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Heterogeneous Ramsey Algebras and Classification of Ramsey Vector Spaces

Carlson introduced the notion of a Ramsey space as a generalization to the Ellentuck space. When a Ramsey space is induced by an algebra, Carlson suggested a study of its purely combinatorial version now called Ramsey algebra. Some basic results for homogeneous algebras have been obtained. In this paper, we introduce the notion of a Ramsey algebra for heterogeneous algebras and derive some basic results. Then, we study the Ramsey-algebraic properties of vector spaces.


Introduction
The notion of a Ramsey space was introduced by Carlson in [2]. In the modern literature, this notion is referred to as a topological Ramsey space. The class of topological Ramsey spaces induced by algebras has been singled out to be studied from a purely combinatorial point of view. An algebra is a structure consisting of a family of sets, called the domain of the algebra, and a family of operations on these sets. Hindman's Theorem states that for each positive integer r and each coloring c ∶ Z + → {1, . . . , r}, there exists an infinite S ⊆ Z + such that c is constant on the set {∑ i∈F i ∶ F ⊆ S, F is finite}. The same theorem can be cast in terms of finite subsets of the natural numbers [8]. Considering questions about infinite sets along the same line, a result of Erdös-Rado [4] shows that not all sets of reals 1 possess similar property. Hence, it is natural to ask which among those definable sets-specifically, sets belonging to the σ-algebra generated by the usual topology on the realshave the analogous property. Galvin-Prikry [5] showed that the Borel sets do, whereas, using the method of forcing, Silver [9] showed that the analytic sets have the desired property.
Ellentuck in [3] showed that the result of Silver could also be proved using topological formulation without having to appeal to metamathematical methods. In his proof, Ellentuck introduced what is to be called the Ellentuck topology. A generalization of the Ellentuck topology is then introduced by Carlson in [2], which he called (topological) Ramsey space. Carlson showed in the same paper that classical results such as the Hales-Jewett Theorem and Hindman's Theorem can be derived as corollaries to his results on the Ramsey space of variable words.
If one is to ask whether a given structure is a Ramsey space, the definition requires that one checks some topological properties of the space. However, Carlson's abstract version of Ellentuck's Theorem (cf. [2]) turns such topological question into a combinatorial one. Carlson had, in fact, suggested that one could embark on a specifically combinatorial study of spaces induced by algebras; this study was then initiated by Teh [cf. [10] & [11]] and the topic became known as Ramsey algebra.
There is already quite a number of existing Ramsey algebraic results on algebras commonly encountered in the mathematical literature; in particular, a result on infinite integral domains, which will be applied to our work in this paper, can be found in [10]. Other results including the existence of idempotent ultrafilters can be found in [12] and [13]. In this paper, we extend the study of Ramsey algebras to many-sorted structures. We will first formulate a notion of Ramsey algebra for manysorted algebras, then we will derive some results pertaining to vector spaces. A latter section is then dedicated to the study of the homogeneous structure over a family of functions induced by the nonzero scalars of a vector space.

Preliminary
We begin with the notion of an algebra. From a logical point of view, an algebra is a structure interpreting a many-sorted, purely functional language. That is, an algebra A consists of a family (A ξ ) ξ∈I of sets and a family F of functions, where each f ∈ F is a function with domain some Cartesian product A ξ 1 × ⋯ × A ξn (n ∈ ω) of some members of (A ξ ) ξ∈I and codomain some member A of (A ξ ) ξ∈I . We write A = ((A ξ ) ξ∈I , F ) for the algebra just described. The family of sets pertaining to a given algebra is called the universe or domain of the algebra and each set in the family a phylum as per [1]. Throughout this paper, we reserve the symbol I for the index set of the algebra under discussion.
When I is a singleton, the algebra is said to be homogeneous. Examples of homogeneous algebras are groups, rings, Boolean algebras, etc.. For a homogeneous algebra, the notation ((A ξ ) ξ∈I , F ) will be simplified to (A, F ), where A is the only phylum of the domain. A vector space is a natural example of a heterogeneous algebra consisting of two phyla.
The set of natural numbers, inclusive of 0, is denoted by ω. If f is a function, Dom(f ) denotes the domain of f , Cod(f ) denotes the codomain of f , and Rn(f ) denotes the range of f . We will identify any n-tuple (x 1 , . . . , x n ) with the finite sequence ⟨x 1 , . . . , x n ⟩ and, ifx denote the n-tuple, ⃗ x will denote the associated sequence, and vice versa. If ⃗ e is an infinite sequence, the notation ⃗ e−n denotes the tail ⟨⃗ e(n), ⃗ e(n+1), . . .⟩ and ⃗ e ↾ n denotes the initial segment ⟨⃗ e(0), . . . , ⃗ e(n)⟩. If σ is a finite sequence, then σ denotes the length of the sequence. If A is a nonempty set, an infinite sequence of A is an element of ω A. If D is a set, then id D denotes the identity function on D.
While the family of phyla (A ξ ) ξ∈I in a given heterogeneous algebra ((A ξ ) ξ∈I , F ) need not be pairwise disjoint in general, we shall always assume that the family is indeed pairwise disjoint in the study of heterogeneous Ramsey algebra following the assumption made by Carlson in [2].

The Basic Notions of Heterogeneous Ramsey Algebras
The study of Ramsey algebra is a fairly recent endeavor. In this section, we define the notion of a Ramsey algebra and the necessary definitions leading up to it. We will begin with the notion of an orderly composition due to Carlson.
Definition 3.1 (Orderly Composition & Orderly Terms). Let F be a family of operations on (A ξ ) ξ∈I . An n-ary function f is called an orderly composition of F if there exists h 1 , . . . , h k , g ∈ F such that (1) g is a k-ary function, (2) h j is an n j -ary function for each j ∈ {1, . . . , k}, The collection OT(F ) of orderly terms over F is the smallest collection of functions containing F ∪ id A ξ ξ∈I and is closed under orderly compositions.
The collection of orderly terms over F is in fact the collection of operations on (A ξ ) ξ∈I which can be generated by an application of finitely many of the following rules: (1) for each ξ ∈ I, the identity function id A ξ is an orderly term, (2) every operation in F is an orderly term, (3) if f is an operation on (A ξ ) ξ∈I given by f (x 1 , . . . ,x k ) = g(h 1 (x 1 ), . . ., h k (x k )) for some g ∈ F and some orderly terms h 1 , . . . , h k , then f is an orderly term. We denote the concatenation operation by * in this paper. Definition 3.2 (Reduction ≤ F ). Let ((A ξ ) ξ∈I , F ) be an algebra and let ⃗ a and ⃗ b be members of ω ⋃ ξ∈I A ξ . Then ⃗ a is said to be a reduction of ⃗ b if there exist orderly terms f j over F and finite subsequences The relation ≤ F is a preorder 2 on ω ⋃ ξ∈I A ξ . Note that the inclusion of the identity functions in the set of orderly terms is necessary to ensure that the relation ≤ F is reflexive.
for each n ∈ ω and that ⃗ e is the sort of ⃗ a.
Note that the sort of a sequence is unique under our assumption that the sets in the family of phyla (A ξ ) ξ∈I are pairwise disjoint.
We end this section with the central notion of the paper.
Definition 3.5 (⃗ e-Ramsey Algebra). Let ⃗ e be a sort. An algebra ((A ξ ) ξ∈I , F ) is said to be an ⃗ e-Ramsey algebra if, for each ⃗ e-sorted sequence ⃗ b and each X ⊆ A ⃗ e(0) , there exists an ⃗ e-sorted reduction ⃗ a such that FR ⃗ e F (⃗ a) is either contained in or disjoint from X.
Such a sequence ⃗ a is said to be homogeneous for X (with respect to F ).
In the case where the algebra is homogeneous, the set of sorts would be a singleton and we shall drop the reference to ⃗ e and refer any such ⃗ e-Ramsey algebra as a Ramsey algebra. Also, the set defined by Eq. 3.4 can be characterized by is a finite subsequence of ⃗ b}, in which case the notation will be simplified to FR F ( ⃗ b).
Remark 3.1. If A is an ⃗ e-Ramsey algebra for a given ⃗ e, then for each finite partition Q 1 ∪ ⋯ ∪ Q N of A ⃗ e(0) and for each ⃗ e-sorted ⃗ b, there exists an ⃗ e-sorted ⃗ a ≤ F ⃗ b and an i * ∈ {1, . . . , N} such that FR ⃗ e F (⃗ a) ⊆ Q i * . A classic result of Ramsey algebra is that of Hindman: Theorem 3.1 (Hindman). Every semigroup is a Ramsey algebra.

From Ramsey Spaces to Ramsey Algebras
The origin of Ramsey algebras has its roots in the notion a Ramsey space introduced by Carlson. As mentioned in the introductory section, largely due to the ensuing work of Todorcevic [14], this notion is now referred to as a topological Ramsey space. This section is intended to give a short account of the connection between Ramsey algebras and topological Ramsey spaces.
If R is a nonempty set of infinite sequences equipped with a preorder ⪯, then for all ⃗ b ∈ R and all n ∈ ω, the sets form a neighborhood basis of the natural topology on R.
Definition 4.1. Let R be a nonempty set of infinite sequences equipped with a preorder ⪯ and let X ⊆ R. The set X is said to be Ramsey if, The set X is said to be Ramsey null in the event there exists an ⃗ a ∈ [n, ⃗ b] for every given ⃗ b and n ∈ ω such that [n, ⃗ a] ⊆ X C .
Definition 4.2. If R is a nonempty set of infinite sequences equipped with the preorder ⪯, then (R, ⪯) endowed with the natural topology is called a topological Ramsey space if every set which has the Property of Baire is Ramsey and every meager set is Ramsey null 3 .
The neighborhood sets given in Eq. 2 and the natural topology so generated are defined with the preorder induced by the reduction operation within an algebra ((A ξ ) ξ∈I , F ) in mind. We recall from Section 2 that the relation ≤ F for any given algebra ((A ξ ) ξ∈I , F ) is a preordering on the infinite sequences of ω ⋃ ξ∈I A ξ ; we will be chiefly interested in the natural topology on subsets of ω ⋃ ξ∈I A ξ determined by any given sort ⃗ e as these are the topologies connected to the notion of a ⃗ e-Ramsey algebra. For each sort ⃗ e, define R ⃗ e to be the set consisting of ⃗ e-sorted sequences of ⋃ ξ∈I A ξ equipped with the relation ≤ F ↾ (R ⃗ e × R ⃗ e ). We will abuse notation and denote such restriction by the same symbol ≤ F when no confusion arises. Given an algebra ((A ξ ) ξ∈I , F ) and a sort ⃗ e, we write R ⃗ e ((A ξ ) ξ∈I , F ) for (R ⃗ e , ≤ F ) to make explicit the algebra involved. For emphasis, we also add ⃗ e as superscript to the members of the neighborhood basis [n, The next two theorems of Carlson [2], adapted to the current context, are key to relating the notion of a Ramsey algebra to that of a topological Ramsey space. The next theorem can be thought of as an abstract Ellentuck Theorem.
e and X is a set of initial segments of length n + 1 of ⃗ e-sorted sequences , then there exists ⃗ a ∈ [n, ⃗ b] ⃗ e such that the set consisting of all the initial segments of sequences in [n, ⃗ a] ⃗ e of length n + 1 is either a subset of X or is disjoint from X.
The preceding theorem is the key to relating topological Ramsey spaces to Ramsey algebra. When paraphrased, it becomes a statement in the language of Ramsey algebra.

Some Elementary Results
Given any algebra A with the indexing set I, we single out the following class of sorts: We will mainly be studying ⃗ e-Ramsey algebras for ⃗ e belonging in this class. The reason for this choice is that sequences of sorts belonging in Ω bear resemblance to properties familiar from homogeneous Ramsey algebras. For example, if ⃗ e ∈ Ω, then Compare this to Eq. 1. The reader can go through the definition of a reduction to verify Eqv. 3.
As the first term of a sort is of critical importance in reductions, we break the set Ω down as follows: For each ξ ∈ I, To do this, note that ⃗ c ′ ≤ F ⃗ a by the transitivity of ≤ F . Now, apply Eq. 3 and we see that A classification of homogeneous algebras all of whose operations are unary can be found in [10]. We state it here for reference purposes: Theorem 5.1. If F is a collection of unary operations, then (A, F ) is a Ramsey algebra if and only if every element of A can be sent into an element of S = {a ∈ A ∶ f (a) = a for all f ∈ F } by finitely many applications of the members of F . Theorem 5.1's heterogeneous analogue is given in the next theorem, but before then we will content ourselves with two lemmas. In Lemma 5.2 and Theorem 5.2 below, we consider algebras whose family F of operations are disjoint in the following sense: Apart from tedious definition chasing and an appeal to Eqv. 3, these two parts of the lemma are intuitive.
(3) Let η ∈ I be an index that occurs only finitely often in ⃗ e and let N ∈ ω be such that ⃗ e(N) = η and ⃗ e(i) ≠ η for all i > N. Then, since F is a disjoint union of the G ξ 's, we see from Lemma 5.3 (Katetǒv [7]). Let A be a set and suppose that T ∶ A → A does not have fixed points. Then, there exists a partition A = P 1 ∪P 2 ∪P 3 such that T [P i ] ∩ P i = ∅ for each i = 1, 2, 3.
Then, A is an ⃗ e-Ramsey algebra if and only if every element of A ⃗ e(0) can be sent into the set of fixed points S ⃗ does not have unary operations, then S ⃗ e(0) = A ⃗ e(0) and so A is an ⃗ e-Ramsey algebra.
Proof. (1)(⇒) Suppose that A is an ⃗ e-Ramsey algebra for ⃗ e ∈ Ω η and let ⃗ β be an infinite sequence of A η and X ⊆ A η . Now pick an ⃗ e-sorted sequence ⃗ b such that its subsequence all of whose terms are members of be ⃗ e-sorted and homogeneous for X. Thus, define the infinite sequence ⃗ α of A η as the subsequence of ⃗ a all of whose terms lie in A η . Observe that ⃗ α ≤ Gη ⃗ β, a property again guaranteed by the disjoint composition of F . We may now apply Part 2 of Lemma 5.2 to obtain FR Gη (⃗ α) = FR ⃗ e F (⃗ a). As a result, the homogeneity of ⃗ α for X entails. (⇐) Suppose that (A η , G η ) is a Ramsey algebra and let ⃗ b be an ⃗ e-sorted sequence and X ⊆ A η . To see that A is an ⃗ e-Ramsey algebra, it suffices to show that, if ⃗ β is the subsequence of ⃗ b consisting of terms lying in A η and ⃗ α ≤ Gη ⃗ β is homogeneous for X, then there exists an ⃗ We then construct the desired ⃗ a recursively as follows: the subsequence of ⟨⃗ a(0), . . . , ⃗ a(N)⟩ whose terms are members of A η is a subsequence of ⃗ α.
We then decide the value of ⃗ a(N + 1) as follows: then we pick the first term belonging in A ξ in the tail of ⃗ b without σ and let it be the value of ⃗ a (N + 1). The sequence ⃗ a so constructed is ⃗ e-sorted and is such that ⃗ a ≤ F ⃗ b. In addition, its subsequence ⃗ α ′ of terms that belong in A η forms a subsequence of ⃗ α. As such, we have FR Gη (⃗ α ′ ) ⊆ FR Gη (⃗ α). Now, by Part 2 of Lemma 5.2, we have FR ⃗ e F (⃗ a) = FR Gη (⃗ α ′ ). The homogeneity of ⃗ a for X then follows from the homogeneity of ⃗ α for X.
≠ a for some f ∈ G η . Such T does not have fixed points and so, by Lemma 5.3, let P 1 ∪P 2 ∪P 3 be a partition of A η ∪{α} such that T [P i ]∩P i = ∅ for each i = 1, 2, 3. This induces a partition Q 1 ∪ Q 2 ∪ Q 3 of A η such that whenever a ∈ A ∩ S C η , if a ∈ Q i , then f (a) ∈ Q i for some f ∈ G η . Now, suppose that A is an ⃗ e-Ramsey algebra. Towards a contradiction, assume that there exists c ∈ A η such that c cannot be sent into S η by finitely many applications of the unary operations of G η . Fix one such element c and pick any ⃗ e-sorted ⃗ b such that ⃗ b(0) = c. Referencing Remark 3.1, there exists an ⃗ e-sorted ⃗ a ≤ F ⃗ b and an i * ∈ {1, 2, 3} such that FR ⃗ e F (⃗ a) ⊆ Q i * . By Part 3 of Lemma 5.2, ⃗ a(0) is obtained from ⃗ b(0) = c by finitely many applications of the unary operations of G η , therefore ⃗ a(0) ∈ S η by our choice of c. The operation T now furnishes an f ∈ G η such that f (⃗ a(0)) ∈ Q i * , which contradicts f (⃗ a(0)) ∈ FR F (⃗ a) ⊆ Q i * .
A simple example will show us that, if heterogeneous operations are present, then the conclusion for Part 1 above need not hold: for which a 0 ≠ a 1 . Let α 0 ∈ A 1 and let ⃗ e ∈ Ω 0 be nonconstant. Note that (A 0 , ○) is a Ramsey algebra. However, for each ⃗ e-sorted ⃗ a ≤ F ⃗ b, the set FR ⃗ e F (⃗ a) has at least two elements a 0 , a 1 and so no such ⃗ c can be homogeneous for {a 0 } ⊆ A 0 . Now, let us call by Ω J the set of those ⃗ e ∈ Ω such that all indices ξ ∈ I appearing in ⃗ e form the subset J ⊆ I. Also, let Ω J η = {⃗ e ∈ Ω J ∶ ⃗ e ∈ Ω η }.
Theorem 5.3. For any family F of operations, if A = ⋃ ξ∈I A ξ , F is an ⃗ e-Ramsey algebra for an ⃗ e ∈ Ω J η , then it is an ⃗ e-Ramsey algebra for all ⃗ e ∈ Ω J η .
Proof. Let ⃗ e, ⃗ e ′ ∈ Ω J η and suppose that A is an ⃗ e-Ramsey algebra. It suffices to prove that if A is an ⃗ e-Ramsey algebra, then it is an ⃗ e ′ -Ramsey algebra.
Let the ⃗ e ′ -sorted sequence ⃗ b ′ and X ⊆ A η be given. We obtain an ⃗ e-sorted subsequence ⃗ b of ⃗ b ′ , which is possible since ⃗ e ′ ∈ Ω J . Then, choose an ⃗ e-sorted ⃗ a ≤ F ⃗ b such that ⃗ a is homogeneous for X as we have assumed that A is an ⃗ e-Ramsey algebra. Next, we obtain an ⃗ e ′ -sorted subsequence ⃗ a ′ of ⃗ a, possible since ⃗ e ∈ Ω J . By the transitivity of We want to emphasize that the preceding theorem holds because, as we see in its proof, the sorts ⃗ e and ⃗ e ′ share the same first term and the same set J of indices. Part 1 of Theorem 5.2 illustrates the dependence on the first term, whereas one can easily cook up an example to demonstrate that sharing the same J is indeed essential.

Classification of Ramsey Vector Spaces
We now begin the study of vector spaces in the context of Ramsey algebra. For completeness sake, we carry out an analysis for all possible sorts in this study. A vector space is a structure (V, F, + F , × F , + V , ⋅), where (V, + V ) forms an abelian group, (F, + F , × F ) a field whose elements are called scalars, and ⋅ ∶ F × V → V is scalar multiplication such that 1 ⋅ v = v for all v ∈ V, and the following axioms of distributivity hold: Vector spaces are two-sorted heterogeneous structure: (V, F, + V , + F , × F , ⋅). In this section, F is reserved for the family {+ V , + F , × F , ⋅} and we always let A 0 denote the underlying field of a vector space and A 1 the set of vectors. The uppercase "oh" O is reserved for the zero vector in any vector space under discussion.
A few observations regarding the orderly terms of F will be helpful. (1) The only unary orderly terms over F are the identity functions id F and id V ; the only binary orderly terms over F are the binary operations in F . (2) If Dom(f ) = F n for some nonzero n ∈ ω, then Rn(f ) ⊆ F. Proof. Carrying out an induction argument on the generation of the orderly terms will establish Facts 1 and 2. Facts 3 through 5 can also be established by induction on the generation of orderly terms. We will only give the proof of Fact 4 and a sketch of the proof of Fact 5.
Fact 4: The implication holds if f is either id F or id V . Similarly, if f ∈ F , then the antecedent of the implication holds provided f is either scalar multiplication or vector addition. In both cases, the consequent of the implication holds.
Now, suppose f (x 1 ,x 2 ) = g(h 1 (x 1 ), h 2 (x 2 )) for some g ∈ F and orderly terms h 1 and h 2 , where the arity of g owes to the fact that every g ∈ F is binary. If g is field addition or field multiplication, then both h 1 and h 2 are scalar-valued and hencex 1 andx 2 are all scalar terms, contradicting ξ n = 1. Thus, g is either vector addition or scalar multiplication. In both cases, h 2 is vector valued. Hence,x 2 involves some vector term. Then, by the induction hypothesis, the trailing term ofx 2 is a vector.
Fact 5: For the base cases, the conclusion is obvious. Thus, suppose f (x) = g(h 1 (x 1 ), h 2 (x 2 )) with g ∈ F and with h 1 , h 2 ∈ OT(F ) both satisfying the conclusion of Fact 5. Now assumex satisfies the hypothesis of Fact 5. If g is vector addition, then f (x) = h 1 (x 1 ) + h 2 (x 2 ), and so f clearly satisfies the conclusion of Fact 5; this is the case where the result is the nonempty sum of some vectors fromx. Proofs of the other cases of g are similarly obvious and will, hence, be omitted.
The next lemma will be helpful in proving various Ramsey-algebraic properties of vector spaces. Lemma 6.2. Let (V, F, F ) be a vector space, let ⃗ b be an ⃗ e-sorted sequence of elements of F ∪ V, and let ⃗ a be an ⃗ e-sorted reduction of ⃗ b. If ⃗ e ∈ Ω with n * ∈ ω being least such that ⃗ e − (n * + 1) is constant, then ⃗ a(i) = ⃗ b(i) for each i ≤ n * .
Proof. For any such sort ⃗ e, the eventually constant value can be either 0 or 1. The proofs of both cases are similar and we will only provide for the case that is eventually constant with value 0. In such a case, ⃗ e(n * ) = 1 and ⃗ e(i) = 0 for all i > n * . To begin the proof, for each i ≤ n * , let ⃗ b i and f i ∈ OT(F ) be such that ⃗ a(i) = f i (b i ) and ⃗ b 1 * ⋯ * ⃗ b n * is a subsequence of ⃗ b. Let the leading term of ⃗ b n * be ⃗ b(j). We claim that n * = j. Clearly, n * ≤ j. Now assume that n * < j. Then, we would have f n * operating on scalars, yielding a scalar (Fact 2, Lemma 6.1), contradicting ⃗ a(n * ) being a vector. That f n * is unary follows from Fact 4 of Lemma 6.1.
It now follows by the pigeonhole principle that f i for each i < n * is also unary, consequently each of the f i for i ≤ n * , is an identity function. Clearly, the equality ⃗ a(i) = ⃗ b(i) holds for each i ≤ n * .
The case where ⃗ e ∈ Ω is less trivial. The first proposition for such a case concerns vector spaces over finite fields. In the following proof, we say that a finite sequence σ is a reduction of the finite sequence τ if, for each i ∈ {0, . . . , σ − 1}, there exist f i ∈ OT(F ) and a subsequence τ i of τ such that (1) τ 0 * ⋯ * τ σ −1 is a subsequence of τ and Proposition 6.2. Every vector space over a finite field is an ⃗ e-Ramsey algebra for all ⃗ e ∈ Ω.
Proof. Fix ⃗ e. Let X ⊆ A ⃗ e(0) be given and let ⃗ b be an ⃗ e-sorted sequence of elements of F ∪ V. Since there are infinitely many scalars in ⃗ b and the field F is finite, there exists a scalar, call it ρ, which occurs infinitely often in ⃗ b. The scalar ρ has a finite order, meaning there exists an s ∈ Z + such that sρ = 0.
We construct an ⃗ e-sorted reduction ⃗ a of ⃗ b recursively. ⟨⃗ a(0), . . . , ⃗ a(N)⟩ has been constructed as a reduction of some initial segment that ⟨⃗ a(0), . . . , ⃗ a(N + 1)⟩ is a reduction of some initial segment ⃗ b extending σ (in the case ⃗ e(N +1) = 0, we use the fact that there are infinitely many, and thus at least s many, ρ's in ⃗ b − σ). Now, since all scalar terms are 0 in ⃗ a, by Facts 3 and 5 of Lemma 6.1, it follows that we have FR ⃗ e F (⃗ a) = {0} and the homogeneity of ⃗ a for X clearly holds.
(Case ⃗ e(0) = 1.) Our plan is to start from the given ⃗ b and obtain ⃗ a ≤ F ⃗ c ≤ F ⃗ b of the same sort ⃗ e so that ⃗ a is homogeneous for X. In the process, we will make use of an auxiliary sequence ⃗ a ′ . The first step is, in a manner similar to the previous case, to reduce ⃗ b to the sequence ⃗ c having the property that its scalar terms are all the zero scalar.
The next step is to obtain an auxiliary sequence ⃗ a ′ homogeneous for X with respect to {+ V } ⊆ F , obtained by considering the subsequence of vector terms of ⃗ c and applying Proposition 3.1 (recall that (V, + V ) is a semigroup).
To obtain ⃗ a, set ⃗ a(0) = ⃗ a ′ (0). Suppose ⟨⃗ a(0), . . . , ⃗ a(N)⟩ has been constructed from ⃗ c as a reduction of an initial segment σ of ⃗ c and such that the subsequence of vector terms of ⟨⃗ a(0), . . . , ⃗ a(N)⟩ forms a sub- c) and such that the subsequence of vector terms of ⟨⃗ a(0), . . . , ⃗ a(N), ⃗ a ′ (N ′ )⟩ forms a subsequence of ⃗ a ′ . This extended sequence is yet another reduction of an extended initial segment of ⃗ c. Thus, ⃗ a is an ⃗ e-sorted reduction of ⃗ c. Now, all the scalar terms of ⃗ a are 0, whereas its vector terms form a subsequence of ⃗ a ′ . Thus, by Fact 5 of Lemma 6.1, if f ∈ OT(F ) is vector-valued and ⃗ q is a finite subsequence of ⃗ a, then f (q) is a sum of some vectors in ⃗ q. It follows by the homogeneity of ⃗ a ′ with respect to X that f (q) ∈ X for all such ⃗ q and f ∈ OT(F ), or f (q) ∈ X for all such ⃗ q and f ∈ OT(F ). Therefore, FR ⃗ e F (⃗ a) ⊆ X or FR ⃗ e F (⃗ a) ⊆ X C by Eq. 3. The story is different when F is infinite. We begin with two lemmas. Lemma 6.3. Suppose (F, + F , × F ) is an infinite field. There exists a sequence ⃗ β ∈ ω F such that for every orderly terms f, g, f ′ , g ′ over {+ F , × F } and for every finite subsequences ⃗ β 0 * ⃗ β 1 and ⃗ β 2 * ⃗ β 3 of ⃗ β, the following holds: Such sequence ⃗ β exists owing to the growth rate of the orderly terms over the field operations. To construct ⃗ β recursively, we can begin with two distinct field elements almost arbitrarily. The orderly terms that can apply to these elements are limited to the field operations themselves. To construct the third element, we only need to pick a field element outside the image set of the first two elements under the field operations with an added requirement, namely the element picked must satisfy Inequation (4). Such procedure is carried out ad infinitum where, at each step, the next element is picked from outside the image set of the previous elements under the orderly terms that can operate on them, the number of available orderly terms which is finite, and explicitly requiring Inequation (4) to hold thus far. Each recursive step can proceed because of the assumption that the field is infinite. For a complete proof, see Lemma 5.4 of [10].
Corollary 6.1. No infinite field is a Ramsey algebra.
The proof of this corollary hinges upon a sequence ⃗ β given in Lemma 6.3 and the associated subset Y of the field F: where Ψ(f, g) is the statement "f, g ∈ OT({+ F , × F }) and ⃗ β 0 * ⃗ β 1 is a finite subsequence of ⃗ β." A moment's reflection reveals that, for each f, g ∈ OT({+ F , × F }) and each finite subsequence Thus, given an arbitrary β is homogeneous for Y , whence the corollary follows.
Lemma 6.4. Let F be infinite and ⃗ e a given sort. Suppose ⃗ β ∈ ω F and suppose v ∈ V is a fixed nonzero vector. Define Then the following holds: If F is an orderly term over F and ⃗ b 0 is some finite subsequence of ⃗ b, then there exists some ⃗ b 0 -ary orderly term f over {+ F , × F } such that Proof. When Rn(F ) ⊆ F, Lemma 6.1 implies that Dom(F ) must be the Cartesian product F n . We may let f = F . As for the case Rn(F ) ⊆ V, we prove the conclusion of the lemma by induction on the length of the subsequence ⟨ ⃗ b(i 1 ), . . . , ⃗ b(i n )⟩. Starting from n = 1, by Fact 1 of Lemma 6.1 and clearly id F ∈ OT({+ F , × F }).
For the inductive step, let F be of arity N + 1 and ⟨ ⃗ b(i 1 ), . . . , ⃗ b(i N +1 )⟩ some subsequence of ⃗ b of length N +1. Breaking up F into its components gives for some g ∈ F and h 1 , h 2 ∈ OT(F ). Either g is vector addition or g is scalar multiplication (g cannot be a field operation because Rn(f ) ⊆ V by case assumption).
If g is vector addition, then h 1 and h 2 are vector-valued. The induction hypothesis applies to give On the other hand, if g is scalar multiplication, then h 1 is scalar-valued and h 2 is vector-valued, and the induction hypothesis applies to give Call a vector space nontrivial if V ≠ {O}.
Proposition 6.3. Suppose that ⃗ e ∈ Ω. Then, no nontrivial vector space over an infinite field is an ⃗ e-Ramsey algebra.
Proof. Let ⃗ β be a fixed sequence guaranteed by Lemma 6.3 and let Y be the associated set given in Eq. 5. Pick a nonzero v ∈ V and define ⃗ b by , an application of the preceding lemma then shows that for each i ∈ ω, there exist some orderly term h i over {+ F , × F } and some finite subsequence Fix an arbitrary ⃗ e-sorted reduction ⃗ a of ⃗ b. It suffices to show that ⃗ a is not homogeneous with respect to X or Y depending on whether ⃗ e ∈ Ω 0 or ⃗ e ∈ Ω 1 , respectively: (When ⃗ e ∈ Ω 1 .) First, pick m 2 > m 1 > 0 such that ⃗ e(m 1 ) = 0 and ⃗ e( (Such ⃗ c 1 and ⃗ c 2 exist as ⃗ e-sorted reductions of ⃗ a since there are infinitely many of both 0's and 1's among the terms of ⃗ e: each of ⃗ c 1 − 1 and ⃗ We dovetail the results above into a classification of vector spaces, thus summarizing the discussion of this section: Theorem 6.1. Let (V, F, + V , + F , × F , ⋅) be a vector space.
(1) In the case where F is a finite field, the vector space is an ⃗ e-Ramsey algebra for all sorts ⃗ e. (2) In the case where F is an infinite field, the vector space is not an ⃗ e-Ramsey algebra for any sorts ⃗ e except when ⃗ e is a nonconstant eventually constant sequence or when it is constant with value 1.

A Related Homogeneous Structure
Scalar multiplication in a vector space (V, F, + V , + F , × F , ⋅) gives rise to a collection of unary functions given by f r (v) = r ⋅ v for each scalar r ∈ F and vector v ∈ V. We may, therefore, consider the algebra (V, F ), where F consists of the vector operation + and each of the functions f r . Notice that the function f 0 induced by the zero scalar sends every vector to the zero vector O. As a consequence, if ⃗ b is an infinite sequence of vectors, we can trivially obtain the infinite sequence consisting of the zero vector as a reduction. The implication of this observation is that (V, F ) is trivially a Ramsey algebra.
To avoid this trivial situation, we consider the algebra (V, K), where K = F ∖ {f 0 }. The symbol K will be reserved for this class of functions in this section.
We proceed with a discussion of the orderly terms over K. First, note that the composition h(x, y) = +(f r (x), f s (y)) is an orderly term. The same is true if we have n functions composed iteratively in this fashion. In particular, if r 1 , r 2 , . . . , r n are nonzero scalars, then is an orderly term over K. The first lemma of this section states that orderly terms over K are of the form given by Eq. 6. Proof. We have seen that ∑ n i=1 r i ⋅x i defines an orderly term over K. As for the converse, we do induction on the generation of the orderly terms over K. Thus, first observe that the claim is clearly true for the orderly term f r of any nonzero scalar r; it is also true when f is the identity function since id V = f 1 . The conclusion is equally trivial for vector addition. Now, suppose g ∈ K and h 1 , h 2 are orderly terms satisfying Eq. 6, i.e. for some nonzero scalars r i , s j , i ∈ {1, . . . , n}, j ∈ {1, . . . , m}, If f (x 1 , . . . , x n , y 1 , . . . , y m ) = g(h 1 (x 1 , . . . , x n ), h 2 (y 1 , . . . , y m )), where g is vector addition, then f (x 1 , . . . , x n , y 1 , . . . , which is clearly of the desired form. On the other hand, given that f (x 1 , . . . , x n ) = f r (h 1 (x 1 , . . . , x n )) for some nonzero scalar r, we are led to f (x 1 , . . . , (rr i ) ⋅ x i and none of the scalars rr i is 0. Hence, such f also assumes the form of Eq. 6. This concludes the proof of the lemma.
A short discussion on linear independence is helpful. Suppose u 1 , . . . , u n are linearly independent vectors and v = r 1 ⋅ u 1 + ⋯ + r n ⋅ u n , then it is easy to check that, if f is an orderly term over K and f (u 1 , . . . , u n ) = v, then f is unique, namely f (x 1 , . . . , x n ) = ∑ n i=1 r i x 1 . We stress that this uniqueness is not met should the vectors u 1 , u 2 , . . . , u n be linearly dependent. Specifically, the zero vector O is not the image of any orderly term over K applied to a tuple consisting of linearly independent vectors.
The first result in this section concerns the structure (V, K) for finite dimensional vector space.
Theorem 7.1. (V, K) is a Ramsey algebra for every finite dimensional vector space.
Proof. Let the vector space be n-dimensional and let ⃗ b ∈ ω V. The proof is by recursion; we will wave our hand on this proof.
Since ⃗ b(0), ⃗ b(1), . . . , ⃗ b(n) are n + 1 vectors, they are linearly dependent and so for some scalars r 0 , r 1 , . . . , r n , not all of which are 0, we obtain ∑ n i=0 r i ⋅ ⃗ b(i) = O (this need not correspond to an orderly term because, if it did, then none of the scalars could be 0). If i 0 < ⋯ < i k , k ≤ n, are indices corresponding to the nonzero scalars, then we have ∑ k j=0 r i j ⋅ ⃗ b(i j ) = O as the image of ( ⃗ b(i 0 ), . . . , ⃗ b(i k )) under the corresponding orderly term.
We repeat the above procedure on the subsequent n + 1 vectors recursively using appropriate nonzero scalars. This generates a sequence of zero vectors O as a reduction of ⃗ b, which is homogeneous for any given X ⊆ V.
We now turn our attention to infinite dimensional vector spaces. To obtain our next theorem, we will be applying Corollary 4.3 of [10]; we state the result here for convenience. For our purpose, the set A stated in the lemma will be the set V of vectors in question and H = {f r ∶ r ∈ F, r ≠ 0}, G = {+} (thus K = H ∪ G). It is easy to verify that the set S = {v ∈ V ∶ f (v) = v for all f ∈ H} of fixed points of H is either V or {O}, depending on whether the underlying field is respectively F 2 or not.
Theorem 7.2. For no infinite dimensional vector space over F ≠ F 2 is (V, K) a Ramsey algebra.
Proof. First, note that (V, G) is a Ramsey algebra since it is a group.
We let ⃗ b = ⟨u 0 , u 1 , . . .⟩, where u 0 , u 1 , . . . are linearly independent vectors. Recall that the zero vector is not the image of any orderly term over K operated on a tuple of linearly independent vectors. As such, for any ⃗ a ≤ K ⃗ b, the element ⃗ a(0), which is not the zero vector, is not in S and hence FR G (⃗ a) ⊆ S for any such ⃗ a. This, by Lemma 7.2, means that the algebra is not a Ramsey algebra.
A direct proof can also be given. The intuition is to pick out a fine bit of vectors to form X. Again, let ⃗ b = ⟨u 0 , u 1 , . . .⟩, where u 0 , u 1 , . . . are linearly independent vectors. Recall from Lemma 7.1 that for each f ∈ OT(K), we have f (x 1 , . . . , x n ) = ∑ n i=1 r i ⋅ x i for some nonzero scalars r 1 , . . . , r n . Let X be defined by where Φ(v) is the statement "v = u i 0 + ∑ n j=1 r j ⋅ u i j , r 1 , . . . , r n are some nonzero scalars, and i 0 < ⋯ < i n ." Given ⃗ a ≤ K ⃗ b, we have ⃗ a(0) = ∑ n j=0 r j ⋅u i j for some i 0 < ⋯ < i n by Lemma 7.1. If r 0 = 1, then ⃗ a(0) ∈ X ∩ FR K (⃗ a), while a simple application of f 2 would put f 2 (⃗ a(0)) ∈ X C ∩ FR K (⃗ a). On the other hand, if r 0 ≠ 1, then we have ⃗ a(0) ∈ X C ∩ FR K (⃗ a), whereas f r −1 0 (⃗ a(0)) ∈ X ∩ FR K (⃗ a). In either case, we have X ∩ FR K (⃗ a) ≠ ∅ and X C ∩ FR K (⃗ a) ≠ ∅. Thus, the conclusion of the theorem holds.
Finally, we also have the following result for vector spaces, finite or infinite, over the field F 2 . Proof. Note that when F = F 2 , we have K = {+, f 1 } = {+, id V }, whence OT(K) is the same as OT({+}). The theorem follows by the fact that (V, +) is a semigroup, hence a Ramsey algebra (Proposition 3.1).

Conclusion
Much of the algebraic and analytical aspects of vector spaces have been studied and utilized ever since the notion of a vector space came into prominence in mathematics and science. This paper offers a glimpse at the less studied combinatorics of vector spaces in the context of a combinatorial notion which is at its infancy stage.
We wrap up with a summary of the results of Section 5 in the language of Ramsey space: The space R ⃗ e (V, F, + V , + F , × F , ⋅) is a Ramsey space if the field F is a finite field; otherwise, it is a Ramsey space only if ⃗ e is constant with value 1. acknowledgements