M ≥ A with the following property. For each d -coloring of { s : [ M ] → [ K ] | s ∈ RS and s ↾ [ A ] = id [ A ] } there is a rigid surjection t : [ M ] → [ L ] , with t ↾ [ A ] = id [ A ] , such that { s ◦ t | s : [ L ] → [ K ] , s ∈ RS and s ↾ [ A ] = id [ A ] } is monochromatic. Graham–Rothschild Theorem, Voigt’s version. Given d > , A ≤ K ,and A ≤ L , there exists M ≥ A with the following property. For each d -coloring of { s : [ M ′ ] → [ K ] | A ≤ M ′ ≤ M, s ∈ RS , and s ↾ [ A ] = id [ A ] } there exist M ′ and a rigid surjection t : [ M ′ ] → [ L ] , with A ≤ M ′ ≤ M and t ↾ [ A ] = id [ A ] , such that { s ◦ t | s : [ L ′ ] → [ K ] , A ≤ L ′ ≤ L, s ∈ RS , and s ↾ [ A ] = id [ A ] } is monochromatic. Translation of rigid surjections into parameter sets. We showhere how to translate statements involving parameter sets (sometimes calledcombinatorial cubes) into statements about rigid surjections. This latterlanguage was proposed by Pr¨omel and Voigt [13]. It has been used in papers[15] and [16] in the context of structural Ramsey theory.With each A -parameter set of dimension l on n as in (1.1), we associatea rigid surjection s V : [ A + n ] → [ A + l ] as follows. We enumerate the setsin G as Y , . . . , Y l so that min Y i < min Y j if i < j . We let s V ↾ [ A ] = id [ A ] and s V ( A + x ) = ( A + i, if x ∈ Y i ; g ( x ) , if x ∈ [ n ] \ S li =1 Y i .This association is a bijection between all A -parameter sets of dimension l on n and all rigid surjections s : [ A + n ] → [ A + l ] with the property s ↾ [ A ] = id [ A ] . Moreover, it is not difficult to check, and we leave it tothe reader, that an A -parameter set U of dimension k on n is a subobjectof an A -parameter set V of dimension l on n if and only if there is a rigidsurjection r : [ A + l ] → [ A + k ] with r ↾ [ A ] = id [ A ] and such that(2.4) s U = r ◦ s V , AMSEY THEORY 13 and for each rigid surjection r : [ A + l ] → [ A + k ] with r ↾ [ A ] = id [ A ] thereis an A -parameter set U of dimension k on n that is a subobject of V suchthat (2.4) holds.These remarks give translations between the statements of the Hales–Jewett theorem, the Graham–Rothschild theorem, their Voigt’s versions,and the self-dual Ramsey theorem phrased in terms of parameter sets as inSubsection 1.2 and the statements of these results phrased in terms of rigidsurjections as in Subsection 2.3.3. Algebraic structures We introduce here the main algebraic notions needed in the abstract ap-proach to Ramsey theory. We illustrate the new notions with two series ofexamples, one related to the classical Ramsey theorem, the other one to theHales–Jewett theorem.3.1. Actoids. The notion of actoid defined below is the most rudimentaryversion of an action much like a semigroup action on a set. Definition 3.1. By an actoid we understand two sets A and Z , a partialbinary function from A × A to A : ( a, b ) → a · b, and a partial binary function from A × Z to Z : ( a, z ) → a . z such that for a, b ∈ A and z ∈ Z if a . ( b . z ) and ( a · b ) . z are both defined,then (3.1) a . ( b . z ) = ( a · b ) . z. The binary operation · on an actoid as above will be called multiplication and the binary operation . will be called action . Unless otherwise stated,the multiplication will be denoted by a · b and the action by a . z . Note thatin the case when A = Z and the multiplication coincides with the action, anactoid becomes what sometimes is called a partial semigroup. With someabuse of notation, we denote an actoid as in the definition above by ( A, Z ).To gain some intuitions about actoids, one may think of both A and Z assets of functions with multiplication a · b on A corresponding to composition a ◦ b that is defined only when the range of b is included in the domain of a .Similarly, the action of A on Z , a . z , corresponds to composition a ◦ z thatis defined when the range of z is included in the domain of a .We will have two sequences of examples illustrating the main notions thatare being introduced: sequence A leads to the classical Ramsey theorem,sequence B leads to the Hales–Jewett theorem. Example A1. Recall the definition of increasing injections and their classII from Subsection 2.1. We let B = Y = II and we make ( B, Y ) intoa composition space as follows. For i, j ∈ II, j · i and j . i are defined ifrange( i ) ⊆ domain( j ) and then j · i = j . i = j ◦ i. Example B1. Fix K ∈ N . Recall the set of increasing surjections IS fromSubsection 2.1. Let X K = { f | ∃ L ∈ N ( f : [ L ] → { } ∪ [ K ]) } For p : [ N ] → [ M ], p ∈ IS and f : [ L ] → { } ∪ [ K ], f ∈ X K , declare p . f tobe defined precisely when the canonical composition f ◦ p is defined, and let p . f = f ◦ p. It is easy to see that (IS , X K ) is an actoid.3.2. Set actoids. We will need to lift the operations on a given actoid tofamilies of sets. This is done by restricting the obvious pointwise liftings andis the content of the following definition of set actoid . There is no harm,as far as applications in this paper go, in thinking about F and S in thedefinition below as consisting of finite non-empty sets. Definition 3.2. Let ( A, Z ) be an actoid. Let F be a family of subsets of A and S a family of subsets of Z . Let ( F, G ) → F • G be a partial function from F × F to F and let ( F, S ) → F • S be a partial function from F × S to S . We say that ( F , S ) with these twooperations is an set actoid over ( A, Z ) provided that whenever F • G isdefined, then f · g is defined for all f ∈ F and g ∈ G and F • G = { f · g | f ∈ F, g ∈ G } , and whenever F • S is defined, then f . x is defined for all f ∈ F and x ∈ S and F • S = { f . x | f ∈ F, x ∈ S } . Because of (3.1), for F, G, S such that both F • ( G • S ) and ( F • G ) • S are defined, one has F • ( G • S ) = ( F • G ) • S .We adopt the following conventions for an actoid ( A, Z ). For sets F, G ⊆ A , we say that F · G is defined if f · g is defined for all f ∈ F and g ∈ G ,and we let(3.2) F · G = { f · g | f ∈ F, g ∈ G } . AMSEY THEORY 15 Similarly, for F ⊆ A and S ⊆ Z , we say that F . S is defined, if f . x isdefined for all f ∈ F and x ∈ S , and we let(3.3) F . S = { f . x | f ∈ F, x ∈ S } . If F = { f } for some f ∈ A , we write f . S for { f } . S , if it is defined. A set actoid as in Definition 3.2 is a restrictionof this natural pointwise operations on sets.We record the following easy lemma. Lemma 3.3. Let ( A, Z ) be an actoid. For F, G ⊆ A and S ⊆ Z if ( F · G ) . S and F . ( G . S ) are both defined, then they are equal and, moreover, for f ∈ F, g ∈ G, x ∈ S ( f · g ) . x = f . ( g . x ) . The lemma above says, in particular, that the pair consisting of the familyof all subsets of A and the family of all subsets of Z with the operationsdefined by (3.2) and (3.3) is an actoid in its own right. Example A2. We continue with Example A1. Recall the definition of (cid:0) LK (cid:1) from Subsection 2.1. We let F = S consist of all subsets of II of the form (cid:0) LK (cid:1) with 1 ≤ K ≤ L or K = L = 0. Note that (cid:0) (cid:1) contains only oneelement—the empty function. For (cid:0) LK (cid:1) , (cid:0) NM (cid:1) ∈ F = S , we let (cid:0) NM (cid:1) • (cid:0) LK (cid:1) and (cid:0) NM (cid:1) • (cid:0) LK (cid:1) be defined if and only if L = M , and then we let (cid:18) NL (cid:19) • (cid:18) LK (cid:19) = (cid:18) NL (cid:19) • (cid:18) LK (cid:19) = (cid:18) NK (cid:19) . One easily checks that ( F , S ) with • and • is a set actoid over ( B, Y ). Notethat (cid:18) NM (cid:19) · (cid:18) LK (cid:19) = (cid:18) NM (cid:19) . (cid:18) LK (cid:19) are defined under a weaker assumption that M ≥ L and in that case theyare both equal to (cid:0) N − ( M − L ) K (cid:1) . Example B2. We continue with Example B1; in particular, K ∈ N re-mains fixed. We assume K ≥ 1. For K ≥ 2, let S K ⊆ X K consist of all h : [ K ] → { } ∪ [ K ] such that for some 1 ≤ a ≤ b < K and 0 ≤ c ≤ K , wehave(3.4) h ( x ) = K , if x ≤ a ; c, if a + 1 ≤ x ≤ b ;max(1 , K − , if b + 1 ≤ x . Formula (3.4) always gives h (1) = K and h ( K ) = max(1 , K − S ′ K ⊆ X K consist of all h : [ K ] → { } ∪ [ K ] such that for some 1 ≤ a ≤ b < K and 0 ≤ c ≤ max(1 , K − h ( x ) = ( c, if a + 1 ≤ x ≤ b ;max(1 , K − , if x ≤ a or b + 1 ≤ x. Formula (3.5) implies that the image of h is included in { }∪ [max(1 , K − S K = { S K | K ≥ } ∪ { S ′ K | K ≥ } . For 0 < K ≤ L , let F L,K = { p ∈ IS | p : [ L ] → [ K ] } . Let F = { F L,K | < K ≤ L } . We let F N,M • F L,K be defined if L = M and then we let F N,L • F L,K = F N,K . We let F M,L • S K and F M,L • S ′ K be defined precisely when K = L and, inthis case, we let F M,K • S K = S M and F M,K • S ′ K = S ′ M . It is now easy to check that ( F , S K ) with the operations defined above isa set actoid over (IS , X K ).3.3. Composition spaces. To formulate the pigeonhole principle, we needadditional structure on actoids. Definition 3.4. A composition space is an actoid ( A, Z ) together with afunction ∂ : Z → Z such that for a ∈ A and z ∈ Z , if a . z is defined, then a . ∂z is defined and (3.6) a . ∂z = ∂ ( a . z ) . This additional function on Z will be called truncation and it will alwaysbe denoted by ∂ possibly with various subscripts and superscripts.Condition (3.6) in the above definition states that in a composition space( A, Z ) the action of A on Z is done by partial homomorphisms of the struc-ture ( Z, ∂ ). If we continue to think of an actoid ( A, Z ) as a family of func-tions A acting by composition on a family of functions Z , then we canview truncation as a “restriction operator” on functions from Z . So, con-dition (3.6) can be translated to say that if the composition of a and z isdefined, then so is the composition of a and the restriction ∂z of z and itsresult is a restriction of the composition of a and z , which we require to be AMSEY THEORY 17 given by the operator ∂ . Truncation can also be thought as producing outof an object z a simpler object ∂z of the same kind. In proofs, this point ofview leads to inductive arguments.We write ∂ t z for the element obtained from z after t ∈ N applications of ∂ . For a subsets S ⊆ Z , we write(3.7) ∂S = { ∂z | z ∈ S } . Again, for t ∈ N , we write ∂ t S for the result of applying the operation ∂ to S t times.We record the following obvious lemma. Lemma 3.5. Let ( A, Z ) be a composition space. Then for F ⊆ A and S ⊆ Z , if F . S is defined, then F . ∂S is defined and ∂ ( F . S ) = F . ( ∂S ) . It follows from the above lemma that if ( A, Z ) is a composition space,then the pair consisting of the family of all subsets of A and the family ofall subsets of Z becomes a composition space with the operations definedby (3.2), (3.3), and (3.7). Example A3. We continue with Examples A1 and A2. For i : [ K ] → N in Y = II, let ∂i = i ↾ [max(0 , K − . It is easy to check that ( B, Y ) with ∂ is a composition space. Example B3. We continue with Examples B1 and B2. Fix K ≥ 1. On X K , we define the following truncation. For f ∈ X K , f : [ L ] → { } ∪ [ K ],let ( ∂f )( x ) = ( max(1 , K − , if f ( x ) = K ; f ( x ) , if f ( x ) ≤ K − p ∈ IS, if p . f is defined, then ∂ ( p . f ) = ∂ ( f ◦ p ) = ( ∂f ) ◦ p = p . ( ∂f ) , and, therefore, (IS , X K ) with ∂ defined above is a composition space. Ramsey domains. The additional structure on composition spaceswill allow us to formulate conditions on set actoids that are needed forour Ramsey result. A set actoid fulfilling these conditions will be called aRamsey domain. But, first we need to introduce the following notion. Definition 3.6. Let ( A, Z ) be an actoid. For a, b ∈ A , we say that b extends a if for each x ∈ Z for which a . x is defined, we have that b . x is definedand a . x = b . x . Now we introduce the main notion of this subsection. Definition 3.7. A set actoid ( F , S ) over a composition space is called a Ramsey domain if each set in S is non-empty and the following conditionshold for all F, G ∈ F and S ∈ S , (i) if F • ( G • S ) is defined, then so is ( F • G ) • S ; (ii) ∂S ∈ S ; (iii) if F • ∂S is defined, then there is H ∈ F such that H • S is definedand for each f ∈ F there is h ∈ H extending f . Condition (i) is crucial in the proof of the abstract Ramsey theorem. Weintroduced the restrictions • and • of pointwise multiplication and pointwiseaction as in (3.2) and (3.3) to make sure that condition (i) is fulfilled inconcrete examples. It says that F • ( G • S ) is not defined by chance; if itis defined, then the product F • G must be defined and it acts on S . Notethat, by Lemma 3.5, this condition implies that F • ( G • S ) = ( F • G ) • S. Condition (ii) is just a closure property. As for condition (iii), in manysituations, it would be useful to be able to deduce from f . ∂x being definedthat f . ∂x = ∂ ( f . x ). Such an implication fails as f . x may be undefined.To counter this failure, condition (iii) ensures that if f . x is defined, for f ∈ F and x ∈ S , then there is h ∈ H such that h . x is defined and f . ∂x = ∂ ( h . x ) , and that this happens for all x ∈ S with the same h . We also point outthat a careful reading of the proofs below shows that one can weaken theconclusion of condition (iii) to “ H • S is defined and for each f ∈ F there is h ∈ H such that h extends f on ∂S , that is, for each x ∈ ∂S , h . x is definedand h . x = f . x .” However, this refinement has not turned out to be usefulso far. Example A4. We continue with Examples A1–A3. We check that the setactoid defined in Example A2 is a Ramsey domain. Point (i) of the definition AMSEY THEORY 19 is clear since (cid:18) QP (cid:19) • ( (cid:18) NM (cid:19) • (cid:18) LK (cid:19) )is defined precisely when M = L and P = N , in which case( (cid:18) QP (cid:19) • (cid:18) NM (cid:19) ) • (cid:18) LK (cid:19) is defined as well. Let (cid:0) LK (cid:1) from the set actoid be given. Then L ≥ K ≥ L = K = 0. If K ≥ 2, then ∂ (cid:0) LK (cid:1) = (cid:0) L − K − (cid:1) ; if K = 1 or K = 0, then ∂ (cid:0) LK (cid:1) = (cid:0) (cid:1) . Point (ii) of the definition follows immediately. To see point(iii), assume that (cid:0) NM (cid:1) • ∂ (cid:0) LK (cid:1) is defined. If K ≥ 2, then M = L − (cid:0) N +1 L (cid:1) witnesses that point (iii) holds. If K = 1, then N = M = 0 and (cid:0) LL (cid:1) witnesses that (iii) holds. If L = K = 0, then again N = M = 0 and (cid:0) (cid:1) witnesses (iii). Example B4. We continue with Examples B1–B3. We check that the setactoid defined in Example B2 is a Ramsey domain. Point (i) of the definitionof Ramsey domain is immediate from the definition of • and • . Point (ii)is clear since ∂S K = S ′ K and ∂S ′ K = S ′ K . To see point (iii), note that if F M,L • ∂S K or F M,L • ∂S ′ K is defined, then L = K and F M,L itself witnessesthat point (iii) holds. Lemma 3.8. Let ( F , S ) be a Ramsey domain. Let S ∈ S and F , . . . , F n ∈F . Assume that z = F n • ( F n − • · · · ( F • ( F • S ))) is defined. Then z = ( F n • ( F n − · · · ( F • F ))) • S and z = ((( F n • F n − ) · · · F ) • F ) • S are defined and z = z = z .Proof. One proves the existence of z and z = z and the existence of z and z = z by separate inductions. To run the inductive argument for z = z , note that by (3.1) and point (i) of Definition 3.7 F n • ( F n − • · · · ( F • ( F • S ))) = F n • ( F n − • · · · ( F • (( F • F ) • S )))and apply the inductive assumption. Similarly, to run the induction for z = z , note that by (3.1) and point (i) of Definition 3.7 F n • ( F n − • · · · ( F • ( F • S ))) = ( F n • F n − ) • ( F n − • · · · ( F • ( F • S ))) , and apply the inductive assumption. (cid:3) Ramsey and pigeonhole conditions and the first abstractRamsey theorem Ramsey condition (R) for set actoids. At this point, we can statethe abstract Ramsey property alluded to in the introduction. Note that itcan be stated for set actoids (truncation is not needed). So let ( F , S ) be aset actoid. Condition (R). For each d > and each S ∈ S there exists F ∈ F suchthat F • S is defined, and for each d -coloring of F • S there exists f ∈ F with f . S monochromatic. Condition (R) from the definition above, when interpreted for the setactoid from Example A2 becomes just the classical Ramsey theorem.Condition (R) is a very abstract property. Essentially all finite Ramseytheorems, even those which we cannot prove using the methods of this pa-per, like for example structural Ramsey theorems, can be seen as particularinstances of (R). (One should point out that possibilities of phrasing certainRamsey statements using actions of groups have been explored in [6, Sec-tion 4], [11, Section 1.5], and, most recently, in [1, Section 4].) Of course,at this point, the main problem is: are there general settings in which (R)can actually be proved? The structure of set actoids over actoids is notrich enough to support such proofs. For this we will need the structure ofRamsey domains over composition spaces and normed composition spaces.We will also need to formulate pigeonhole principles from which condition(R) can be deduced and which can be formulated only using these richerstructures. We proceed to the formulation of such a pigeonhole principleright now.4.2. Pigeonhole principle for Ramsey domains. We formulate two pi-geonhole principles: one here called (P) and a localized version of it called(LP) in Section 5. They are not straightforward abstractions of the classi-cal Dirichlet’s pigeonhole principle. Rather they are conditions that makeit possible to carry out inductive arguments proving the Ramsey property,they are easy to verify in concrete situations and are flexible enough to ac-commodate in applications many concrete statements as special cases. Forexample, the abstract pigeonhole principle (LP) reduces to Dirichlet’s pi-geonhole principle in the case of the classical Ramsey theorem; however,in different situations a variety of other statements, like the Hales–Jewetttheorem or the Graham–Rothschild theorem serve as pigeonhole principles.The pigeonhole principle (P) below can be thought of in the followingway. The Ramsey condition requires, upon coloring of F . S , fixing of acolor on f . S for some f ∈ F . In condition (P), we consider the equivalence AMSEY THEORY 21 relation on S that identifies x and x from S if ∂x = ∂x . The pigeonholeprinciple (P) requires fixing of a color on each equivalence class separately,rather than on the whole S , after acting by an element of F .Let ( F , S ) be a Ramsey domain. Condition (P). For every d > and S ∈ S there exists F ∈ F such that F • S is defined and for each d -coloring c of F • S there exists f ∈ F suchthat for all x , x ∈ S we have ∂x = ∂x = ⇒ c ( f . x ) = c ( f . x ) . It is convenient to illustrate the above definition by sequence B of exam-ples. The localized pigeonhole principle from Section 5 will be illustrated bysequence A. Example B5. Before we continue with Examples B1–B4, we state Dirich-let’s pigeonhole principle phrased here in a surjective form.( ∗ ) For every d > K ≥ L ≥ d -coloring of all q : [ L ] → [2], q ∈ IS, there exists q : [ L ] → [ K ], q ∈ IS, such that { p ◦ q | p : [ K ] → [2] , p ∈ IS } is monochromatic.One can take L = d ( K − 2) + 2 . We claim that the Ramsey domain ( F , S K ) defined in Example B2 andchecked to be a Ramsey domain in Example B4 fulfills (P). Let S K , S ′ K ∈S K . Note that for h , h ∈ S ′ K , ∂h = ∂h implies h = h . Therefore,we only need to check condition (P) for S K . Note that if h ∈ S K , then ∂h uniquely determines h among functions in S K unless h is of the followingform: for some 0 < K < K ,(4.1) h ↾ [ K ] ≡ K and h ↾ ([ K ] \ [ K ]) ≡ max(1 , K − . It follows that given d > 0, we need to find L ≥ K so that for each d -coloring c of F L,K • S K there is p ∈ F L,K such that the color c ( h ◦ p ) is constant for h ∈ S K of the form (4.1) as K runs over [ K − L exists by thevirtue of the basic pigeonhole principle ( ∗ ) stated above. Pigeonhole implies Ramsey. We continue to adhere to the followingconvention: the three operations on a composition space ( A, Z ) are denotedby · , . , and ∂ , respectively, while the operations on a Ramsey domain over( A, Z ) are denoted by • and • . We also use the notation set up in (3.2),(3.3), and (3.7).Theorem 4.1 and Corollary 4.3 give general Ramsey statements derivedfrom the pigeonhole principle (P). Corollary 4.3 is simpler to state thanTheorem 4.1 and is all that is needed from this theorem in most, but notall, situations. Theorem 4.1. Let ( F , S ) be a Ramsey domain fulfilling condition (P). For d > , t ≥ , and S ∈ S , there exists F ∈ F such that F • S is defined andfor each d -coloring c of F • S there exists f ∈ F such that for x , x ∈ S (4.2) ∂ t x = ∂ t x = ⇒ c ( f . x ) = c ( f . x ) . Proof. We will prove the conclusion of the theorem assuming that for every d > , t ≥ , and S ∈ S there is F ∈ F such that F • S is defined and forevery d -coloring c of ∂ t ( F • S ) there is f ∈ F such that for x , x ∈ S (4.3) ∂ t +1 x = ∂ t +1 x = ⇒ c ( ∂ t ( f . x )) = c ( ∂ t ( f . x )) . Making this assumption is justified since it follows from condition (P) asshown by the following argument carried out by induction on t . For t = 0 theassumption above is simply (P). Now we go from t to t + 1. Let S ∈ S . Since( F , S ) is a Ramsey domain, we have ∂S ∈ S . So by the above assumptionfor t there is F ∈ F such that F • ∂S is defined and for every d -coloring c of ∂ t ( F • ∂S ) there is f ∈ F such that for x , x ∈ ∂S (4.3) holds. Now, againsince ( F , S ) is a Ramsey domain, there is G ∈ F such that G • S is definedand every element of F is extended by an element of G . We claim that G makes the assumption true for t + 1. Let c be a d -coloring of ∂ t +1 ( G • S ) = ∂ t ( G . ∂S ) . Since F • ∂S ⊆ G . ∂S , c gives a d -coloring of ∂ t ( F • ∂S ). Thus, by ourchoice of F , there is f ∈ F such that for x , x ∈ ∂S (4.3) holds. Let g ∈ G extend this f . Then for y , y ∈ S with ∂ t +2 y = ∂ t +2 y , we have ∂ t +1 ( ∂y ) = ∂ t +1 ( ∂y ), so c ( ∂ t ( f . ∂y )) = c ( ∂ t ( f . ∂y )) . Since f . ∂y = g . ∂y and f . ∂y = g . ∂y , it follows that c ( ∂ t +1 ( g . y )) = c ( ∂ t ( g . ∂y )) = c ( ∂ t ( g . ∂y )) = c ( ∂ t +1 ( g . y )) , as required.Now, we prove the theorem making the assumption from the beginning ofthe proof. Fix d > 0. The argument is by induction on t ≥ S ∈ S . AMSEY THEORY 23 For t = 0, the conclusion is clear since it requires only that there be a non-empty F ∈ F with F • S defined, which is guaranteed by our assumption.Now we suppose that the conclusion of the theorem holds for t and we showit for t + 1. Apply our assumption stated at the beginning of the proof to d , t , and S obtaining F ∈ F . Note that F • S ∈ S . Apply the inductiveassumption for t to F • S obtaining F ∈ F . Note that F • ( F • S ) isdefined, hence, since ( F , S ) is a Ramsey domain, ( F • F ) • S is definedand, by Lemma 3.3,(4.4) ( F • F ) • S = F • ( F • S )Note that F • F ∈ F , and we claim that it works for t + 1.Let c be a d -coloring of ( F • F ) • S . By (4.4), we can consider it to be acoloring of F • ( F • S ). By the choice of F there exists f ∈ F such thatfor x, y ∈ S and f, g ∈ F ,(4.5) ∂ t ( f . x ) = ∂ t ( g . y ) = ⇒ c ( f . ( f . x )) = c ( f . ( g . y )) . Define a d -coloring ¯ c of ∂ t ( F • S ) by letting for f ∈ F and x ∈ S (4.6) ¯ c ( ∂ t ( f . x )) = c ( f . ( f . x )) . The coloring ¯ c is well-defined by (4.5). By our choice of F , there exists f ∈ F such that for x, y ∈ S (4.7) ∂ t +1 x = ∂ t +1 y = ⇒ ¯ c ( ∂ t ( f . x )) = ¯ c ( ∂ t ( f . y )) . Combining (4.7) with (4.6), we see that for x, y ∈ S∂ t +1 x = ∂ t +1 y = ⇒ c ( f . ( f . x )) = c ( f . ( f . y )) . Now f = f · f is as required since by (4.4) and Lemma 3.3, we have f . ( f . x ) = ( f · f ) . x and f . ( f . y ) = ( f · f ) . y, and the proof is completed. (cid:3) Definition 4.2. A Ramsey domain ( F , S ) is called vanishing if for every S ∈ S there is t ∈ N such that ∂ t S consists of one element. Corollary 4.3. Let ( F , S ) be a vanishing Ramsey domain. If ( F , S ) fulfillscondition (P), then it fulfills condition (R).Proof. The conclusion follows from Theorem 4.1 since for each S ∈ S thereis t ∈ N with ∂ t S having at most one element. For this t , the left hand sidein (4.2) holds for all x , x ∈ S . (cid:3) Localizing the pigeonhole condition and the secondabstract Ramsey theorem We formulate here a localized version (LP) of condition (P) and prove inTheorem 5.3 that, under mild assumptions, it implies (P), making checking(P) much easier. Even though condition (LP) can be stated for Ramseydomains over composition spaces, the proof of Theorem 5.3 requires intro-duction in Subsection 5.2 of a new piece of structure on a composition space,which is nevertheless found in almost all concrete situations.5.1. Localized version (LP) of (P). One can think of condition (LP) inthe following way. In condition (P), we are given a coloring of F • S andare asked to find f ∈ F making the coloring constant on each equivalenceclass of the equivalence relation on S that identifies y , y ∈ S if ∂y = ∂y .Obviously, it is easier to fulfill the requirement of making the coloring con-stant, by multiplying by some f ∈ F , on a single, fixed equivalence classof this equivalence relation. Condition (LP) makes just such a requirement.The price for this weakening of the pigeonhole principle is paid by puttingan additional restriction on the element f ∈ F fixing the color. We willcomment on this restriction after the condition is stated. First, we intro-duce a piece of notation for equivalence classes of the equivalence relationmentioned above. For S ⊆ Z and x ∈ Z , put S x = { y ∈ S | ∂y = x } . Also, for F ⊆ A , let F a = { b ∈ F | b extends a } . Let ( F , S ) be a Ramsey domain over a composition space ( A, Z ). Thefollowing criterion on ( F , S ) turns out to be the right formalization of thelocal version of (P). Condition (LP). For d > , S ∈ S , and x ∈ ∂S , there is F ∈ F and a ∈ A such that F • S is defined, a . x is defined, and for every d -coloring of F a . S x there is f ∈ F a such that f . S x is monochromatic. The equivalence relation on S given by ∂y = ∂y obviously has ∂S as itsset of invariants, that is, two elements of S are equivalent if and only if theirimages in ∂S under the function y → ∂y are the same. In condition (LP),we consider the equivalence class given by x ∈ ∂S and we ask for a ∈ A thatacts on a part of the set of invariants ∂S including x and is such that each d -coloring can be stabilized on S x by multiplication by some f ∈ F that actsin a manner compatible with a . AMSEY THEORY 25 Normed composition spaces. We introduce here a new piece ofstructure on composition spaces. Definition 5.1. Let ( A, Z ) be a composition space. We say that ( A, Z ) is normed if there is a function | · | : Z → D , where ( D, ≤ ) is a partial order,such that for x, y ∈ Z , | x | ≤ | y | implies that for all a ∈ Aa . y defined ⇒ ( a . x defined and | a . x | ≤ | a . y | ) . A function | · | as in the above definition will be called a norm . In mostcases, for example in all cases considered in this paper, ( D, ≤ ) will be a linearorder. However, there are natural situations, occurring in a forthcomingwork, in which ( D, ≤ ) is not linear. A remnant of linearity will always beretained as explained in Definition 5.2 below. Example A5. We continue with Examples A1–A4. Define | · | : II → N by | i | = max range( i ) , for i ∈ II. It is easy to see that the function defined above is a norm on( B, Y ); thus, ( B, Y ) becomes a normed composition space.Checking that ( F , S ) defined in Example A2 fulfills (LP) amounts to anapplication of the standard pigeonhole principle. Clearly S is vanishing. Itfollows from Corollary 5.4 below that ( F , S ) is a Ramsey domain fulfilling(R), which gives the classical Ramsey theorem.5.3. Localized pigeonhole implies Ramsey. We need one more defini-tion. Definition 5.2. A Ramsey domain ( F , S ) over a normed composition spaceis called linear if the image of S under the norm is linear for each S ∈ S . Here is the main theorem of this section. Theorem 5.3. Let ( F , S ) be a linear Ramsey domain over a normed com-position space. Assume that S consists of finite sets. If ( F , S ) fulfills (LP),then ( F , S ) fulfills (P).Proof. Let ( A, Z ) with · , . , ∂, | · | be the normed composition space overwhich ( F , S ) is defined. For the sake of clarity, in this proof, expressions ofthe form F k F k − · · · F S and f k f k − · · · f x stand for F k • ( F k − • · · · ( F • S )) and f k . ( f k − . · · · ( f . x )) , respectively. In particular, f x stands for f . x . Fix d > 0. Let S ∈ S . Since ∂S is finite, ∂S ∈ S and ( F , S ) is linear, wecan list ∂S as x , x , . . . , x n with | x n | ≤ | x n − | ≤ · · · ≤ | x | . We produce F , . . . , F n ∈ F and b , . . . , b n ∈ A as follows. For 1 ≤ k ≤ n + 1, after k − F , . . . , F k − , b , . . . , b k − . They have the following properties:(a) F k − F k − · · · F S is defined;(b) b k − b k − · · · b x l is defined for k − ≤ l ≤ n ;(c) for 1 ≤ j ≤ k − 1, for every d -coloring of F j . ( F j − · · · F S ) b j − ··· b x j there is f j ∈ F j extending b j such that f j . ( F j − · · · F S ) b j − ··· b x j is monochromatic;(d) for 1 ≤ j ≤ k − ≤ l ≤ n , if f j ∈ F j extends b j and ˜ x ∈ S is suchthat ∂ ˜ x = x l , then ∂ ( f k − f k − · · · f ˜ x ) = b k − b k − · · · b x l . We make step k ≤ n of the recursion. With the fixed d , we apply (LP)to F k − F k − · · · F S , which exists by (a) and obviously is in S , and to b k − b k − · · · b x k ∈ Z , which exists by (b). This is permissible. Indeed, x k ∈ ∂S and, by (c), there are f j ∈ F j extending b j for 1 ≤ j ≤ k − 1, andso, by (d) taken with l = k , we get b k − · · · b x k ∈ ∂ ( F k − F k − · · · F S ) . This application of (LP) gives F k ∈ F and b k ∈ A . Now (a), (b), and(c) follow immediately from our choice of F k and b k and the assumption | x l | ≤ | x k | for l ≥ k . Point (d) is a consequence of (a) and (b) for k and (d)for k − k ≤ l ≤ n . Let f j ∈ F j extend b j ,for each 1 ≤ j ≤ k , and let ˜ x ∈ S be such that ∂ ˜ x = x l . Note that using (d)for k − l , we get(5.1) ∂ ( f k − · · · f ˜ x ) = b k − · · · b x l . Thus, since, by (b) for k , b k b k − · · · b x l is defined, so is b k ∂ ( f k − · · · f ˜ x ).Now, since f k extends b k , we see that f k ∂ ( f k − · · · f ˜ x ) exists and(5.2) f k ∂ ( f k − · · · f ˜ x ) = b k ∂ ( f k − · · · f ˜ x ) . Putting (5.1) and (5.2) together, we get (d) for k since ∂ ( f k f k − · · · f ˜ x ) = f k ∂ ( f k − · · · f ˜ x )= b k ∂ ( f k − · · · f ˜ x ) = b k b k − · · · b x l . Note that above f k f k − · · · f ˜ x is defined by (a) for k . AMSEY THEORY 27 So the recursive construction has been carried out. Note that by (a)(5.3) F n F n − · · · F S is defined. We can apply Lemma 3.8 to the Ramsey domain ( F , S ) to seethat ( F n • ( F n − • · · · • F )) • S is defined as well. Now, F n • ( F n − • · · · • F )is an element of F , and we claim that for each d -coloring c of( F n • ( F n − • · · · • F )) • S there are f ∈ F , . . . , f n ∈ F n such that for x , x ∈ S we have(5.4) ∂x = ∂x = ⇒ c (( f n · ( f n − · · · f )) . x ) = c (( f n · ( f n − · · · f )) . x ) . This will verify that ( F , S ) fulfills (P).Fix, therefore, a d -coloring c of ( F n • ( F n − • · · · • F )) • S . We recursivelyproduce f n ∈ F n , . . . , f ∈ F . Note first that since (5.3) is defined, byLemmas 3.8, we have that for each 1 ≤ k ≤ n ( F n • ( F n − • · · · • F )) • S = F n • ( F n − • · · · ( F k • ( F k − F k − · · · F S ))) . Therefore, having produced f n , . . . , f k +1 , we can consider the d -coloring of F k • ( F k − F k − · · · F S ) given by f y → c ( f n · · · f k +1 f y ) , for f ∈ F k and y ∈ F k − F k − · · · F S . By (c), we get f k ∈ F k such that c ( f n · · · f k +1 f k y ) is constant for y ∈ ( F k − F k − · · · F S ) b k − b k − ··· b x k (5.5)and f k extends b k .We claim that f n , . . . , f produced this way witness that (5.4) holds. Let y , y ∈ S be such that ∂y = ∂y , and let this common value be x k for some1 ≤ k ≤ n . For i = 1 , 2, condition (d) gives b k − b k − · · · b x k = ∂ ( f k − f k − · · · f y i ) . and so f k − f k − · · · f y i ∈ ( F k − F k − · · · F S ) b k − b k − ··· b x k , which in light of (5.5) implies that(5.6) c ( f n · · · f k +1 f k ( f k − · · · f y )) = c ( f n · · · f k +1 f k ( f k − · · · f y )) . Since (5.3) is defined, by Lemma 3.8, applied to the Ramsey domain ( F , S ),and by inductively applying Lemma 3.3, we get that for i = 1 , f n f n − · · · f y i = ( f n · ( f n − · · · f )) y i . From this equality and from (5.6) the conclusion follows. (cid:3) The following corollary is an immediate consequence of Theorem 5.3 andCorollary 4.3. Corollary 5.4. Let ( F , S ) be a linear vanishing Ramsey domain over anormed composition space. Assume that S consists of finite sets. If ( F , S ) fulfills (LP), then it fulfills (R). Remarks on normed composition spaces. The most involved al-gebraic notion introduced in the paper is the notion of normed compositionspace. Below, we give a list of conditions that are more symmetric thanthose defining normed composition spaces. We then prove in Lemma 5.5that the new conditions define a structure that is essentially equivalent to anormed composition space. It is worth remarking that all the normed com-position spaces in the present paper, in [17], and in [18] fulfill the conditionsbelow.Let ( A, Z, · , . , ∂, | · | ) be such that · is a partial function from A × A to A , . is a partial function from A × Z to Z , ∂ is a function from Z to Z and | · | is a function from Z to a set with a partial order ≤ . We consider thefollowing set of conditions:(a) if a . ( b . z ) and ( a · b ) . z are defined for a, b ∈ A and z ∈ Z , then a . ( b . z ) = ( a · b ) . z ;(b) if a . z and a . ∂z are defined for a ∈ A and z ∈ Z , then ∂ ( a . z ) = a . ∂z ;(c) | ∂z | ≤ | z | for each z ∈ Z ;(d) if | y | ≤ | z | and a . z is defined for a ∈ A and y, z ∈ Z , then a . y isdefined and | a . y | ≤ | a . z | .Condition (a) is just the action property. Conditions (b), (c) and (d) saythat each pair of functions constituting the structure interact in a naturalway: action with truncation in (b), truncation with norm in (c), and normwith action in (d). Lemma 5.5. If ( A, Z, · , . , ∂, | · | ) fulfills conditions (a)–(d), then ( A, Z ) with · , . , ∂ and | · | is a normed composition spaceProof. Almost all the properties defining a normed composition space arealready explicit among (a)–(d). One only needs to check that for a ∈ A and z ∈ Z if a . z is defined, then so is a . ∂z , and this property follows from (c)and (d). (cid:3) AMSEY THEORY 29 Propagating the pigeonhole principle In this section, we prove two results that make it possible to propagatecondition (P) to new examples. In the first result, we show how to obtaincondition (P) on appropriately defined finite products assuming it holds onthe factors. The second result involves the notion of interpretation of setsfrom a Ramsey domain in another Ramsey domain. This result shows thatif each set from a Ramsey domain is interpretable in some Ramsey domainfulfilling (P) then that Ramsey domain fulfills (P) as well.6.1. Products. We prove here a consequence of Theorem 4.1 that extendsthis theorem to products. First, we set up a general piece of notation. Let X i , 1 ≤ i ≤ l , be sets, and let U i be a family of subsets of X i . Let O i ≤ l U i = { l Y i =1 U i | U i ∈ U i for i = 1 , . . . , l } . When U i = U for all 1 ≤ i ≤ l , we write U ⊗ l for N i ≤ l U i . Note that N i ≤ l U i consists of subsets of Q i ≤ l X i .Let ( A i , Z i ), 1 ≤ i ≤ l , be composition spaces. The multiplication andaction on each of them is denoted by the same symbols · and . ; the trunca-tion on Z i is denoted by ∂ i . The product of ( A i , Z i ), 1 ≤ i ≤ l , is defined inthe natural coordinatewise way. Its underlying sets are l Y i =1 A i and l Y i =1 Z i ;the multiplication ( a i ) · ( b i ), for ( a i ) , ( b i ) ∈ Q i ≤ l A i , is declared to be definedprecisely when a i · b i is defined for each i ≤ l and then( a i ) · ( b i ) = ( a i · b i );the action ( a i ) . ( z i ), for ( a i ) ∈ Q i ≤ l A i and ( z i ) ∈ Q i ≤ l Z i , is defined pre-cisely when a i . z i is defined for each i ≤ l and then( a i ) . ( z i ) = ( a i . z i );the truncation ∂ π of ( z i ) is given by ∂ π ( z i ) = ( ∂ i z i ) . It is easy to check, and we leave it to the reader, that the definitions abovedescribe a composition space. We call it the product composition space( Q i ≤ l A i , Q i ≤ l Z i ). If ( A i , Z i ) = ( A, Z ) for each i ≤ l , we write ( A l , Z l ) for( Q i ≤ l A i , Q i ≤ l Z i ).Let now ( F i , S i ) be Ramsey domains over ( A i , Z i ), 1 ≤ i ≤ l . We de-fine the operations on N i ≤ l F i , N i ≤ l S i as follows. We declare ( Q i ≤ l F i ) • ( Q i ≤ l G i ) to be defined precisely when F i • G i is defined for each 1 ≤ i ≤ l and then we let ( l Y i =1 F i ) • ( l Y i =1 G i ) = l Y i =1 ( F i • G i ) , and ( Q i ≤ l F i ) • ( Q i ≤ l S i ) is defined if F i • S i is defined for each i and then( l Y i =1 F i ) • ( l Y i =1 S i ) = l Y i =1 ( F i • S i ) . Lemma 6.1. For ≤ i ≤ l , let ( F i , S i ) be Ramsey domains over composi-tion spaces ( A i , Z i ) . Then ( N i ≤ l F i , N i ≤ l S i ) is a Ramsey domain over thecomposition space ( Q i ≤ l A i , Q i ≤ l Z i ) .Proof. All the points from the definition of Ramsey domain are clear. Forexample, to check point (iii), assume that ( Q i ≤ l F i ) • ∂ π ( Q i ≤ l S i ) is definedfor some Q i ≤ l F i ∈ N i ≤ l F i and Q i ≤ l S i ∈ N i ≤ l S i . This means that foreach 1 ≤ i ≤ l , F i • ∂ i S i is defined, so we can find H i ∈ F i as requiredby point (iii) of the definition of Ramsey domain for ( F i , S i ). Then clearlypoint (iii) holds for Q i ≤ l H i ∈ N i ≤ l F i . (cid:3) The following proposition propagates the pigeonhole principle from factorsto products. In the proof of this proposition Theorem 4.1 is used even thoughwe are checking only condition (P). Proposition 6.2. Let ( F i , S i ) , ≤ i ≤ l , be Ramsey domains fulfilling (P).Assume that each S i consists of finite sets. Then ( N i ≤ l F i , N i ≤ l S i ) is aRamsey domain fulfilling (P).Proof. By Lemma 6.1, it suffices to check (P).We define a composition space structure on A ∗ = l Y i =1 A i , Z ∗ = { , . . . , l − } × l Y i =1 Z i as follows. The multiplication on A ∗ is the same as in the product compo-sition space ( Q i ≤ l A i , Q i ≤ l Z i ). For ( a i ) ∈ A ∗ and ( p, ( z i )) ∈ Z ∗ , we make( a i ) . ( p, ( z i )) be defined if a i . z i is defined for all i and( a i ) . ( p, ( z i )) = ( p, ( a i . z i )) . For ( p, ( z i )) ∈ Z ∗ , let ∂ ∗ ( p, ( z i )) = ( p + 1 (mod l ) , ( y i )) , where y i = z i if i = p + 1 and y p +1 = ∂ p +1 z p +1 . It is easy to see that ( A ∗ , Z ∗ )is a composition space. AMSEY THEORY 31 Define F ∗ = O i ≤ l F i and let S ∗ consist of all sets of the form { p } × S, where p ∈ { , . . . , l − } , and S ∈ N i ≤ l S i . Define • on F ∗ to coincide with • on N i ≤ l F i . Declare F • ( { p } × S ) to be defined if and only if F • S isdefined in ( N i ≤ l F i , N i ≤ l S i ), and let F • ( { p } × S ) = { p } × ( F • S ) . It is easy to check that ( F ∗ , S ∗ ) is a Ramsey domain over the compositionspace ( A ∗ , Z ∗ ).We claim that ( F ∗ , S ∗ ) fulfills (P). To prove it, fix d > { p } × l Y i =1 S i ∈ S ∗ , for some S i ∈ S i . For i = p + 1, pick F i ∈ F i such that F i • S i is defined.Such F i exists by condition (P) with the number of colors equal to 1 forthe Ramsey domains ( F i , S i ). Now, we apply condition (P) to ( F p +1 , S p +1 )with the following number of colors:(6.1) d Q i = p +1 | F i • S i | . (Note that the number defined above is finite since F i • S i is finite, as itbelongs to S i .) This application gives us F p +1 ∈ F p +1 such that F p +1 • S p +1 is defined and for each coloring of F p +1 • S p +1 with the number of colorsgiven by (6.1) there is f ∈ F p +1 such that for any two x, y ∈ S p +1 fulfilling(6.2) ∂ p +1 x = ∂ p +1 y,f . x and f . y get the same color. Having defined F i , 1 ≤ i ≤ l , note that l Y i =1 F i ∈ F ∗ , and that ( l Y i =1 F i ) • ( { p } × l Y i =1 S i )is defined. Given a d -coloring c of( l Y i =1 F i ) • ( { p } × l Y i =1 S i ) , which set is equal to { p } × F • S × · · · × F p +1 • S p +1 × · · · × F l • S l , consider the coloring of F p +1 • S p +1 defined by(6.3) h → ( c ( p, h , . . . , h p , h, h p +2 , . . . , h l ) | ( h i ) i = p +1 ∈ Y i = p +1 F i • S i ) . This is a coloring with the number of colors equal to (6.1). Therefore, thereexists f p +1 ∈ F p +1 such that for any two x, y ∈ S p +1 fulfilling (6.2), f p +1 . x and f p +1 . y get the same color. Pick f i ∈ F i for i = p + 1 arbitrarily. Withthese choices ( f i ) is an element of Q i ≤ l F i . Note now that for( p, ( x i )) , ( p, ( y i )) ∈ { p } × l Y i =1 S i we have(6.4) ∂ ∗ ( p, ( x i )) = ∂ ∗ ( p, ( y i ))precisely when x i = y i for i = p + 1 and (6.2) holds for x p +1 and y p +1 . Thisobservation allows us to say that the definition of the coloring in (6.3) andour choice of f p +1 imply that if (6.4) holds, then c (( f i ) . ( p, ( x i ))) = c (( f i ) . ( p, ( y i ))) . Thus, indeed, ( F ∗ , S ∗ ) is a Ramsey domain with (P).Now apply Theorem 4.1 (with t = l ) to the Ramsey domain ( F ∗ , S ∗ ),which has (P), while keeping in mind that N i ≤ l F i = F ∗ and that for x ∈ Q i ≤ l Z i , we have (0 , ∂ π x ) = ∂ l ∗ (0 , x ) . The proposition follows. (cid:3) Example B6. We continue with Examples B1–B5. Let l ∈ N . By Proposi-tion 6.2 and Example B5, the Ramsey domain ( F ⊗ l , S ⊗ lK ) over the productcomposition space (IS l , X lK ) fulfills (P). This fact will be used in Subsec-tion 8.1.6.2. Interpretations. We introduce here a notion of interpretability. Itsimportance is contained in Proposition 6.4. It is a general notion and wewill need its full generality when proving the Hales–Jewett theorem. Definition 6.3. Let ( F , R ) and ( G , S ) be Ramsey domains over composi-tion spaces ( A, X ) and ( B, Y ) , respectively. Let S ∈ S . We say that S is interpretable in ( F , R ) if there exists R ∈ R and a function α : S → R suchthat AMSEY THEORY 33 (i) for y , y ∈ S , ∂y = ∂y = ⇒ ∂α ( y ) = ∂α ( y );(ii) if F • R is defined for some F ∈ F , then there exist G ∈ G , with G • S defined, and a function φ : F → G such that for f , f ∈ F and y , y ∈ S (6.5) f . α ( y ) = f . α ( y ) = ⇒ φ ( f ) . y = φ ( f ) . y . Proposition 6.4. Let ( G , S ) be a Ramsey domain. If each S ∈ S is in-terpretable in a Ramsey domain fulfilling condition (P), then ( G , S ) fulfillscondition (P).Proof. Let S ∈ S and d > F , R ) be a Ramsey domain with(P) over ( A, X ) in which S is interpretable. Find R ∈ R and α : S → R as in the definition of interpretability. Since ( F , R ) fulfills (P), we can find F ∈ F such that F • R is defined and for each d -coloring c ′ of F • R thereexists f ∈ F such that for all x , x ∈ S we have(6.6) ∂x = ∂x = ⇒ c ′ ( f . x ) = c ′ ( f . x ) . For F given above, find G ∈ G such that G • S is defined and φ : F → G for which (6.5) holds. Assume we have a d -coloring c of G • S . Define a d -coloring c ′ of F • R as an arbitrary extension to F • R of the function givenby c ′ ( f . α ( y )) = c ( φ ( f ) . y ) , where f ∈ F and y ∈ S . Note that c ′ is well defined by (6.5). For this c ′ ,find f ∈ F for which (6.6) holds. Let now y , y ∈ T be such that(6.7) ∂y = ∂y . Since condition (i) in the definition of interpretability holds for α , (6.7) gives ∂α ( y ) = ∂α ( y ) . Therefore, by the definition of c ′ , by the choice of f and since α ( y ) , α ( y ) ∈ S , we get c ( φ ( f ) . y ) = c ′ ( f . α ( y )) = c ′ ( f . α ( y )) = c ( φ ( f ) . y ) . Thus, we see that (6.7) implies the above equality. It follows that φ ( f ) ∈ G is as required by condition (P) for the coloring c . (cid:3) The proposition above will be applied in Section 8. Examples of composition spaces and Ramsey domains The remainder of the paper illustrates the theoretical results of the earliersections. It contains applications of the general results proved so far to par-ticular cases. For the most part, these applications involve only formulatingnew notions and interpreting some statements as other statements.7.1. Examples of truncations. Forgetful truncation of rigid surjections. The first type of truncationwe introduce is obtained by forgetting the largest value of a function. It isdefined on rigid surjections. We call it the forgetful truncation and we defineit as follows. Let s : [ L ] → [ K ] be a rigid surjection. If K > 0, then L > L = min { y ∈ [ L ] | s ( y ) = K } . Define(7.1) ∂ f s = s ↾ [ L − . If K = 0, then L = 0 and s is the empty function, and we let(7.2) ∂ f ∅ = ∅ . Thus, unless s is empty, ∂ f s forgets the largest value of s while remaining arigid surjection. Unless s is empty, ∂ f s is a proper restriction of s .7.1.2. Confused truncation of surjections. Another way of truncating a sur-jection is obtained by confusing the largest value with the one directly belowit. This type of truncation is defined on non-empty surjections. We definethe confused truncation as follows. Let v : [ L ] → [ K ] be a surjection with K > 0. Define for y ∈ [ L ](7.3) ( ∂ c v )( y ) = ( v ( y ) , if v ( y ) < K ;max(1 , K − , if v ( y ) = K .Note that ∂ c v : [ L ] → [max(1 , K − Examples of composition spaces. Three composition spaces weredefined in Examples A3, B3, and B6. In this section, we describe a numberof new composition spaces. They are used in the proofs of Ramsey-typeresults later on. One more composition spaces are given in Section 9. Normed composition space ( A , X ) . Let A = RS and X = { v ∈ S | v = ∅} . AMSEY THEORY 35 For s , s ∈ A let s · s be defined when the canonical composition s ◦ s is defined, and let s · s = s ◦ s . For s ∈ A and v ∈ X , let s . v be defined precisely when the canonicalcomposition v ◦ s is defined and let s . v = v ◦ s. We equip X with the confused truncation ∂ c given by (7.3). Define a norm | · | : X → N by letting | v | = L for v ∈ X with v : [ L ] → [ K ].The following lemma is straightforward to prove. Lemma 7.1. ( A , X ) with the operations defined above is a normed com-position space. Normed composition space ( A , X ) . Let A = X = RS. We definethe multiplication on A by the same formula s · s = s ◦ s , for s , s ∈ A , and the action of A on X by t . s = s ◦ t, for t ∈ A and s ∈ X , where all the compositions are canonical compositionsof rigid surjections and they are taken under the assumptions under whichcanonical composition is defined. We equip X with the forgetful truncation ∂ f given by (7.1) and (7.2). Define | · | : X → N by letting for t : [ L ] → [ K ], | t | = L .The following lemma is again straightforward to prove. Lemma 7.2. ( A , X ) with the operations defined above is a normed com-position space. Normed composition space ( A , X ) . Let A = X = AS. Given( s, p ) , ( t, q ) ∈ AS, s, p : [ L ] → [ K ] and t, q : [ N ] → [ M ], we let ( t, q ) · ( s, p )and ( t, q ) . ( s, p ) be defined if M ≥ L and in that case we let( t, q ) · ( s, p ) = ( t, q ) . ( s, p ) = (( s ◦ t ) ↾ dom( p ◦ q ) , p ◦ q ) , where all ◦ on the right hand side are canonical compositions, and the lefthand side is defined under the conditions under which the canonical compo-sitions on the right hand side are defined. We also define a truncation ∂ on X by(7.4) ∂ ( s, p ) = ( s ↾ dom( ∂ f p ) , ∂ f p ) , where ∂ f is the forgetful truncation. Furthermore, we define | · | : AS → N by | ( s, p ) | = L if s, p : [ L ] → [ K ]. We leave checking the following easy lemma to the reader. Lemma 7.3. ( A , X ) with the operations defined above is a normed com-position space. Examples of Ramsey domains. We give here examples of Ramseydomains that are relevant to further considerations. Three Ramsey domainswere already defined in Examples A2, B2, and B6, and one more will bedefined in Section 9. Ramsey domain ( F , S ) over ( A , X ) . The family F consists of all setsof the form(7.5) F N,M,L = { s ∈ RS | s : [ N ] → [ M ] , s ↾ [ L ] = id [ L ] } , for 0 < L ≤ M ≤ N . The family S consists of sets of the form S L,v that are defined as follows. Let v : [ L ] → [ K ] be a surjection for some0 < K ≤ L and let L ≤ L . Put S L,v = { v ∈ S | v : [ L ′ ] → [ K ] for some L ≥ L ′ ≥ L , and v ↾ [ L ] = v } . We let F Q,P,N • F M,L,K be defined if K = N and P = M and in thatcase we put F Q,M,K • F M,L,K = F Q,L,K . We let F P,N,M • S L,v be defined, where v : [ L ] → [ K ], if L = M and L = N and in that case we set F P,L,L • S L,v = S P,v . We leave checking the following easy lemma to the reader. Lemma 7.4. (i) ∂ c S L,v = S L,∂ c v . (ii) ( F , S ) is a Ramsey domain over ( A , X ) . Ramsey domain ( F , S ) over ( A , X ) . Both F and S consist of twotypes of sets. We start with defining F . Its elements are sets F N,M,L givenby (7.5) and sets G N,M,L = { s ∈ RS | s : [ N ′ ] → [ M ] for L ≤ N ′ ≤ N, s ↾ [ L ] = id [ L ] } for 0 < L ≤ M ≤ N . Also let G , , = {∅} . The operation • on F is defined only in the following situations: F N,M,K • F M,L,K and G N,M,K • G M,L,K , AMSEY THEORY 37 and we let F N,M,K • F M,L,K = F N,L,K and G N,M,K • G M,L,K = G N,L,K . Define the family S to consist of all sets of the following two forms.Let s : [ L ] → [ K ] be a rigid surjection, for some L ≥ K > 0, and let K ≥ K , L ≥ L , and L ≥ K . Put S L,K,s = { s ∈ RS | s : [ L ] → [ K ] and s ↾ [ L ] = s } , and T L,K,s = { s ∈ RS | s : [ L ′ ] → [ K ] for some L ≥ L ′ ≥ L , and s ↾ [ L ] = s } . Put also T , , ∅ = {∅} . Let the operation • be defined only in the following two situations: F M,L,L • S L,K,s and G M,L,L • T L,K,s , where s : [ L ] → [ K ]. In these situations, we let F M,L,L • S L,K,s = S M,Ks and G M,L,L • T L,K,s = T M,K,s . The proof of the following lemma amounts to easy checking. We leave itto the reader. In point (i) of this lemma, parameters L, K, s vary over thevalues for which the appropriate sets ( S L,K,s and T L,K,s ) are defined. Alsoin connection with the closure of S under truncation, note that for eachrigid surjection t : [ L ] → [ K ], { t } = T L,K,t . Lemma 7.5. (i) Let [ K ] be the image of s . Then ∂ f S L,K,s = T L − ,K − ,s if K > K > ∂ f S L,K ,s = { ∂ f s } ; ∂ f T L,K,s = T L − ,K − ,s if K > K > ∂ f T L,K ,s = { ∂ f s } . (ii) ( F , S ) is a Ramsey domain over ( A , X ) . Ramsey domain ( F , S ) over ( A , X ) . In the definitions below, we areslightly less general than in the definitions of Ramsey domains described sofar. As before, the Ramsey domains consist of sets of elements of A and X that map a given [ L ] or its initial segment to a given [ K ], but we refrainfrom considering such sets with the additional requirement that elements inthem start with a fixed augmented surjection. This additional generalitycan be easily achieved, but it is not needed in our application.For L ≥ K > L = K = 1, let F L,K = { ( s, p ) ∈ AS | s, p : [ L ] → [ K ] and s − ( K ) = { L }} , and, for L ≥ K > L = K = 0, let G L,K = { ( s, p ) ∈ AS | s, p : [ L ′ ] → [ K ] for some 0 < L ′ ≤ L } . Note that G , = { ( ∅ , ∅ ) } . Let F = S consist of all these sets definedabove.The two operations • and • are equal to each other and they are definedprecisely in the situations described below and with the results specifiedbelow: F M,L • F L,K = F M,L • F L,K = F M,K . and G M,L • G L,K = G M,L • G L,K = G M,K . Recall definition (7.4) of the truncation ∂ on ( A , X ). The followinglemma is straightforward to check. Lemma 7.6. (i) For < K ≤ L , ∂F L,K = G L − ,K − and ∂G L,K = G L − ,K − , and, for ≤ L , ∂F , = G , and ∂G L, = ∂G , = G , . (ii) ( F , S ) with the operations defined above is a Ramsey domain over ( A , X ) . Applications In this section, we give applications of the methods developed in thepaper. We give two proofs in detail, that of the Hales–Jewett theorem, inSubsection 8.1, and that of the self-dual Ramsey theorem, in Subsection 1.2,as these two proofs are of more interest than the other ones. These two proofsillustrate how the results of Sections 5 and 6 can be applied: the proof of theHales–Jewett theorem uses Propositions 6.2 and 6.4, the proof of the self-dual Ramsey theorem uses Theorem 5.3. Additionally, in Subsection 8.2,we sketch how to obtain the Graham–Rothschild theorem and, in Section 9,we describe a limiting case that is related to the considerations of [5].We prove these theorems here in the form in which they are stated in Sub-section 2.3, that is, in terms of injections and surjections. The statements interms of parameter sets, partitions and subsets are given in Subsection 1.2.In Subsection 2.4, we described the way of translating these statements fromone form to the other.8.1. The Hales–Jewett theorem. We prove below a theorem that com-bines into one the usual Hales–Jewett theorem [4] and Voigt’s version of thistheorem for partial functions [22, Theorem 2.7] as phrased in Subsection 2.3.One gets the classical Hales–Jewett theorem from the statement below bysetting L = L + 1, L = K , and v = id [ K ] in the assumption and L ′ = L in the conclusion. Indeed, with the notation as in the statement below, if AMSEY THEORY 39 the domains of both v and v are equal to [ L ], then v ◦ s and v ◦ s getthe same color. One derives the Voigt version for the same values of theparameters in the assumption and for L ′ < L in the conclusion. Indeed,the color of v ◦ s for v : [ L ′ ] → [ K ] depends only on L ′ , so one applicationof Dirichlet’s pigeonhole principle gives Voigt’s version of the Hales–Jewetttheorem. Hales–Jewett, combined version. Given d > , < K ≤ L ≤ L and asurjection v : [ L ] → [ K ] , there exists M ≥ L with the following property.For each d -coloring c of { v : [ M ′ ] → [ K ] | L ≤ M ′ ≤ M and v ↾ [ L ] = v } there exists a rigid surjection s : [ M ] → [ L ] such that s ↾ [ L ] = id [ L ] and c ( v ◦ s ) = c ( v ◦ s ) whenever v , v : [ L ′ ] → [ K ] , for the same L ≤ L ′ ≤ L , and v ↾ [ L ] = v ↾ [ L ] = v . We will use the Ramsey domain ( F , S ) defined in Subsection 7.3 andproved to be a Ramsey domain in Lemma 7.4.The proof of the following lemma is an application of the notion of inter-pretability. Lemma 8.1. ( F , S ) fulfills condition (P).Proof. Recall first the conclusion of Example B6. In this example, we have afamily of Ramsey domain fulfilling (P) ( F ⊗ l , S ⊗ lK ) parametrized by naturalnumbers K and l . We claim that S L,v , with v : [ L ] → [ K ] for some L ≤ L , is interpretable in ( F ⊗ ( L − L )0 , S ⊗ ( L − L ) K ), which will prove the lemmaby Proposition 6.4.Set l = L − L . Take ( S ) l ∈ S ⊗ lK and define α : S L,v → ( S ) l as follows. For a natural number 0 ≤ k ≤ K , let ˜ k ∈ S be the function e k ( x ) = K , if x = 1; k, if x = 2;max(1 , K − , if x = 3.Let v ∈ S L,v be such that v : [ L ′ ] → [ K ]. Define α ( v ) = ( ^ v ( L + 1) , ^ v ( L + 2) , . . . , ] v ( L ′ ) , e , . . . , e , where we put L − L ′ entries e ∂ c v = ∂ c v for v , v ∈ S L,v , then ∂ π ( α ( v )) = ∂ π ( α ( v )). So condition(i) from the definition of interpretation holds.We now check condition (ii). Note that ( Q i ≤ l F N i ,M i ) • ( S ) l is definedprecisely when M i = 3 for all i ≤ l . Fix therefore Q i ≤ l F N i , ∈ F ⊗ l . For N = P i ≤ l N i , F L + N,L,L • S L,v is defined in ( F , S ). We need to describea function φ : Y i ≤ l F N i , → F L + N,L,L . Let ¯ p = ( p , . . . , p l ) ∈ Y i ≤ l F N i , . Since v is a surjection, we can fix l , l ∈ [ L ] so that v ( l ) = K and v ( l ) = max(1 , K − . For x ∈ [ L ], let φ (¯ p )( x ) = x, for x ∈ [ L + N + · · · + N i ] \ [ L + N + · · · + N i − ], let φ (¯ p )( x ) = l , if p i ( x − ( L + N + · · · + N i − )) = 1; L + i, if p i ( x − ( L + N + · · · + N i − )) = 2; l , if p i ( x − ( L + N + · · · + N i − )) = 3.We check that (6.5) of the definition of interpretability holds. Note that,for v ∈ S L,v , with v : [ L ′ ] → [ K ], the sequence φ (¯ p ) . v = v ◦ φ (¯ p )is the concatenation of the sequences v , ^ v ( L + 1) ◦ p , ^ v ( L + 2) ◦ p , . . . , ] v ( L ′ ) ◦ p L ′ − L , ( K , . . . , K ) , where the sequence of K -s at the end has length equal to the size of p − L ′ − L +1 (1); in particular, it has length 0 if L ′ = L . On the other hand,¯ p . α ( v ) =( ^ v ( L + 1) ◦ p , ^ v ( L + 2) ◦ p , . . . , ] v ( L ′ ) ◦ p L ′ − L , e ◦ p L ′ − L +1 , . . . , e ◦ p l ) . Thus, (6.5) follows since the size of p − L ′ − L +1 (1) is the number of entriesequal to K at the beginning of e ◦ p L ′ − L +1 , and therefore the above formuladetermines φ (¯ p ) . v . (cid:3) AMSEY THEORY 41 Theorem 4.1 applied to the Ramsey domain ( F , S ), which has (P) byLemma 8.1, gives directly the Hales–Jewett theorem as stated at the begin-ning of this subsection; we apply Theorem 4.1 with t = K − 1, to S K,v ∈ S with v : [ L ] → [ K ].8.2. The Graham–Rothschild theorem. We outline here a proof of theGraham–Rothschild theorem, both the original version [3] and the partialrigid surjection version isolated by Voigt [22, Theorem 2.9]. Here are thetwo statements that were already recalled in Subsection 2.3. Graham–Rothschild. Given d > , K ≤ K , L ≤ L and a rigid surjec-tion s : [ L ] → [ K ] , there exists M ≥ L with the following property. Foreach d -coloring of { s : [ M ] → [ K ] | s ∈ RS and s ↾ [ L ] = s } there is a rigid surjection t : [ M ] → [ L ] , with t ↾ [ L ] = id [ L ] , such that { s ◦ t | s : [ L ] → [ K ] , s ∈ RS and s ↾ [ L ] = s } is monochromatic. Graham–Rothschild, Voigt’s version. Given d > , K ≤ K , L ≤ L and a rigid surjection s : [ L ] → [ K ] , there exists M ≥ L with thefollowing property. For each d -coloring c of { s : [ M ′ ] → [ K ] | L ≤ M ′ ≤ M and s ↾ [ L ] = s } there exist M ′ and a rigid surjection t : [ M ′ ] → [ L ] , with L ≤ M ′ ≤ M and t ↾ [ L ] = id [ L ] , such that { s ◦ t | s : [ L ′ ] → [ K ] , L ≤ L ′ ≤ L, s ∈ RS , and s ↾ [ L ] = s } is monochromatic. We use the Ramsey domain ( F , S ) over the composition space ( A , X )as defined in Subsection 7.3. It is not difficult to check from the Hales–Jewett theorem as stated in Subsection 8.1 that property (LP) holds for( F , S ). So we have the following lemma. Lemma 8.2. ( F , S ) fulfills (LP). In the proof of the above lemma, the following obvious observation playsa crucial role. If s and t are two non-empty rigid surjections with ∂ f s = ∂ f t : [ L ] → [ K ] , then not only s ↾ [ L ] = t ↾ [ L ], but also s ↾ [ L + 1] = t ↾ [ L + 1]as s ( L + 1) = t ( L + 1) = K + 1. We note the obvious facts that ( F , S ) is vanishing and linear. Now anapplication of Corollary 5.4 to ( F , S ) (using Lemmas 7.2 and 7.5) yields thetwo versions of the Graham–Rothschild theorem as stated at the beginningof this subsection.8.3. The self-dual Ramsey theorem. We prove Theorem 2.1. Recallthe definition of connections and their multiplication from Subsection 1.2.First we state a reformulation of Theorem 2.1 and the usual partial functionversion of this reformulation. We follow these reformulations with an expla-nation of how the first one implies Theorem 2.1. Then, we give argumentsfor the two statements. Self-dual Ramsey theorem. Given d > , < K ≤ L , there exists M with the following property. For each d -coloring of { ( s, p ) ∈ AS | ( s, p ) : [ M ] → [ K ] and s − ( K ) = { M }} there is an augmented surjection ( t , q ) : [ M ] → [ L ] such that t − ( L ) = { M } and { ( t , q ) · ( s, p ) | ( s, p ) : [ L ] → [ K ] , ( s, p ) ∈ AS and s − ( K ) = { L }} is monochromatic. Self-dual Ramsey theorem; partial augmented surjection version. Given d > , K ≤ L , there exists M with the following property. For each d -coloring of { ( s, p ) ∈ AS | ( s, p ) : [ M ′ ] → [ K ] for some M ′ ≤ M } there is an augmented surjection ( t , q ) : [ M ′ ] → [ L ] for some M ′ ≤ M such that { ( t , q ) · ( s, p ) | ( s, p ) ∈ AS , ( s, p ) : [ L ′ ] → [ K ] for some L ′ ≤ L } is monochromatic. To obtain Theorem 2.1 from the first of the above statements, associatewith an increasing surjection p : [ L ] → [ K ] an increasing injection i p : [ K ] → [ L ] given by i p ( x ) = max p − ( x ). If ( s, p ) is an augmented surjection with s, p : [ L ] → [ K ], for some 0 < K ≤ L , and s − ( K ) = { L } , then( s ↾ [ L − , i p ↾ [ K − L − 1] and [ K − 1] and each connection between[ L − 1] and [ K − 1] is uniquely representable in this way. Moreover, if ( t, q ) is AMSEY THEORY 43 another augmented surjection with t, q : [ M ] → [ L ] and with t − ( L ) = { M } ,then(( s ◦ t ) ↾ [ M − , i p ◦ q ↾ [ K − t ↾ [ M − , i q ↾ [ L − · ( s ↾ [ L − , i p ↾ [ K − , where the multiplication on the right hand side is the multiplication of con-nections. These observations show that the first of the above statementsimplies Theorem 2.1.The following lemma will turn out to be an immediate consequence ofthe Hales–Jewett theorem and the Voigt version of the Graham–Rothschildtheorem. Lemma 8.3. ( F , S ) fulfills (LP).Proof. A moment of thought (using the way ∂ acts on subsets of X ) con-vinces us that to see (LP) it suffices to show Conditions 1 and 2 below,for L ≥ K > d > 0. To state these conditions, fix ( s , p ) ∈ AS, s , p : [ L ] → [ K − 1] for some L < L . The role of the elements x and a in (LP) is played by ( s , p ) and (id [ L ] , id [ L ] ), respectively. Note that ( t, q )from F M,L or G M,L extends (id [ L ] , id [ L ] ) precisely when t ↾ [ L ] = id [ L ] and q ↾ [ L + 1] = id [ L +1] . Condition 1. There exists M ≥ L such that for each d -coloring of F M,L . { ( s, p ) ∈ F L,K | ∂ ( s, p ) = ( s , p ) } there exists ( t , q ) ∈ F M,L such that t ↾ [ L ] = id [ L ] and q ↾ [ L + 1] = id [ L +1] and { ( t , q ) . ( s, p ) | ( s, p ) ∈ F L,K , ∂ ( s, p ) = ( s , p ) } is monochromatic. This statement amounts to proving the following result. There exists M > L such that for each d -coloring of all rigid surjections t : [ M − → [ K − with t ↾ [ L ] = s there exists a rigid surjection t : [ M − → [ L − such that t ↾ [ L ] = id [ L ] and { s ◦ t | s : [ L − → [ K − a rigid surjection and s ↾ [ L ] = s } is monochromatic. This is a special case of the Hales–Jewett theorem, as stated and provedin Subsection 8.1. Condition 2. There exists M ≥ L such that for each d -coloring of G M,L . { ( s, p ) ∈ G L,K | ∂ ( s, p ) = ( s , p ) } there exists ( t , q ) ∈ G M,L such that t ↾ [ L ] = id [ L ] and q ↾ [ L + 1] = id [ L +1] and { ( t , q ) . ( s, p ) | ( s, p ) ∈ G L,K , ∂ ( s, p ) = ( s , p ) } is monochromatic. We will produce a sequence of statements the last of which will be Con-dition 2.Fix L ≥ K . We will specify later how large L should be. Statement(A) below follows from the Graham–Rothschild theorem as stated in Sub-section 8.2 in the same way as Condition 1 above follows the Hales–Jewetttheorem; we obtain M − d , K , L and s .(A) There exists M > L such that for each d -coloring of all augmented sur-jections from [ M ] to [ K ] there exists an augmented surjection ( t , q ) : [ M ] → [ L ] such that t ↾ [ L ] = id [ L ] , q ↾ [ L + 1] = id [ L +1] , and { ( t , q ) . ( s, p ) | ( s, p ) : [ L ] → [ K ] and ∂ ( s, p ) = ( s , p ) } is monochromatic. Using statement (A), we show below how to obtain the following state-ment.(B) There exists M > L with the following property. For each d -coloringof all augmented surjections from [ M ′ ] to [ K ] , where M ′ is such that L By the choice of M , there exists ( t ′ , q ′ ) : [ M ] → [ L ] such that the conclu-sion of (B) holds for it. Let ( t ′′ , q ′′ ) : [ M + 1] → [ L + 1] be given by t ′′ ↾ [ M ] = t ′ , q ′′ ↾ [ M ] = q ′ , and t ′′ ( M + 1) = q ′′ ( M + 1) = L + 1 . Then it is easy to see that the augmented surjection ( t , q ) . ( t ′′ , q ′′ ) witnessesthe conclusion of (B) for coloring c .(C) There exists M > L with the following property. For each d -coloringof all augmented surjections from [ M ′ ] to [ K ] , where L < M ′ ≤ M , thereexists an augmented surjection ( t , q ) : [ M ] → [ L + 1] such that t ↾ [ L ] =id [ L ] , q ↾ [ L + 1] = id [ L +1] , and the set { ( t , q ) . ( s, p ) | ( s, p ) : [ L ′ ] → [ K ] , L < L ′ ≤ L, and ∂ ( s, p ) = ( s , p ) } is monochromatic. To see (C), first pick L so that for each d -coloring of [ L − L ], there is asubset of [ L − L ] with L + 1 − L elements that is monochromatic. Apply(B) to L obtaining M . This M works for (C). Indeed, for a d -coloring asin (C) (and in (B)), we get ( t , q ) : [ M ] → [ L ] as in the conclusion of (B).Now we use the choice of L to find ( t , q ) : [ L ] → [ L + 1] with t = q and q ↾ [ L + 1] = id [L +1] , and so that ( t , q ) defined to be ( t , q ) . ( t , q ) isas required by (C).Now, (C) is easily seen to be just a reformulation of Condition 2. Thus,(LP) holds and the lemma follows. (cid:3) Since ( F , S ) is clearly vanishing, by Lemmas 7.3 and 7.6, Corollary 4.3can be applied to ( F , S ) yields the statements from the beginning of thissubsection. 9. Walks, a limiting case In this section, we give a natural limiting example of the extent of condi-tion (R). The motivation for this example comes from [5] and is related toa problem of Uspenskij [21].Recall that walks were defined in Subsection 2.1, and the set of all walkswas denoted there by W . Let C = Z = W . We note that C ⊆ A and Z ⊆ X , as defined in Subsection 7.2. We equip C with the multiplicationinherited from A and we take the action of C on Z to be the one inheritedfrom ( A , X ). Note also that Z is closed under the forgetful truncation ∂ f with which ( A , X ) is equipped. We take it as the truncation on ( C, Z ).We also consider the function | · | defined on X , we restrict it to Z , anddenote it again by | · | . The following lemma is an immediate consequenceof Lemma 7.2. Lemma 9.1. ( C, Z ) with the operations defined above is a normed compo-sition space. Let H = W consist of all finite non-empty subsets F of W such that forsome K > F is a walk with range [ K ]. We write r ( F ) = K .We write d ( F ) = L if L is the largest natural number such that there is awalk in F whose domain is [ L ]. We call F tidy if each walk in F has domain[ d ( F )]. For F , F ∈ H = W , F • F and F • F are defined if and only if d ( F ) = r ( F ) and F being tidy implies that F is tidy. Then we let F • F = F • F = F · F = F . F . One easily checks the following lemma. Lemma 9.2. ( H , W ) with • and • is a Ramsey domain over ( C, Z ) . The Ramsey statement equivalent to this Ramsey domain fulfilling con-dition (R) was motivated by a question of Uspenskij [21], which asked ifthe universal minimal flow of the homeomorphism group of the pseudo-arcis the pseudo-arc itself together with the natural action of the homeomor-phism group. Uspenskij’s question would have had a positive answer if thisRamsey statement were true. (The Ramsey statement would imply such ananswer by [5] and by a dualization of the techniques of [6].) However, thetheorem below implies that the Ramsey statement is false. Theorem 9.3. For every M ≥ there exists a -coloring of all walks from [ M ] to [3] such that for each walk t : [ M ] → [6] the set { s ◦ t | s : [6] → [3] a walk } is not monochromatic.Proof. We show a bit more: to contradict monochromaticity we only need aset of walks s : [6] → [3] that differ at two elements of their common domain.Let S = { s : [6] → [3] | s (1) = 1 , s (2) = s (5) = 2 , s (6) = 3 , s (3) , s (4) ∈ { , }} . Clearly each element of S is a walk from [6] to [3]. We claim that for each M ≥ M ] to [3] such that for eachwalk t : [ M ] → [6] the set S ◦ t is not monochromatic.Let M ≥ 3. For a walk u : [ M ] → [3] define a ( u ) = |{ y ∈ [ M ] | u ( x ) ≤ x ≤ y, u ( y ) = 1 , and u ( y + 1) = 2 }| . Define a 2-coloring c by letting c ( u ) = a ( u ) mod 2 . We claim that this coloring is as required. AMSEY THEORY 47 Let t : [ M ] → [6] be a walk. We analyze t in order to compute a ( s ◦ t )for s ∈ S in terms of certain numbers associated with t . Let M ≤ M bethe smallest natural number with t ( M ) = 6. There exist unique, pairwisedisjoint intervals I ⊆ [ M ] that are maximal with respect to the property t ( I ) ⊆ { , } . For such an I , let I − and I + be (min I ) − I ) + 1,respectively. Note that t ( I − ) , t ( I + ) ∈ { , } . We distinguish four types ofsuch intervals I : I ∈ P ⇐⇒ t ( I − ) = 2 and t ( I + ) = 5; I ∈ P ⇐⇒ t ( I − ) = 5 and t ( I + ) = 2; I ∈ Q ⇐⇒ t ( I − ) = t ( I + ) = 2; I ∈ Q ⇐⇒ t ( I − ) = t ( I + ) = 5 . Note right away that since t is a walk, t (1) = 1, t ( M − 1) = 5, and t ( x ) ∈{ , . . . , } for x ∈ [ M − | P | − | P | = 1 and therefore(9.1) | P | + | P | is odd.For each I ∈ P ∪ P ∪ Q ∪ Q , define a t ( I ) as follows. a t ( I ) = ( |{ x ∈ I | t ( x ) = 3 , t ( x + 1) = 4 }| , if I ∈ P ∪ Q ; |{ x ∈ I | t ( x ) = 4 , t ( x + 1) = 3 }| , if I ∈ P ∪ Q .Note that for I in Q or Q , the two cases in the above definition give thesame value for a t ( I ). Further, let a t ( ∗ ) = |{ x ∈ [ M ] | t ( x ) = 1 , t ( x + 1) = 2 }| . Recall now the set S introduced in the beginning of the proof. We canwrite S = { s , s , s , s } , where s , s , s and s are determined by theconditions s (3) = s (4) = 1 ,s (3) = 2 , s (4) = 1 ,s (3) = 1 , s (4) = 2 , and s (3) = s (4) = 2 . An inspection convinces us that a ( s i ◦ t )= a t ( ∗ ) + P I ∈ P ∪ P ∪ Q ∪ Q , if i = 1; a t ( ∗ ) + P I ∈ P ∪ P a t ( I ) + P I ∈ Q a t ( I ) + P I ∈ Q ( a t ( I ) + 1) , if i = 2; a t ( ∗ ) + P I ∈ P ∪ P a t ( I ) + P I ∈ Q ( a t ( I ) + 1) + P I ∈ Q a t ( I ) , if i = 3; a t ( ∗ ) , if i = 4. Assume towards a contradiction that a walk t : [ M ] → [6] is such that S ◦ t is monochromatic. It follows from the above expressions for a ( s i ◦ t )that the numbers X I ∈ P ∪ P ∪ Q ∪ Q , X I ∈ P ∪ P a t ( I ) + X I ∈ Q a t ( I ) + X I ∈ Q ( a t ( I ) + 1) , X I ∈ P ∪ P a t ( I ) + X I ∈ Q ( a t ( I ) + 1) + X I ∈ Q a t ( I ) , and 0have the same parity, that is, since the last number is 0, they must all beeven. Now it follows from the first line that(9.2) | P | + | P | + | Q | + | Q | is evenand, by subtracting the second line from the third one, that | Q | − | Q | iseven, and so(9.3) | Q | + | Q | is even.Equations (9.2) and (9.3) imply that the natural number | P | + | P | is evencontradicting (9.1). (cid:3) A problem Observe that the failure of a Ramsey result from Section 9, the classicalRamsey theorem, and the Graham–Rothschild theorem can be viewed in auniform way as statements about composition spaces that are closely relatedto each other. Fix A so that IS ⊆ A ⊆ RS, where the families of increasingsurjections IS and of rigid surjections RS are defined in Subsection 2.1.Assume that for s, t ∈ A with s : [ L ] → [ K ] and t : [ N ] → [ M ] with L ≤ M , the canonical composition s ◦ t , as defined in Subsection 2.2, is in A .Assume further that ∂ f s ∈ A for s ∈ A , where the forgetful truncation ∂ f isdefined in Subsection 7.1. Consider the composition space ( A, A ) in whichmultiplication on A is the same as the action of A on A , both are definedprecisely when s ◦ t is defined and are given by t · s = t . s = s ◦ t, and in which the truncation operator is given by ∂ f . The following problempresents itself. Find all sets A as above such that the following Ramsey statement holds:given d > and K, L there is M such that for each d coloring of { s ∈ A | AMSEY THEORY 49 s : [ M ] → [ K ] } there exists t ∈ A with t : [ M ] → [ L ] and such that { s ◦ t | s ∈ A, s : [ L ] → [ K ] } . is monochromatic. Note that if A = RS or A = IS, then the answer is positive. In thefirst case, the resulting Ramsey theorem is the dual Ramsey theorem (seeSubsection 8.2) and in the second case it is a reformulation of the classicalRamsey theorem (see Example A5 in Subsection 5.2). Note also that, byTheorem 9.3, if A is the set of all walks W, then the answer is negative. Acknowledgement. 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