A Dual Ramsey Theorem for Permutations

In 2012 M. Soki\'c proved that the class of all finite permutations has the Ramsey property. Using different strategies the same result was then reproved in 2013 by J. B\"ottcher and J. Foniok, in 2014 by M. Bodirsky and in 2015 yet another proof was provided by M. Soki\'c. Using the categorical reinterpretation of the Ramsey property in this paper we prove that the class of all finite permutations has the dual Ramsey property as well. It was Leeb who pointed out in 1970 that the use of category theory can be quite helpful both in the formulation and in the proofs of results pertaining to structural Ramsey theory. In this paper we argue that this is even more the case when dealing with the dual Ramsey property.


Introduction
Generalizing the classical results of F. P. Ramsey from the late 1920's, the structural Ramsey theory originated at the beginning of 1970's in a series of * Supported by the Grant No. 174019 of the Ministry of Education, Science and Technological Development of the Republic of Serbia. papers (see [12] for references). We say that a class K of finite structures has the Ramsey property if the following holds: for any number k 2 of colors and all A, B ∈ K such that A embeds into B there is a C ∈ K such that no matter how we color the copies of A in C with k colors, there is a monochromatic copy B ′ of B in C (that is, all the copies of A that fall within B ′ are colored by the same color). Many classes of structures were shown to have the Ramsey property: finite linearly ordered graphs [1,14], finite posets together with an additional linear extension of the poset ordering [15], finite linearly ordered metric spaces [13], and so on.
In 2012 M. Sokić proved that the class of all finite permutations has the Ramsey property [18]. Using different strategies the same result was then reproved in 2013 by J. Böttcher and J. Foniok [4], in 2014 by M. Bodirsky [3] and in 2015 yet another proof was provided by M. Sokić. Discussing the Ramsey property in the context of permutations relies on P. J. Cameron's reinterpretation of permutations in model-theoretic terms [5] as follows. From a traditional point of view a permutation of a set A is any bijection f : A → A. If A is finite, say A = {a 1 , a 2 , . . . , a n }, then each permutation f : A → A can be represented as f = a 1 a 2 . . . a n a i 1 a i 2 . . . a in . So, in order to specify a permutation it suffices to specify two linear orders on A: the "standard" order a 1 < a 2 < . . . < a n on A, and the permuted order a i 1 ⊏ a i 2 ⊏ . . . ⊏ a in . In this paper we adopt P. J. Cameron's point of view and say that a permutation is a triple (A, <, ⊏) where < and ⊏ are linear orders on A.
Using the categorical reinterpretation of the Ramsey property as proposed in [11] we prove in this paper that the class of finite permutations has the dual Ramsey property. Instead of embeddings, which are crucial for the notion of a subpermutation in the above "direct" Ramsey result, we shall consider special surjective maps that we refer to as quotient maps for permutations. These quotient maps are strongly motivated by the notion of minors for permutations suggested, in a different context, by E. Lehtonen in [10].
It was Leeb who pointed out in 1970 [9] that the use of category theory can be quite helpful both in the formulation and in the proofs of results pertaining to structural Ramsey theory. In this paper we argue that this is even more the case when dealing with the dual Ramsey property. Our strategy is to take a "direct" Ramsey result, provide a purely categorical proof of the result and then capitalize on the Duality Principle (an intrinsic principle of category theory) which states that if a statement is true in a category C then the dual of the statement is true in the opposite category C op .
In Section 2 we give a brief overview of standard notions referring to finite linearly ordered sets and category theory, and conclude with the reinterpretation of the Ramsey property in the language of category theory. In Section 3 we consider two ways to transfer the Ramsey property from a category to another category. We first show a Ramsey-type theorem for products of categories generalizing thus the Finite Product Ramsey Theorem for Finite Structures of M. Sokić [19], and then prove a simple result which enables us to transfer the Ramsey property from a category to its (not necessarily full) subcategory. Using these two "transfer principles", starting from a categorical reinterpretation of the Finite Dual Ramsey Theorem we infer in Section 4 a dual Ramsey theorem for the category of finite permutations.

Preliminaries
In this section we give a brief overview of standard notions referring to linearly ordered sets and category theory, and conclude with the reinterpretation of the Ramsey property in the language of category theory.

Chains and permutations
A chain is a pair (A, <) where < is a linear (= total) order on A. In case A is finite, instead of (A, <) we shall simply write A = {a 1 < a 2 < . . . < a n }.
It is easy to see that a map f : A → B between two chains (A, <) and (B, <) is an embedding if and only if x < y ⇒ f (x) < f (y) for all x, y ∈ A.
Let (A, <) and (B, ⊏) be chains such that A ∩ B = ∅. Then (A ∪ B, < ⊕ ⊏) denotes the concatenation of (A, <) and (B, ⊏), which is a chain on A∪B such that every element of A is smaller then every element of B, the elements in A are ordered linearly by <, and the elements of B are ordered linearly by ⊏.
A permutation on a set A is a triple (A, <, ⊏) where < and ⊏ are linear orders on A [5]. Again it is easy to see that an embedding of a permutation (A, <, ⊏) into a permutation (B, <, ⊏) is a map f : A → B such that x < y ⇒ f (x) < f (y), and x ⊏ y ⇒ f (x) ⊏ f (y), for all x, y ∈ A.

Categories and functors
In order to keep the paper self-contained, in this section we provide a brief overview of some elementary category-theoretic notions. For a detailed account of category theory we refer the reader to [2].
In order to specify a category C one has to specify a class of objects Ob(C), a set of morphisms hom C (A, B) for all A, B ∈ Ob(C), the identity morphism id A for all A ∈ Ob(C), and the composition of morphisms · so that Example 1. Finite chains and embeddings constitute a category that we denote by Ch.
Example 2. Following [16] we say that a surjection f : (In other words, a rigid surjection maps an initial segment of a chain onto an initial segment of the other chain. Other than that, a rigid surjection is not required to respect the linear orders in question.) The composition of two rigid surjections is again a rigid surjection, so finite chains and rigid surjections constitute a category which we denote by Ch rs .
For a category C, the opposite category, denoted by C op , is the category whose objects are the objects of C, morphisms are formally reversed so that hom C op (A, B) = hom C (B, A), and so is the composition: A functor F : C → D from a category C to a category D maps Ob(C) to Ob(D) and maps morphisms of C to morphisms of D so that F Categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are inverses of one another both on objects and on morphisms.
The product of categories C 1 and C 2 is the category C 1 × C 2 whose objects are pairs ( An oriented multigraph ∆ consists of a collection (possibly a class) of vertices Ob(∆), a collection of arrows Arr(∆), and two maps dom, cod : Arr(∆) → Ob(∆) which assign to each arrow f ∈ Arr(∆) its domain dom(f ) and its codomain cod(f ). If dom(f ) = γ and cod(f ) = δ we write briefly f : γ → δ. Intuitively, an oriented multigraph is a "category without composition". Therefore, each category C can be understood as an oriented multigraph whose vertices are the objects of the category and whose arrows are the morphisms of the category. A multigraph homomorphism between oriented multigraphs Γ and ∆ is a pair of maps (which we denote by the same symbol) F : Ob(Γ) → Ob(∆) and F : Arr Let C be a category. For any oriented multigraph ∆, a diagram in C of shape ∆ is a multigraph homomorphism F : ∆ → C. Intuitively, a diagram in C is an arrangement of objects and morphisms in C that has the shape of ∆. A diagram F : ∆ → C is commutative if morphisms along every two paths between the same nodes compose to give the same morphism.
A diagram F : ∆ → C is has a commutative cocone in C if there exists a C ∈ Ob(C) and a family of morphisms (e δ : F (δ) → C) δ∈Ob(∆) such that for every arrow g : δ → γ in Arr(∆) we have e γ · F (g) = e δ : Fig. 1 for an illustration). We say that C together with the family of morphisms (e δ ) δ∈Ob(∆) is a commutative cocone in C over the diagram F .

The Ramsey property in the language of category theory
Let C be a category and S a set. We say that For an integer k 2 and A, B, C ∈ Ob(C) we write C −→ (B) A k to denote that for every k-coloring hom C (A, C) = Σ 1 ∪ . . . ∪ Σ k there is an i ∈ {1, . . . , k} and a morphism w ∈ hom C (B, C) such that w · hom C (A, B) ⊆ Σ i .  For all positive integers k, a, m there is a positive integer n such that for every n-element set C and every k-coloring of the set C a of all partitions of C with exactly a blocks there is a partition β of C with exactly m blocks such that the set of all partitions from C a which are coarser than β is monochromatic.
Namely, it was observed in [16] that each partition of a finite linearly ordered set can be uniquely represented by the rigid surjection which takes each element of the underlying set to the minimum of the block it belongs to.  [19] where instead of finite tuples of finite sets we deal with finite tuples of finite structures. We shall now provide a further generalization of this result in the form of a Ramsey theorem for products of categories. The benefit of such a general result is that we can then invoke Duality Principle of category theory to automatically infer statements about the dual Ramsey property.

Transferring the Ramsey property between categories
Theorem 6. Let C 1 and C 2 be categories such that hom C i (A, B) is finite for each i ∈ {1, 2} and all A, B ∈ Ob(C i ). (a) If C 1 and C 2 both have the Ramsey property then C 1 × C 2 has the Ramsey property.
(b) If C 1 and C 2 both have the dual Ramsey property then C 1 × C 2 has the dual Ramsey property.
Since hom C 1 ×C 2 (Ã,C) = hom C 1 (A 1 , C 1 ) × hom C 2 (A 2 , C 2 ), the coloring χ uniquely induces the k t -coloring be the k-coloring defined by for some e ∈ w 2 · hom C 2 (A 2 , B 2 ). Note that χ ′′ is independent of the choice of e because w 2 · hom C 2 (A 2 , B 2 ) is χ ′ -monochromatic. Since C 1 has the Ramsey property there is a morphism w 1 : (b) Assume now that both C 1 and C 2 have the dual Ramsey property. Then C op 1 and C op 2 have the Ramsey property, whence C op 1 × C op 2 has the Ramsey property by (a). By definition, the category (C op 1 ×C op 2 ) op = C 1 ×C 2 then has the dual Ramsey property.
Another way of transferring the Ramsey property is from a category to its subcategory. (For many deep results obtained in this fashion see [8].) We shall now present a simple result which enables us to transfer the Ramsey property from a category to its (not necessarily full) subcategory.
Consider a finite, acyclic, bipartite digraph where all the arrows go from one class of vertices into the other and the out-degree of all the vertices in the first class is 2: Such a digraph will be referred to as a binary digraph. A binary diagram in a category C is a diagram F : ∆ → C where ∆ is a binary digraph, F takes the bottom row of ∆ onto the same object, and takes the top row of ∆ onto the same object, Fig. 2. A subcategory D of a category C is closed for binary diagrams if every binary diagram F : ∆ → D which has a commuting cocone in C has a commuting cocone in D. Proof. Take any k 2 and A, B ∈ Ob(D) such that hom D (A, B) = ∅. Since D is a subcategory of C and C has the Ramsey property, there is a C ∈ Ob(C) such that C −→ (B) A k . Let hom C (B, C) = {e 1 , e 2 , . . . , e n }. Let us now construct a binary diagram in D as follows. Intuitively, for each e i ∈ hom C (B, C) we add a copy of B to the diagram, and whenever e i · u = e j · v for some u, v ∈ hom D (A, B) we add a copy of A to the diagram together with two arrows: one going into the ith copy of B labelled by u and another one going into the jth copy of B labelled by v (note that, by the construction, this diagram has a commuting cocone in C): Formally, let ∆ be the binary diagram whose objects are and whose arrows are of the form u : (u, v, i, j) → i and v : (u, v, i, j) → j.
Let F : ∆ → D be the following diagram whose action on objects is: and whose action on morphisms is F (g) = g: / / C As we have already observed in the informal discussion above, the diagram F : ∆ → D has a commuting cocone in C, so, by the assumption, it has a commuting cocone in D. Therefore, there is a D ∈ Ob(D) and morphisms f i : B → D, 1 i n, such that the following diagram in D commutes: and define a k-coloring as follows. For j ∈ {2, . . . , k} let Σ ′ j = {e s · u : 1 s n, u ∈ hom D (A, B), f s · u ∈ Σ j }, and then let Let us show that Σ ′ i ∩ Σ ′ j = ∅ whenever i = j. By the definition of Σ ′ 1 it suffices to consider the case where i 2 and j 2. Assume, to the contrary, that there is an h ∈ Σ ′ i ∩ Σ ′ j for some i = j, i 2, j 2. Then h = e s · u for some s and some u ∈ hom D (A, B) such that f s · u ∈ Σ i and h = e t · v for some t and some v ∈ hom D (A, B) such that f t · v ∈ Σ j . Then e s · u = h = e t · v. Clearly, s = t and we have that (u, v, s, t) ∈ Ob(∆). (Suppose, to the contrary, that s = t. Then e s · u = e t · v implies u = v because e s = e t and all the morphisms in C are monic. But then Σ i ∋ f s · u = f t · v ∈ Σ j , which contradicts the assumption that and Σ i ∩ Σ j = ∅.) Consequently, f s · u = f t · v because D and morphisms f i : B → D, 1 i n, form a commuting cocone over F : ∆ → D in D. Therefore, f s · u = f t · v ∈ Σ i ∩ Σ j , which is not possible.
Since, by construction, C −→ (B) A k , there is an e ℓ ∈ hom C (B, C) and a j such that e ℓ · hom C (A, B) ⊆ Σ ′ j . Let us show that Assume, first, that j 2 and take any u ∈ hom D (A, B). Since e ℓ · u ∈ Σ ′ j it follows by the definition of Σ ′ j that f ℓ · u ∈ Σ j . Assume now that j = 1 and take any u ∈ hom D (A, B). Suppose that f ℓ · u / ∈ Σ 1 . Then f ℓ · u ∈ Σ m for some m 2. But then e ℓ · u ∈ Σ ′ m . On the other hand, e ℓ · u ∈ Σ ′ 1 by assumption (j = 1). This is in contradiction with the construction of Σ ′ 1 .

A dual Ramsey theorem for permutations
Every dual Ramsey theorem relies on some notion of a "surjective structure map". In case of permutations an appropriate notion has been suggested, in a different context, by E. Lehtonen in [10, Section 6] as follows. Let σ = a i 1 ⊏ a i 2 ⊏ . . . ⊏ a in be a permutation of a finite linearly ordered set A = {a 1 < a 2 < . . . < a n } and let Π be a partition of A.
Π denotes the block of Π that contains x and min < means that the minimum is taken with respect to < (the "standard" ordering of A). Then take the tuple f Π (σ) = (f Π (a i 1 ), f Π (a i 2 ), . . . , f Π (a in )) and remove the repeated elements leaving only the first occurrence of each. What remains is a permutation of f Π (A) that we refer to as the quotient of σ by Π.
Example 8. Let A = {0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9} be a finite linearly ordered set and let be a permutation of A. Let Π be the following partition of A: and u • g i = v • g j (because f · C op u = u · C f = u • f in this case). Now, take any x ∈ C ∪ D.
This concludes the proof.