A New Proof of the Ne\v{s}et\v{r}il-R\"odl Theorem

TL;DR

This paper presents a novel, category theory-based proof of the Nešetřil-R"odl Theorem, simplifying the traditional combinatorial approach by using categorical constructions to derive the result from basic building blocks.

Contribution

It introduces a new proof method for the Nešetřil-R"odl Theorem utilizing category theory, offering a more conceptual and streamlined approach.

Findings

Proof leverages categorical constructions instead of combinatorial complexity

Simplifies understanding of the Nešetřil-R"odl Theorem

Potentially broadens applicability of proof techniques in structural Ramsey theory

Abstract

In this paper we give a new proof of the Ne\v{s}et\v{r}il-R\"odl Theorem, a deep result of discrete mathematics which is one of the cornerstones of the structural Ramsey theory. In contrast to the well-known proofs which employ intricate combinatorial strategies, this proof is spelled out in the language of category theory and the main result follows by applying several simple categorical constructions. The gain from the approach we present here is that, instead of giving the proof in the form of a large combinatorial construction, we can start from a few building blocks and then combine them into the final proof using general principles.

Equations

$$A{<}_{lex}B\text{ iff }A\subset B,$$

$$\text{ A and B are incomparable};$$

$$A{<}_{alex}B\text{ iff }A\subset B,$$

$$\text{ A and B are incomparable};$$

$$A{<}_{\overline{lex}}B\text{ iff }A\supset B,$$

$$\text{ A and B are incomparable}.$$

$$u\cdot v=u[v_1/x_1,v_2/x_2,\dots ,v_m/x_m]\in W_k^n(A)$$

$${\Phi }_{Y,X}:{\mathrm{hom}}_{\mathbf{C}}(F(Y),X)\to {\mathrm{hom}}_{\mathbf{D}}(Y,G(X))$$

$$F(D)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$C$$

$$\bullet \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\bullet$$

$$C$$

$$D$$

$$F:\mathrm{Ob}(\mathbf{H}(b))\rightleftarrows \mathrm{Ob}(\mathbf{GR}(\{0\},X)):G,$$

$$\mathcal{E}_{n}=\big{\{}\{Y_{1},\ldots,Y_{b}\}:Y_{1},\ldots,Y_{b}\in\mathcal{P}(\{1,\ldots,n\})\text{ and }Y_{1}\cap\ldots\cap Y_{b}\neq\varnothing\big{\}}.$$

$${\Phi }_{\mathcal{A},n}:{\mathrm{hom}}_{\mathbf{GR}(\{0\},X)}(F(\mathcal{A}),n)\to {\mathrm{hom}}_{\mathbf{H}(b)}(\mathcal{A},G(n))$$

$$a_i=\bigcup \{X_{\alpha }:i\in D_{\alpha }\},$$

$$\min (a_i\setminus a_j)⩽\min X_{\eta }<\min (a_j\setminus a_i).$$

$$h_i=\{\begin{array}{ll}{x}_j,& f^{-1}(D_i)=D_j^{\prime }\\ 0,& \text{otherwise.}\end{array}$$

$$\text{if }(a_1,a_2,\dots ,a_n)\in R^{\mathcal{A}}\text{ then }{a}_1{<}^{\mathcal{A}}{a}_2{<}^{\mathcal{A}}\dots {<}^{\mathcal{A}}{a}_n.$$

$$({\mathcal{C}}_1,\dots ,{\mathcal{C}}_n)$$

$$R_s^{\mathcal{D}}=\{(d_1,d_2,\dots ,d_{r_s})\in D:(d_1^s,d_2^s,\dots ,d_{r_s}^s)\in R_s^{{\mathcal{C}}_s}\text{ and }{d}_1^1<\dots <d_{r_s}^1\}.$$

$${\phi }_i(b)=(e_i^1(b),\dots ,e_i^n(b))\in D.$$

$${\phi }_i(b)=(e_i^1(b),\dots ,e_i^n(b)){<}_{lex}(e_i^1(b^{\prime }),\dots ,e_i^n(b^{\prime }))={\phi }_i(b^{\prime }).$$

$$(e_i^1\circ u,e_i^2\circ u,\dots ,e_i^n\circ u)=(e_j^1\circ v,e_j^2\circ v,\dots ,e_j^n\circ v).$$

$${\phi }_i\circ u$$

$$=(e_j^1\circ v,e_j^2\circ v,\dots ,e_j^n\circ v)$$

$$={\phi }_j\circ v$$

$$\mathcal{C}⟶(\mathcal{B}{|}_{\Sigma \cup \{<\}}{)}_k^{\mathcal{A}{|}_{\Sigma \cup \{<\}}}.$$

$$\mathrm{tp}(\overline{a})=\{(i,j):a_i⩽a_j\}$$

$$\mathrm{mat}(\overline{a})=(a_{i_1},\dots ,a_{i_s})$$

$$\mathrm{tup}(\sigma ,\overline{b})=(a_1,\dots ,a_r)\in A^r$$

$$a_{\eta }=b_{\xi }\text{ if and only if }\eta {\equiv }_{\sigma }{i}_{\xi }.$$

Full text

Key Words: Ramsey property, relational strucures, categorical constructions

AMS Subj. Classification (2010): 18A10, 05C55

1 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 papers (see [9] for references). We say that a class $\mathbf{K}$ of finite structures has the Ramsey property if the following holds: for any number $k\geqslant 2$ of colors and all $\mathcal{A},\mathcal{B}\in\mathbf{K}$ such that $\mathcal{A}$ embeds into $\mathcal{B}$ there is a $\mathcal{C}\in\mathbf{K}$ such that no matter how we color the copies of $\mathcal{A}$ in $\mathcal{C}$ with $k$ colors, there is a monochromatic copy $\mathcal{B}^{\prime}$ of $\mathcal{B}$ in $\mathcal{C}$ (that is, all the copies of $\mathcal{A}$ that fall within $\mathcal{B}^{\prime}$ are colored by the same color).

One of the cornerstones of the structural Ramsey theory is the Nešetřil-Rödl Theorem which states that the class of all finite linearly ordered relational structures (all having the same, fixed, relational type) has the Ramsey property [1], [10, 11]. The fact that this result has been proved independently by several research teams, and then reproved in various ways and in various contexts [1, 11, 12, 13] clearly demonstrates the importance and justifies the distinguished status this result has in discrete mathematics.

In this paper we give yet another proof of the Nešetřil-Rödl Theorem. In contrast to the well-known proofs which employ intricate combinatorial strategies, our proof starts from (a categorical version of) the Graham-Rothschild Theorem [3] and then transfers the Ramsey property from the Graham-Rothschild category (see Example 2.3 for the definition) to the category of finite linearly ordered relational structures using products of categories, pre-adjunctions (see Section 3 for the definition) and passing to a special subcategory. The gain from the approach we present here is that, instead of giving the proof in the form of a large combinatorial construction, we can start from a few building blocks and then combine them into the final proof using general categorical principles.

In Section 2 we give a brief overview of standard notions referring to first order structures and formulate the Ramsey property in the language of category theory. In Section 3 we discuss the invariance of the Ramsey property under finite products of categories, (a particular form of) pre-adjunctions, and under passing to special subcategories. The corresponding results were proved in [6, 7, 8] but in order to make the paper self-contained we provide brief sketches of the proofs. Finally, in Section 4 we use the four results from Section 3 to present a new, categorical proof of the Nešetřil-Rödl Theorem. Let us underline that in this paper we do not consider the more general version of the theorem which shows the Ramsey property also for classes of structures defined by forbidden substructures.

2 Preliminaries

In order to fix notation and terminology in this section we give a brief overview of standard notions referring to first order structures and formulate the Ramsey property in the language of category theory. For a systematic treatment of category-theoretic notions we refer the reader to [2].

2.1 Structures

Let $\Theta$ be a set of function and relation symbols. A $\Theta$-structure $\mathcal{A}=(A,\Theta^{\mathcal{A}})$ is a set $A$ together with a set $\Theta^{\mathcal{A}}$ of functions and relations on $A$ which are interpretations of the corresponding symbols in $\Theta$. The underlying set of a structure $\mathcal{A}$, $\mathcal{A}_{1}$, $\mathcal{A}^{*}$, … will always be denoted by its roman letter $A$, $A_{1}$, $A^{*}$, … respectively. A structure $\mathcal{A}=(A,\Theta^{\mathcal{A}})$ is finite if $A$ is a finite set.

An embedding $f:\mathcal{A}\hookrightarrow\mathcal{B}$ is an injection $f:A\rightarrow B$ which respects functions, and preserves and reflects the relations. Surjective embeddings are isomorphisms. We write $\mathcal{A}\cong\mathcal{B}$ to denote that $\mathcal{A}$ and $\mathcal{B}$ are isomorphic, and $\mathcal{A}\hookrightarrow\mathcal{B}$ to denote that there is an embedding of $\mathcal{A}$ into $\mathcal{B}$.

A structure $\mathcal{A}$ is a substructure of a structure $\mathcal{B}$ ($\mathcal{A}\leqslant\mathcal{B}$) if the identity map is an embedding of $\mathcal{A}$ into $\mathcal{B}$. Let $\mathcal{A}$ be a structure and $\varnothing\neq B\subseteq A$. Then \hbox{\mathcal{A}}\hbox{\upharpoonright}_{B}=(B,\hbox{\Theta^{\mathcal{A}}}\hbox{\upharpoonright}_{B}) denotes the substructure of $\mathcal{A}$ induced by $B$, where \hbox{\Theta^{\mathcal{A}}}\hbox{\upharpoonright}_{B} denotes the restriction of each function and relation in $\Theta^{\mathcal{A}}$ to $B$. Note that \hbox{\mathcal{A}}\hbox{\upharpoonright}_{B} is not required to exist for every $B\subseteq A$. For example, if $\Theta^{\mathcal{A}}$ contains functions, only those $B$ which are closed with respect to all the functions in $\Theta^{\mathcal{A}}$ qualify for the base set of a substructure.

If $\mathcal{A}$ is a $\Theta$-structure and $\Sigma\subseteq\Theta$ then by \hbox{\mathcal{A}}\hbox{|}_{\Sigma} we denote the $\Sigma$-reduct of $\mathcal{A}$: \hbox{\mathcal{A}}\hbox{|}_{\Sigma}=(A,\Sigma^{\mathcal{A}}).

Let \mathcal{L}=(L,\mbox{<}) be a finite linearly ordered set. For a nonempty $X\subseteq L$ let $\min_{\mathcal{L}}(X)$, resp. $\max_{\mathcal{L}}(X)$, denote the minimum, resp. maximum, of $X$ in $\mathcal{L}$. As a convention we let $\min_{\mathcal{L}}\varnothing=\mathstrut$the top element of $\mathcal{L}$, and $\max_{\mathcal{L}}\varnothing=\mathstrut$the bottom element of $\mathcal{L}$.

Let $\mathrel{<_{\mathit{lex}}}$, $\mathrel{<_{\mathit{alex}}}$ and $\mathrel{<_{\overline{\mathit{lex}}}}$ denote the lexicographic, anti-lexicographic and complemented lexicographic ordering on $\mathcal{P}(L)$, respectively, defined as follows:

[TABLE]

(Note that $A\mathrel{<_{\overline{\mathit{lex}}}}B$ iff $L\setminus A\mathrel{<_{\mathit{lex}}}L\setminus B$, hence the name.) It is easy to see that all these are linear orders on $\mathcal{P}(L)$.

2.2 The Ramsey property in the language of category theory

Let $\mathbf{C}$ be a category and $\mathcal{S}$ a set. We say that $\mathcal{S}=\mathcal{X}_{1}\cup\ldots\cup\mathcal{X}_{k}$ is a $k$-coloring of $\mathcal{S}$ if $\mathcal{X}_{i}\cap\mathcal{X}_{j}=\varnothing$ whenever $i\neq j$. Equivalently, a $k$-coloring of $\mathcal{S}$ is any map $\chi:\mathcal{S}\to\{1,2,\ldots,k\}$. For an integer $k\geqslant 2$ and $A,B,C\in\mathrm{Ob}(\mathbf{C})$, the class of objects of $\mathbf{C}$, we write $C\longrightarrow(B)^{A}_{k}$ to denote that for every $k$-coloring $\hom_{\mathbf{C}}(A,C)=\mathcal{X}_{1}\cup\ldots\cup\mathcal{X}_{k}$ there is an $i\in\{1,\ldots,k\}$ and a morphism $w\in\hom_{\mathbf{C}}(B,C)$ such that $w\cdot\hom_{\mathbf{C}}(A,B)\subseteq\mathcal{X}_{i}$.

Definition 2.1

A category $\mathbf{C}$ has the Ramsey property if for every integer $k\geqslant 2$ and all $A,B\in\mathrm{Ob}(\mathbf{C})$ such that $\hom_{\mathbf{C}}(A,B)\neq\varnothing$ there is a $C\in\mathrm{Ob}(\mathbf{C})$ such that $C\longrightarrow(B)^{A}_{k}$.

Example 2.1

For $b\geqslant 2$ a linearly ordered $b$-uniform hypergraph is a structure \mathcal{A}=(A,E,\mbox{<}) where $A$ is a nonempty set of vertices of $\mathcal{A}$, $E$ is a set of $b$-subsets of $A$ whose elements are called the hyperedges of $\mathcal{A}$ and $<$ is a linear order on $A$. An embedding between two linearly ordered $b$-uniform hypergraphs \mathcal{A}=(A,E,\mbox{<}) and \mathcal{B}=(B,F,\mbox{<}) is an injective map $f:A\to B$ such that $e\in E$ if and only if $f(e)\in F$ for every $e\in E$.

Let ${\mathbf{H}}(b)$, $b\geqslant 2$, denote the category whose objects are finite linearly ordered $b$-uniform hypergraphs and whose morphisms are embeddings. The category ${\mathbf{H}}(b)$ has the Ramsey property for every $b\geqslant 2$ [1, 11].

Example 2.2

Let $\Theta$ be a relational language and let \mbox{<}\notin\Theta be a binary relational symbol. A linearly ordered $\Theta$-structure is a (\Theta\cup\{\mbox{<}\})-structure \mathcal{A}=(A,\Theta^{\mathcal{A}},\mbox{<^{\mathcal{A}}}) where $<^{\mathcal{A}}$ is a linear order on $A$.

By \mathbf{Rel}(\Theta,\mbox{<}) we denote the category whose objects are finite linearly ordered $\Theta$-structures and whose morphisms are embeddings. The category \mathbf{Rel}(\Theta,\mbox{<}) has the Ramsey property. This is the famous Nešetřil-Rödl Theorem [1, 10].

Example 2.3

Let $X=\{x_{1},x_{2},\ldots\}$ be a countably infinite set of variables and let $A$ be a finite alphabet disjoint from $X$. An $m$-parameter word over $A$ of length $n$ is a word $w\in(A\cup\{x_{1},x_{2},\ldots,x_{m}\})^{n}$ satisfying the following:

• •

each of the letters $x_{1},\ldots,x_{m}$ appears at least once in $w$, and

• •

$\min(w^{-1}(x_{i}))<\min(w^{-1}(x_{j}))$ whenever $1\leqslant i<j\leqslant m$.

(Here $w^{-1}(a)$ denotes the set of all the positions in $w$ where $a$ appears.)

Let $W^{n}_{m}(A)$ denote the set of all the $m$-parameter words over $A$ of length $n$. For $u\in W^{n}_{m}(A)$ and $v=v_{1}v_{2}\ldots v_{m}\in W^{m}_{k}(A)$ let

[TABLE]

denote the word obtained by replacing each occurence of $x_{i}$ in $u$ with $v_{i}$, simultaneously for all $i\in\{1,\ldots,m\}$.

Let $\mathbf{GR}(A,X)$ denote the Graham-Rothschild category over $A$ and $X$ whose objects are positive integers 1, 2, …, whose morphisms are given by $\hom(k,n)=W^{n}_{k}(A)$ if $k\leqslant n$ and $\hom(k,n)=\varnothing$ if $k>n$, and where the composition of morphisms is defined in (2.1). For every finite set $A$ and a countably infinite set $X=\{x_{1},x_{2},\ldots\}$ disjoint from $A$ the Graham-Rothschild category $\mathbf{GR}(A,X)$ has the Ramsey property. This is the famous Graham-Rothschild Theorem [3].

3 Transferring the Ramsey property between categories

Our proof of the Nešetřil-Rödl Theorem relies on the idea of transferring the Ramsey property from a category which is known to posses it (such as the Graham-Rothschild category) to the category we are interested in using some “transfer principles”. In this section we collect three such principles proved in [7, 8]. In order to make the paper self-contained we also provide sketches of proofs.

A pair of maps $F:\mathrm{Ob}(\mathbf{D})\rightleftarrows\mathrm{Ob}(\mathbf{C}):G$ is a pre-adjunction between the categories $\mathbf{C}$ and $\mathbf{D}$ [7] provided there is a family of maps

[TABLE]

indexed by the family $\{(Y,X)\in\mathrm{Ob}(\mathbf{D})\times\mathrm{Ob}(\mathbf{C}):\hom_{\mathbf{C}}(F(Y),X)\neq\varnothing\}$ and satisfying the following:

• (PA)

for every $C\in\mathrm{Ob}(\mathbf{C})$, every $D,E\in\mathrm{Ob}(\mathbf{D})$, every $u\in\hom_{\mathbf{C}}(F(D),C)$ and every $f\in\hom_{\mathbf{D}}(E,D)$ there is a $v\in\hom_{\mathbf{C}}(F(E),F(D))$ satisfying $\Phi_{D,C}(u)\cdot f=\Phi_{E,C}(u\cdot v)$.

[TABLE]

(Note that in a pre-adjunction $F$ and $G$ are not required to be functors, just maps from the class of objects of one of the two categories into the class of objects of the other category; also $\Phi$ is just a family of maps between hom-sets satisfying the requirement above.)

Theorem 3.1

[7] Let $\mathbf{C}$ and $\mathbf{D}$ be categories and let $F:\mathrm{Ob}(\mathbf{D})\rightleftarrows\mathrm{Ob}(\mathbf{C}):G$ be a pre-adjunction with $\Phi_{Y,X}:\hom_{\mathbf{C}}(F(Y),X)\to\hom_{\mathbf{D}}(Y,G(X))$ as the corresponding family of maps between hom-sets. Assume that $\mathbf{C}$ has the Ramsey property. Then $\mathbf{D}$ has the Ramsey property.

Proof.

(Sketch) Take any $D,E\in\mathrm{Ob}(\mathbf{D})$ and an integer $k\geqslant 2$. Since $\mathbf{C}$ has the Ramsey property, there is a $C\in\mathrm{Ob}(\mathbf{C})$ such that $C\longrightarrow(F(D))^{F(E)}_{k}$. Let us show that $G(C)\longrightarrow(D)^{E}_{k}$. Take any coloring $\hom_{\mathbf{D}}(E,G(C))=\mathcal{X}_{1}\cup\ldots\cup\mathcal{X}_{k}$ and construct a coloring $\hom_{\mathbf{C}}(F(E),C)=\mathcal{X}^{\prime}_{1}\cup\ldots\cup\mathcal{X}^{\prime}_{k}$ where $\mathcal{X}^{\prime}_{i}=\{u\in\hom_{\mathbf{C}}(F(E),C):\Phi_{E,C}(u)\in\mathcal{X}_{i}\}$. By the choice of $C$ there is a $u\in\hom_{\mathbf{C}}(F(D),C)$ and a $j\in\{1,\ldots,k\}$ such that $u\cdot\hom_{\mathbf{C}}(F(E),F(D))\subseteq\mathcal{X}^{\prime}_{j}$. Then it is easy to show that $\Phi_{D,C}(u)\cdot\hom_{\mathbf{D}}(E,D)\subseteq\mathcal{X}_{j}$. Namely, take any $f\in\hom_{\mathbf{D}}(E,D)$. Since $F:\mathrm{Ob}(\mathbf{D})\rightleftarrows\mathrm{Ob}(\mathbf{C}):G$ is a pre-adjunction, there is a $v\in\hom_{\mathbf{C}}(F(E),F(D))$ such that $\Phi_{D,C}(u)\cdot f=\Phi_{E,C}(u\cdot v)$. But then $u\cdot v\in\mathcal{X}^{\prime}_{j}$, so $\Phi_{E,C}(u\cdot v)\in\mathcal{X}_{j}$. Therefore, $\Phi_{D,C}(u)\cdot f\in\mathcal{X}_{j}$. ∎

In other words, we have just shown that “right pre-adjoints” preserve the Ramsey property. En passant, let us mention that the Ramsey property is invariant under categorical equivalence, and that right adjoints preserve the Ramsey property while left adjoints preserve its dual [6]. (A category $\mathbf{C}$ has the dual Ramsey property if $\mathbf{C}^{\mathrm{op}}$ has the Ramsey property.)

An important transfer principle is the Product Ramsey Theorem for Finite Structures of M. Sokić [14]. We proved this statement in the categorical context in [8] where we used this abstract version to prove that the class of finite permutations has the dual Ramsey property.

Theorem 3.2

[8] Let $\mathbf{C}_{1}$ and $\mathbf{C}_{2}$ be categories such that $\hom_{\mathbf{C}_{i}}(A,B)$ is finite for all $A,B\in\mathrm{Ob}(\mathbf{C}_{i})$, $i\in\{1,2\}$. If $\mathbf{C}_{1}$ and $\mathbf{C}_{2}$ both have the Ramsey property then $\mathbf{C}_{1}\times\mathbf{C}_{2}$ has the Ramsey property.

Consequently, if $\mathbf{C}_{1},\ldots,\mathbf{C}_{n}$ are categories with the Ramsey property then the category $\mathbf{C}_{1}\times\ldots\times\mathbf{C}_{n}$ has the Ramsey property.

Proof.

(Sketch) Take any $k\geqslant 2$ and $\tilde{A}=(A_{1},A_{2})$, $\tilde{B}=(B_{1},B_{2})$ in $\mathrm{Ob}(\mathbf{C}_{1}\times\mathbf{C}_{2})$ such that $\hom(\tilde{A},\tilde{B})\neq\varnothing$. Take $C_{1}\in\mathrm{Ob}(\mathbf{C}_{1})$ and $C_{2}\in\mathrm{Ob}(\mathbf{C}_{2})$ so that $C_{1}\longrightarrow(B_{1})^{A_{1}}_{k}$ and $C_{2}\longrightarrow(B_{2})^{A_{2}}_{k^{t}}$, where $t$ is the cardinality of $\hom_{\mathbf{C}_{1}}(A_{1},C_{1})$. Put $\tilde{C}=(C_{1},C_{2})$. To show that $\tilde{C}\longrightarrow(\tilde{B})^{\tilde{A}}_{k}$ take any coloring $\chi:\hom_{\mathbf{C}_{1}\times\mathbf{C}_{2}}({\tilde{A}},{\tilde{C}})\to\{1,\ldots,k\}$. This coloring uniquely induces the $k^{t}$-coloring $\chi^{\prime}:\hom_{\mathbf{C}_{2}}({A_{2}},{C_{2}})\to\{1,\ldots,k\}^{\hom_{\mathbf{C}_{1}}({A_{1}},{C_{1}})}$. By construction, $C_{2}\longrightarrow(B_{2})^{A_{2}}_{k^{t}}$, so there is a $w_{2}:B_{2}\to C_{2}$ such that $w_{2}\cdot\hom_{\mathbf{C}_{2}}({A_{2}},{B_{2}})$ is $\chi^{\prime}$-monochromatic. Define $\chi^{\prime\prime}:\hom_{\mathbf{C}_{1}}({A_{1}},{C_{1}})\to\{1,\ldots,k\}$ by $\chi^{\prime\prime}(e_{1})=\chi(e_{1},e)$ for some $e\in w_{2}\cdot\hom_{\mathbf{C}_{2}}({A_{2}},{B_{2}})$. (Note that $\chi^{\prime\prime}$ is well defined because $w_{2}\cdot\hom_{\mathbf{C}_{2}}({A_{2}},{B_{2}})$ is $\chi^{\prime}$-monochromatic.) Since $\mathbf{C}_{1}$ has the Ramsey property there is a morphism $w_{1}:B_{1}\to C_{1}$ such that $w_{1}\cdot\hom_{\mathbf{C}_{1}}({A_{1}},{B_{1}})$ is $\chi^{\prime\prime}$-monochromatic. It is now easy to show that for $\tilde{w}=(w_{1},w_{2})$ we have that $\tilde{w}\cdot\hom_{\mathbf{C}_{1}\times\mathbf{C}_{2}}({(A_{1},A_{2})},{(B_{1},B_{2})})$ is $\chi$-monochromatic. ∎

Finally, we shall also need a way to transfer the Ramsey property from a category to its subcategory. (For many deep results obtained in this fashion see [5].) In [8] we devised a simple result which enables us to transfer the Ramsey property from a category to its (not necessarily full) subcategory, as follows.

A diagram in a category $\mathbf{C}$ is a functor $F:\Delta\to\mathbf{C}$ where the category $\Delta$ is referred to as the shape of the diagram. We shall say that a diagram $F:\Delta\to\mathbf{C}$ is consistent in $\mathbf{C}$ if there exists a $C\in\mathrm{Ob}(\mathbf{C})$ and a family of morphisms $(e_{\delta}:F(\delta)\to C)_{\delta\in\mathrm{Ob}(\Delta)}$ such that for every morphism $g:\delta\to\gamma$ in $\Delta$ we have $e_{\gamma}\cdot F(g)=e_{\delta}$:

[TABLE]

We say that $C$ together with the family of morphisms $(e_{\delta})_{\delta\in\mathrm{Ob}(\Delta)}$ forms a compatible cone in $\mathbf{C}$ over the diagram $F$.

A binary category is a finite, acyclic, bipartite digraph with loops 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 (modulo loops):

[TABLE]

A binary diagram in a category $\mathbf{C}$ is a functor $F:\Delta\to\mathbf{C}$ where $\Delta$ is a binary category, $F$ takes the bottom row of $\Delta$ onto the same object, and takes the top row of $\Delta$ onto the same object, Fig. 1. A subcategory $\mathbf{D}$ of a category $\mathbf{C}$ is closed for binary diagrams if every binary diagram $F:\Delta\to\mathbf{D}$ which is consistent in $\mathbf{C}$ is also consistent in $\mathbf{D}$.

Theorem 3.3

[8] Let $\mathbf{C}$ be a category such that every morphism in $\mathbf{C}$ is monic and such that $\hom_{\mathbf{C}}(A,B)$ is finite for all $A,B\in\mathrm{Ob}(\mathbf{C})$, and let $\mathbf{D}$ be a (not necessarily full) subcategory of $\mathbf{C}$. If $\mathbf{C}$ has the Ramsey property and $\mathbf{D}$ is closed for binary diagrams, then $\mathbf{D}$ has the Ramsey property.

Proof.

(Sketch) Take any $k\geqslant 2$ and $A,B\in\mathrm{Ob}(\mathbf{D})$ such that $\hom_{\mathbf{D}}(A,B)\neq\varnothing$. Since $\mathbf{D}$ is a subcategory of $\mathbf{C}$ and $\mathbf{C}$ has the Ramsey property, there is a $C\in\mathrm{Ob}(\mathbf{C})$ such that $C\longrightarrow(B)^{A}_{k}$.

Let us now construct a binary diagram in $\mathbf{D}$ as follows. Let $\hom_{\mathbf{C}}(B,C)=\{e_{1},e_{2},\ldots,e_{n}\}$. Intuitively, for each $e_{i}\in\hom_{\mathbf{C}}(B,C)$ we add a copy of $B$ to the diagram, and whenever $e_{i}\cdot u=e_{j}\cdot v$ for some $u,v\in\hom_{\mathbf{D}}(A,B)$ we add a copy of $A$ to the diagram together with two arrows: one going into the $i$th copy of $B$ labelled by $u$ and another one going into the $j$th copy of $B$ labelled by $v$:

[TABLE]

Note that, by the construction, this diagram is consistent in $\mathbf{C}$, so, by the assumption, it is consistent in $\mathbf{D}$ as well. Therefore, there is a $D\in\mathrm{Ob}(\mathbf{D})$ and morphisms $f_{i}:B\to D$, $1\leqslant i\leqslant n$, such that the following diagram in $\mathbf{D}$ commutes:

[TABLE]

Let us show that in $\mathbf{D}$ we have $D\longrightarrow(B)^{A}_{k}$. Take any $k$-coloring $\hom_{\mathbf{D}}(A,D)=\mathcal{X}_{1}\cup\ldots\cup\mathcal{X}_{k}$, and define a $k$-coloring $\hom_{\mathbf{C}}(A,C)=\mathcal{X}^{\prime}_{1}\cup\ldots\cup\mathcal{X}^{\prime}_{k}$ as follows. For $j\in\{2,\ldots,k\}$ let $\mathcal{X}^{\prime}_{j}=\{e_{s}\cdot u:1\leqslant s\leqslant n,u\in\hom_{\mathbf{D}}(A,B),f_{s}\cdot u\in\mathcal{X}_{j}\}$, and then let $\mathcal{X}^{\prime}_{1}=\hom_{\mathbf{C}}(A,C)\setminus\bigcup_{j=2}^{k}\mathcal{X}^{\prime}_{j}$. Since $C\longrightarrow(B)^{A}_{k}$, there is an $e_{\ell}\in\hom_{\mathbf{C}}(B,C)$ and a $j$ such that $e_{\ell}\cdot\hom_{\mathbf{C}}(A,B)\subseteq\mathcal{X}^{\prime}_{j}$. Then it easily follows that $f_{\ell}\cdot\hom_{\mathbf{D}}(A,B)\subseteq\mathcal{X}_{j}$. ∎

4 The Nešetřil-Rödl Theorem

We are now ready to present a new proof of the Nešetřil-Rödl Theorem.

Let us start by showing that for every $b\geqslant 2$ the category ${\mathbf{H}}(b)$ has the Ramsey property. The proof that we present here is an instance of a more general phenomenon which we addressed in more detail in [7] and where the main idea of the proof comes from.

Theorem 4.1

[1, 11] For every $b\geqslant 2$ the category ${\mathbf{H}}(b)$ has the Ramsey property.

Proof.

Fix a $b\geqslant 2$. In order to prove the theorem it suffices to show that there is a pre-adjunction

[TABLE]

where $X$ is a countably infinite set of variables disjoint from $\{0\}$. The result then follows from Theorem 3.1 and the fact that the category $\mathbf{GR}(\{0\},X)$ has the Ramsey property (Example 2.3).

Let \mathcal{A}=(A,E,\mbox{<}) be a finite linearly ordered $b$-uniform hypergraph. A downset in $\mathcal{A}$ is either a singleton $\{a\}$ where $a\in A$, or a subset $D$ of $A$ such that $|D|\geqslant 2$ and $D\subseteq e$ for some $e\in E$.

For an $\mathcal{A}\in\mathrm{Ob}({\mathbf{H}}(b))$ let $F(\mathcal{A})=\mathstrut$the number of distinct nonempty downsets in $\mathcal{A}$. On the other hand, for a positive integer $n$ let G(n)=\big{(}\mathcal{P}(\{1,\ldots,n\}),\mathcal{E}_{n},\mbox{\mathrel{<_{\overline{\mathit{lex}}}}}\big{)} where

[TABLE]

For a finite linearly ordered $b$-uniform hypergraph $\mathcal{A}$ and a positive integer $n$ define

[TABLE]

as follows. Let \mathcal{A}=(\{1,2,\ldots,k\},E,\mbox{<}) where $<$ is the usual ordering of the integers. Let $D_{1}$, …, $D_{m}$ be all the nonempty downsets in $\mathcal{A}$ and let $D_{1}\mathrel{<_{\mathit{alex}}}D_{2}\mathrel{<_{\mathit{alex}}}\ldots\mathrel{<_{\mathit{alex}}}D_{m}$. For $u\in\hom_{\mathbf{GR}(\{0\},X)}(F(\mathcal{A}),n)=W^{n}_{m}(\{0\})$, let $X_{i}=u^{-1}(x_{i})$, $1\leqslant i\leqslant m$, and let

[TABLE]

$1\leqslant i\leqslant k$. Put $\Phi_{\mathcal{A},n}(u)=\hat{u}$ where $\hat{u}:\mathcal{A}\to G(n):i\mapsto a_{i}$.

To show that the definition of $\Phi$ is corect we have to show that for every $u\in W^{n}_{m}(\{0\})$ the mapping $\hat{u}$ is an embedding $\mathcal{A}\hookrightarrow G(n)$. Take any $i_{1},\ldots,i_{b}\in A$ such that $\{i_{1},\ldots,i_{b}\}\in E$ and let $\{i_{1},\ldots,i_{b}\}=D_{\alpha}$. Then $a_{i_{1}}\cap\ldots\cap a_{i_{b}}\supseteq X_{\alpha}\neq\varnothing$, so $\{a_{i_{1}},\ldots,a_{i_{b}}\}\in\mathcal{E}_{n}$.

On the other hand, assume that $\{a_{i_{1}},\ldots,a_{i_{b}}\}\in\mathcal{E}_{n}$. Then $a_{i_{1}}\cap\ldots\cap a_{i_{b}}\neq\varnothing$. Since each $a_{j}$ is a union of some $X_{\alpha}$’s and all the $X_{\alpha}$’s are pairwise disjoint, it follows that $a_{i_{1}}\cap\ldots\cap a_{i_{b}}$ is also a union of some $X_{\alpha}$’s. Therefore, there is a $\beta$ such that $a_{i_{1}}\cap\ldots\cap a_{i_{b}}\supseteq X_{\beta}$. Then $i_{1},\ldots,i_{b}\in D_{\beta}$. Since $i_{1},\ldots,i_{b}$ are pairwise distinct and $|D_{\beta}|\leqslant b$ by the definition of the downset, it follows that $D_{\beta}=\{i_{1},\ldots,i_{b}\}$. Since the only $b$-elements downsets of $\mathcal{A}$ are hyperedges of $\mathcal{A}$, it follows that $\{i_{1},\ldots,i_{b}\}\in E$.

Finally, let us show that $i<j$ implies $a_{i}\mathrel{<_{\overline{\mathit{lex}}}}a_{j}$. Since both $\{i\}$ and $\{j\}$ are downsets in $\mathcal{A}$ there exist $\eta$ and $\xi$ such that $D_{\eta}=\{i\}$ and $D_{\xi}=\{j\}$. Then $X_{\eta}\subseteq a_{i}$ and $X_{\eta}\cap a_{j}=\varnothing$, while $X_{\xi}\subseteq a_{j}$ and $X_{\xi}\cap a_{i}=\varnothing$, whence follows that $a_{i}$ and $a_{j}$ are incomparable as sets. Note also that $\{i\}\mathrel{<_{\mathit{alex}}}\{j\}\mathrel{<_{\mathit{alex}}}D$ for every downset $D$ in $\mathcal{A}$ such that $D\ni j$, so

[TABLE]

Therefore, $a_{i}\mathrel{<_{\overline{\mathit{lex}}}}a_{j}$.

So, the definition of $\Phi$ is correct. We still have to show that this family of maps satisfies the requirement (PA).

Let \mathcal{B}=(\{1,2,\ldots,\ell\},F,\mbox{<}) be a finite linearly ordered $b$-uniform hypergraph that embeds into $\mathcal{A}$. Let $D^{\prime}_{1}$, …, $D^{\prime}_{d}$ be all the nonempty downsets in $\mathcal{B}$ and let $D^{\prime}_{1}\mathrel{<_{\mathit{alex}}}D^{\prime}_{2}\mathrel{<_{\mathit{alex}}}\ldots\mathrel{<_{\mathit{alex}}}D^{\prime}_{d}$. Take any embedding $f:\mathcal{B}\hookrightarrow\mathcal{A}$ and let us show that there is a word $h=h_{1}h_{2}\ldots h_{m}\in W^{m}_{d}(\{0\})$ such that $\Phi_{\mathcal{A},n}(u)\circ f=\Phi_{\mathcal{B},n}(u\cdot h)$.

Define $h=h_{1}h_{2}\ldots h_{m}\in W^{m}_{d}(\{0\})$ as follows:

[TABLE]

Let us first show that $h$ is indeed a $d$-parameter word. It is easy to see that every downset in $\mathcal{B}$ is an inverse image of a downset in $\mathcal{A}$ so each of the variables $x_{1},\ldots,x_{d}$ appears at least once in $h$.

Let us show that $\min(h^{-1}(x_{\alpha}))<\min(h^{-1}(x_{\beta}))$ whenever $1\leqslant\alpha<\beta\leqslant d$. Take $\alpha$, $\beta$ such that $1\leqslant\alpha<\beta\leqslant d$ and let $\min(h^{-1}(x_{\beta}))=q$. Since $h_{q}=x_{\beta}$ we know that $f^{-1}(D_{q})=D^{\prime}_{\beta}$. Take $p$ so that $D^{\prime}_{\alpha}=f^{-1}(D_{p})$. Then $h_{p}=x_{\alpha}$. If $p<q$ then $\min(h^{-1}(x_{\alpha}))\leqslant p<q=\min(h^{-1}(x_{\beta}))$ and we are done. Assume, therefore, that $p>q$. So, we have that $D_{p}\mathrel{>_{\mathit{alex}}}D_{q}$ (because $p>q$) and $f^{-1}(D_{p})\mathrel{<_{\mathit{alex}}}f^{-1}(D_{q})$ (because $\alpha<\beta$ whence $f^{-1}(D_{p})=D^{\prime}_{\alpha}\mathrel{<_{\mathit{alex}}}D^{\prime}_{\beta}=f^{-1}(D_{q})$). Let us show that in this case there exists a $D_{r}$ such that $D_{r}\mathrel{<_{\mathit{alex}}}D_{q}$ and $f^{-1}(D_{r})=f^{-1}(D_{p})$. Put $D_{r}=f(f^{-1}(D_{p}))$ so that $f^{-1}(D_{r})=f^{-1}(D_{p})$. We know that $f^{-1}(D_{p})\mathrel{<_{\mathit{alex}}}f^{-1}(D_{q})$, so $f(f^{-1}(D_{p}))\mathrel{<_{\mathit{alex}}}f(f^{-1}(D_{q}))$. Since $f(f^{-1}(D_{q}))\subseteq D_{q}$ it follows that $D_{r}=f(f^{-1}(D_{p}))\mathrel{<_{\mathit{alex}}}f(f^{-1}(D_{q}))\mathrel{<_{\mathit{alex}}}D_{q}$ (as $\mathrel{<_{\mathit{alex}}}$ extends $\subseteq$). Now, $D_{r}\mathrel{<_{\mathit{alex}}}D_{q}$ implies that $r<q$, while $f^{-1}(D_{r})=f^{-1}(D_{p})=D^{\prime}_{\alpha}$ means that $h_{r}=x_{\alpha}$. Therefore, $\min(h^{-1}(x_{\alpha}))\leqslant r<q=\min(h^{-1}(x_{\beta}))$, which completes the proof that $h$ is a $d$-parameter word.

Let $X^{\prime}_{i}=(u\cdot h)^{-1}(x_{i})$, $1\leqslant i\leqslant d$, and $a^{\prime}_{j}=\bigcup\{X^{\prime}_{\beta}:j\in D^{\prime}_{\beta}\}$, $1\leqslant j\leqslant\ell$. The following is a strightforward but useful observation: if $h^{-1}(x_{\beta})=\{\alpha_{1},\ldots,\alpha_{s}\}$ then $D^{\prime}_{\beta}=f^{-1}(D_{\alpha_{1}})=\ldots=f^{-1}(D_{\alpha_{s}})$ and $X^{\prime}_{\beta}=(u\cdot h)^{-1}(x_{\beta})=u^{-1}(x_{\alpha_{1}})\cup\ldots\cup u^{-1}(x_{\alpha_{s}})=X_{\alpha_{1}}\cup\ldots\cup X_{\alpha_{s}}$.

In order to complete the proof it suffices to show that $a^{\prime}_{j}=a_{f(j)}$ for all $1\leqslant j\leqslant\ell$.

$(\subseteq)$: Take any $X^{\prime}_{\beta}\subseteq a^{\prime}_{j}$. Then $j\in D^{\prime}_{\beta}$. Let $h^{-1}(x_{\beta})=\{\alpha_{1},\ldots,\alpha_{s}\}$. Then $D^{\prime}_{\beta}=f^{-1}(D_{\alpha_{1}})=\ldots=f^{-1}(D_{\alpha_{s}})$, whence $j\in f^{-1}(D_{\alpha_{i}})$ for all $1\leqslant i\leqslant s$. Consequently, $f(j)\in D_{\alpha_{i}}$ for all $1\leqslant i\leqslant s$, so $X_{\alpha_{i}}\subseteq a_{f(j)}$ for all $1\leqslant i\leqslant s$. Finally, $X^{\prime}_{\beta}=X_{\alpha_{1}}\cup\ldots\cup X_{\alpha_{s}}\subseteq a_{f(j)}$.

$(\supseteq)$: Take any $X_{\alpha}\subseteq a_{f(j)}$. Then $f(j)\in D_{\alpha}$ whence $j\in f^{-1}(D_{\alpha})=D^{\prime}_{\beta}$ so $X^{\prime}_{\beta}\subseteq a^{\prime}_{j}$. By the definition of $h$ we have that $h_{\alpha}=x_{\beta}$ whence $X_{\alpha}\subseteq X^{\prime}_{\beta}$. Therefore, $X_{\alpha}\subseteq a^{\prime}_{j}$. ∎

A finite linearly ordered $\Theta$-structure \mathcal{A}=(A,\Theta^{\mathcal{A}},\mbox{<^{\mathcal{A}}}) is absolutely ordered if the following holds for every $R\in\Theta$:

[TABLE]

Let \overrightarrow{\mathbf{Rel}}(\Theta,\mbox{<}) denote the class of all finite absolutely ordered $\Theta$-structures.

Lemma 4.2

The category \overrightarrow{\mathbf{Rel}}(\Theta,\mbox{<}) has the Ramsey property for every finite relational language $\Theta$.

Proof.

Assume, first, that $\Theta=\{R\}$ where $R$ is an $r$-ary relational symbol. Then it is easy to see that the categories \overrightarrow{\mathbf{Rel}}(\{R\},\mbox{<}) and ${\mathbf{H}}(r)$ are isomorphic, so \overrightarrow{\mathbf{Rel}}(\{R\},\mbox{<}) also has the Ramsey property.

Assume, now, that $\Theta=\{R_{1},R_{2},\ldots,R_{n}\}$ is a finite relational language. Let $\mathbf{C}_{i}$ denote the category \overrightarrow{\mathbf{Rel}}(\{R_{i}\},\mbox{<}), $1\leqslant i\leqslant n$. For an object \mathcal{A}=(A,R_{1}^{\mathcal{A}},\ldots,R_{n}^{\mathcal{A}},\mbox{<})\in\mathrm{Ob}(\overrightarrow{\mathbf{Rel}}(\Theta,\mbox{<})) let \mathcal{A}^{(i)}=(A,R_{i}^{\mathcal{A}},\mbox{<})\in\mathrm{Ob}(\mathbf{C}_{i}). As we have just seen each $\mathbf{C}_{i}$ has the Ramsey property, so the product category $\mathbf{C}_{1}\times\ldots\times\mathbf{C}_{n}$ has the Ramsey property by Theorem 3.2. Let $\mathbf{D}$ be the following subcategory of $\mathbf{C}_{1}\times\ldots\times\mathbf{C}_{n}$:

• •

every \mathcal{A}=(A,R_{1}^{\mathcal{A}},\ldots,R_{n}^{\mathcal{A}},\mbox{<})\in\mathrm{Ob}(\overrightarrow{\mathbf{Rel}}(\Theta,\mbox{<})) gives rise to an object $\overline{\mathcal{A}}=(\mathcal{A}^{(1)},\ldots,\mathcal{A}^{(n)})$ of $\mathbf{D}$, and these are the only objects in $\mathbf{D}$;

• •

every morphism $f:\mathcal{A}\to\mathcal{B}$ in \overrightarrow{\mathbf{Rel}}(\Theta,\mbox{<}) gives rise to a morphism $\overline{f}=(f,\ldots,f):\overline{\mathcal{A}}\to\overline{\mathcal{B}}$ in $\mathbf{D}$, and these are the only morphisms in $\mathbf{D}$.

Clearly, the categories $\mathbf{D}$ and \overrightarrow{\mathbf{Rel}}(\Theta,\mbox{<}) are isomorphic, so in order to complete the proof of the lemma it suffices to show that $\mathbf{D}$ has the Ramsey property.

As $\mathbf{D}$ is a subcategory of $\mathbf{C}_{1}\times\ldots\times\mathbf{C}_{n}$ and the latter one has the Ramsey property, following Theorem 3.3 it suffices to show that $\mathbf{D}$ is closed for binary diagrams. Let $F:\Delta\to\mathbf{D}$ be a binary diagram which is consistent in $\mathbf{C}_{1}\times\ldots\times\mathbf{C}_{n}$ and let $(\mathcal{C}_{1},\ldots,\mathcal{C}_{n})$ together with the morphisms $e_{1},\ldots,e_{k}$ be a compatible cone in $\mathbf{C}_{1}\times\ldots\times\mathbf{C}_{n}$ over $F$:

[TABLE]

Recall that \mathcal{C}_{i}=(C_{i},R_{i}^{\mathcal{C}_{i}},\mbox{<}) and that each $e_{i}$ is a tuple $e_{i}=(e_{i}^{1},\ldots,e_{i}^{n})$ where $e_{i}^{s}:\mathcal{B}^{(s)}\hookrightarrow\mathcal{C}_{s}$. Let $D=C_{1}\times\ldots\times C_{n}$ and let $\mathrel{<_{\mathit{lex}}}$ denote the lexicographic order on $D$ induced by the linear orders (C_{i},\mbox{<}), $1\leqslant i\leqslant n$. Let \mathcal{D}=(D,R_{1}^{\mathcal{D}},\ldots,R_{n}^{\mathcal{D}},\mbox{\mathrel{<_{\mathit{lex}}}}) where

[TABLE]

Here, $r_{s}$ is the arity of $R_{s}$ and $d_{j}=(d_{j}^{1},\ldots,d_{j}^{n})$. To see that $\mathcal{D}\in\mathrm{Ob}(\overrightarrow{\mathbf{Rel}}(\Theta))$ it suffices to note that $(d_{1},d_{2},\ldots,d_{r_{s}})\in R_{s}^{\mathcal{D}}$ implies $d_{1}^{1}<\ldots<d_{r_{s}}^{1}$ whence $d_{1}\mathrel{<_{\mathit{lex}}}d_{2}\mathrel{<_{\mathit{lex}}}\ldots\mathrel{<_{\mathit{lex}}}d_{r_{s}}$. Consequently, $\overline{\mathcal{D}}\in\mathrm{Ob}(\mathbf{D})$.

For each morphism $e_{i}=(e_{i}^{1},\ldots,e_{i}^{n})$ let $\varphi_{i}:B\to D$ be the following mapping:

[TABLE]

Let us show that $\varphi_{i}:\mathcal{B}\to\mathcal{D}$ is an embedding for each $i$.

Assume, first, that $b<b^{\prime}$ in $\mathcal{B}$. Then $e_{i}^{1}(b)<e_{i}^{1}(b^{\prime})$ in $\mathcal{C}_{1}$ so

[TABLE]

Assume, now, that $(b_{1},b_{2},\ldots,b_{r_{s}})\in R_{s}^{\mathcal{B}}$. Then $b_{1}<b_{2}<\ldots<b_{r_{s}}$ in $\mathcal{B}$, whence, as we have just seen, $e_{i}^{1}(b_{1})<e_{i}^{1}(b_{2})<\ldots<e_{i}^{1}(b_{r_{s}})$. Moreover, since $e_{i}^{s}:\mathcal{B}^{(s)}\hookrightarrow\mathcal{C}_{s}$ we have that $(e_{i}^{s}(b_{1}),e_{i}^{s}(b_{2}),\ldots,e_{i}^{s}(b_{r_{s}}))\in R_{s}^{\mathcal{C}_{s}}$. Therefore, $(\varphi_{i}(b_{1}),\varphi_{i}(b_{2}),\ldots,\varphi_{i}(b_{r_{s}}))\in R_{s}^{\mathcal{D}}$.

Conversely, assume that $(\varphi_{i}(b_{1}),\varphi_{i}(b_{2}),\ldots,\varphi_{i}(b_{r_{s}}))\in R_{s}^{\mathcal{D}}$. Then$(e_{i}^{s}(b_{1}),e_{i}^{s}(b_{2}),\ldots,e_{i}^{s}(b_{r_{s}}))\in R_{s}^{\mathcal{C}_{s}}$. Since $e^{s}_{i}$ is an embedding we immediately conclude that $(b_{1},b_{2},\ldots,b_{r_{s}})\in R_{s}^{\mathcal{B}}$.

Therefore, $\varphi_{i}:\mathcal{B}\to\mathcal{D}$ is an embedding for each $i$, whence follows that $\overline{\varphi}_{i}:\overline{\mathcal{B}}\to\overline{\mathcal{D}}$ is a morphism in $\mathbf{D}$ for each $i$. To complete the proof we still have to show that $\overline{\varphi}_{i}\circ\overline{u}=\overline{\varphi}_{j}\circ\overline{v}$ whenever $e_{i}\circ\overline{u}=e_{j}\circ\overline{v}$. Assume that $e_{i}\circ\overline{u}=e_{j}\circ\overline{v}$, i.e.

[TABLE]

Then

[TABLE]

whence easily follows that $\overline{\varphi}_{i}\circ\overline{u}=\overline{\varphi}_{j}\circ\overline{v}$. This concludes the proof. ∎

Theorem 4.3** **(Nešetřil-Rödl)

[1, 11] \mathbf{Rel}(\Theta,\mbox{<}) has the Ramsey property for every relational language $\Theta$.

Proof.

Let us first show that \overrightarrow{\mathbf{Rel}}(\Theta,\mbox{<}) has the Ramsey property for every relational language $\Theta$. Fix an arbitrary relational language $\Theta$ such that \mbox{<}\notin\Theta and take any $k\geqslant 2$ and \mathcal{A},\mathcal{B}\in\mathrm{Ob}(\overrightarrow{\mathbf{Rel}}(\Theta,\mbox{<})) such that $\mathcal{A}\hookrightarrow\mathcal{B}$. Since $\mathcal{B}$ is a finite absolutely ordered $\Theta$-structure we have that $R^{\mathcal{B}}=\varnothing$ for every $R\in\Theta$ such that $\mathrm{ar}(R)>|B|$. Moreover, on a finite set there are only finitely many relations whose arities do not exceed $|B|$. Therefore, there exists a finite $\Sigma\subseteq\Theta$ such that for every $R\in\Theta\setminus\Sigma$ we have $R^{\mathcal{B}}=\varnothing$ or $R^{\mathcal{B}}=S^{\mathcal{B}}$ for some $S\in\Sigma$. Since $\mathcal{A}\hookrightarrow\mathcal{B}$ we have the following: if $R^{\mathcal{B}}=\varnothing$ for some $R\in\Theta\setminus\Sigma$ then $R^{\mathcal{A}}=\varnothing$, and if $R^{\mathcal{B}}=S^{\mathcal{B}}$ for some $R\in\Theta\setminus\Sigma$ and $S\in\Sigma$ then $R^{\mathcal{A}}=S^{\mathcal{A}}$.

The category \overrightarrow{\mathbf{Rel}}(\Sigma,\mbox{<}) has the Ramsey property because $\Sigma$ is finite (Lemma 4.2), so there is a \mathcal{C}=(C,\Sigma^{\mathcal{C}},\mbox{<^{\mathcal{C}}})\in\mathrm{Ob}(\overrightarrow{\mathbf{Rel}}(\Sigma,\mbox{<})) such that

[TABLE]

Define \mathcal{C}^{*}=(C,\Theta^{\mathcal{C}^{*}},\mbox{<^{\mathcal{C}^{*}}})\in\mathrm{Ob}(\overrightarrow{\mathbf{Rel}}(\Theta,\mbox{<})) as follows:

• •

$<^{\mathcal{C}^{*}}$ = $<^{\mathcal{C}}$;

• •

if $S\in\Sigma$ let $S^{\mathcal{C}^{*}}=S^{\mathcal{C}}$;

• •

if $R\in\Theta\setminus\Sigma$ and $R^{\mathcal{B}}=\varnothing$ let $R^{\mathcal{C}^{*}}=\varnothing$;

• •

if $R\in\Theta\setminus\Sigma$ and $R^{\mathcal{B}}=S^{\mathcal{B}}$ for some $S\in\Sigma$, let $R^{\mathcal{C}^{*}}=S^{\mathcal{C}^{*}}$.

Clearly, $\mathcal{C}^{*}$ is a finite absolutely ordered $\Theta$-structure and $\mathcal{C}^{*}\longrightarrow(\mathcal{B})^{\mathcal{A}}_{k}$.

Finally, let us show that \mathbf{Rel}(\Theta,\mbox{<}) has the Ramsey property for every relational language $\Theta$. We start by recalling some basic facts about total quasiorders.

A total quasiorder is a reflexive and transitive binary relation such that each pair of elements of the underlying set is comparable. Each total quasiorder $\sigma$ on a set $I$ induces an equivalence relation $\equiv_{\sigma}$ on $I$ and a linear order $\sqsubset_{\sigma}$ on I/\mbox{\equiv_{\sigma}} in a natural way: $i\mathrel{\equiv_{\sigma}}j$ if $(i,j)\in\sigma$ and $(j,i)\in\sigma$, and (i/\mbox{\equiv_{\sigma}})\mathrel{\sqsubset_{\sigma}}(j/\mbox{\equiv_{\sigma}}) if $(i,j)\in\sigma$ and $(j,i)\notin\sigma$.

Let (A,\mbox{<}) be a linearly ordered set, let $r$ be a positive integer, let $I=\{1,\ldots,r\}$ and let $\overline{a}=(a_{1},\ldots,a_{r})\in A^{r}$. Then

[TABLE]

is a total quasiorder on $I$ which we refer to as the type of $\overline{a}$. Assume that $\sigma=\operatorname{tp}(\overline{a})$. Let s=|I/\mbox{\equiv_{\sigma}}| and let $i_{1}$, …, $i_{s}$ be the representatives of the classes of $\equiv_{\sigma}$ enumerated so that (i_{1}/\mbox{\equiv_{\sigma}})\mathrel{\sqsubset_{\sigma}}\ldots\mathrel{\sqsubset_{\sigma}}(i_{s}/\mbox{\equiv_{\sigma}}). Then

[TABLE]

is the matrix of $\overline{a}$. Note that $a_{i_{1}}<\ldots<a_{i_{s}}$.

Conversely, given a matrix and a total quasiorder we can always reconstruct the original tuple as follows. For a total quasiorder $\sigma$ on $I$ such that |I/\mbox{\equiv_{\sigma}}|=s and an $s$-tuple $\overline{b}=(b_{1},\ldots,b_{s})\in A^{s}$ such that $b_{1}<\ldots<b_{s}$ define an $r$-tuple

[TABLE]

as follows. Let $i_{1}$, …, $i_{s}$ be the representatives of the classes of $\equiv_{\sigma}$ enumerated so that (i_{1}/\mbox{\equiv_{\sigma}})\mathrel{\sqsubset_{\sigma}}\ldots\mathrel{\sqsubset_{\sigma}}(i_{s}/\mbox{\equiv_{\sigma}}). Then put

[TABLE]

(In other words, we put $b_{1}$ on all the entries in i_{1}/\mbox{\equiv_{\sigma}}, we put $b_{2}$ on all the entries in i_{2}/\mbox{\equiv_{\sigma}}, and so on.) Then it is a matter of routine to check that

[TABLE]

Now, for a relational language $\Theta$ such that \mbox{<}\notin\Theta let

[TABLE]

be a relational language where \mathrm{ar}(R,\sigma)=|I/\mbox{\equiv_{\sigma}}|. For \mathcal{A}=(A,\Theta^{\mathcal{A}},\mbox{<^{\mathcal{A}}})\in\mathrm{Ob}(\mathbf{Rel}(\Theta,\mbox{<})) define a \mathcal{A}^{\dagger}=(A,X_{\Theta}^{\mathcal{A}^{\dagger}},\mbox{<^{\mathcal{A}^{\dagger}}}) as follows:

[TABLE]

Clearly, \mathcal{A}^{\dagger}\in\mathrm{Ob}(\overrightarrow{\mathbf{Rel}}(X_{\Theta},\mbox{<})). On the other hand, take any \mathcal{B}=(B,X_{\Theta}^{\mathcal{B}},\mbox{<^{\mathcal{B}}})\in\mathrm{Ob}(\overrightarrow{\mathbf{Rel}}(X_{\Theta},\mbox{<})) and define \mathcal{B}^{*}=(B,\Theta^{\mathcal{B}^{*}},\mbox{<^{\mathcal{B}^{*}}})\in\mathrm{Ob}(\mathbf{Rel}(\Theta,\mbox{<})) as follows:

[TABLE]

Because of (4.1) we have that $(\mathcal{A}^{\dagger})^{*}=\mathcal{A}$ and $(\mathcal{B}^{*})^{\dagger}=\mathcal{B}$ for all \mathcal{A}\in\mathrm{Ob}(\mathbf{Rel}(\Theta,\mbox{<})) and all \mathcal{B}\in\mathrm{Ob}(\overrightarrow{\mathbf{Rel}}(X_{\Theta},\mbox{<})). Therefore, the functor

[TABLE]

is an isomorphism between the categories \mathbf{Rel}(\Theta,\mbox{<}) and \overrightarrow{\mathbf{Rel}}(X_{\Theta},\mbox{<}), its inverse being

[TABLE]

Since \overrightarrow{\mathbf{Rel}}(\Theta,\mbox{<}) has the Ramsey property and \mathbf{Rel}(\Theta,\mbox{<}) is isomorphic to \overrightarrow{\mathbf{Rel}}(X_{\Theta},\mbox{<}), it follows immediately that \mathbf{Rel}(\Theta,\mbox{<}) has the Ramsey property. ∎

5 Acknowledgements

The author would like to thank Miodrag Sokić and Jan Hubička for fruitful discussions concerning earlier versions which improved the clarity, correctness and consistency of the paper.

The author gratefully acknowledges the support of the Grant No. 174019 of the Ministry of Education, Science and Technological Development of the Republic of Serbia.