2 Preliminaries

We present our basic notation around sequences. A tuple $\overline {a}$ is a finite sequence $(a_0,\ldots ,a_{n-1})$ for some natural number n, and $|\overline {a}|=n$ is defined to be the length of the tuple. Given a set X and an integer $s \geq 1$ , $X^s$ denotes the set of all s-tuples from X. We define $\textrm {ran}~\overline {a} :=\{a_0,\ldots ,a_{n-1}\}$ . All tuples $\overline {a}$ are assumed to be finite unless said otherwise. For two tuples $\overline {a}_1, \overline {a}_2$ , by $\overline {a}_1 \subseteq \overline {a}_2$ we mean that $\textrm {ran}~\overline {a}_1 \subseteq \textrm {ran}~\overline {a}_2$ as sets. For an integer $k \geq 0$ , $k:=\{0,1,2,\ldots ,k-1\}$ . For an integer $k \geq 1$ a k-coloring of a set X is any function $c: X \rightarrow k$ . We denote the image $\{c(x) : x \in X\}$ by $c(X)$ . Given a function $f: X^n \rightarrow Y$ and ${\overline {\imath }':=(i_k : k<s) \in X^s}$ , we define $f(\overline {\imath }') = (f(i_k) : k<s)$ . Let $f:\mathrm{X}\rightarrow Y$ be an injective function and let $\psi :A\rightarrow B$ be a function between two subsets of X. We will denote by $f(\psi ):f(A)\rightarrow f(B)$ the function defined by $f(\psi ):=\{(f(a),f(\psi (a))):a\in A\}.$ Moreover, if $\psi $ is injective, $f(\psi )$ is also injective.

A signature L is a list of symbols that must be interpreted in any L-structure as either relations or functions of the specified arity. The signature is relational if it consists only of relation symbols, and functional if it contains at least one function symbol. We do not assume that signatures are either finite or relational, unless explicitly stated. The cardinality of a set S is denoted by $|S|$ . Given a structure $\mathcal {A}$ , $L({\mathcal {A}})$ refers to the signature of $\mathcal {A}$ , $|\mathcal {A}|$ refers to the underlying set of $\mathcal {A}$ , and $||\mathcal {A}||$ to the cardinality of the set $|\mathcal {A}|.$ We denote by $\textrm {Th}(\mathcal {A})$ the theory of $\mathcal {A}$ – the set of all $L({\mathcal {A}})$ -sentences true in $\mathcal {A}$ . Given $\overline {a}$ from $\mathcal {A}$ , $\left < \overline {a} \right>_{\mathcal {A}}$ denotes the substructure generated by $\overline {a}$ in $\mathcal {A}$ (it will be the closure of $\overline {a}$ under the n-ary function symbols of $L({\mathcal {A}})$ for all $n \geq 0$ ). Given a (possibly infinite) tuple $\overline {x}$ in 1–1 correspondence with an enumeration $\overline {a}$ of $\mathcal {A}$ , by $\textrm {Diag}_{\mathcal {A}}(\overline {x})$ we mean the set of all $R(x_{i_0},\ldots ,x_{i_{n-1}})$ for relation symbols $R \in L({\mathcal {A}})$ such that $\mathcal {A} \vDash R(a_{i_0},\ldots ,a_{i_{n-1}})$ . For an integer $n \geq 1$ , an n-ary L-formula is a first-order formula with free variables included in the list $x_0,\ldots ,x_{n-1}$ with parameters in $\mathcal{A}$ . We say that a set $X \subseteq {|\mathcal {A}|}^n$ is definable if there exists n and an n-ary $L({\mathcal {A}})$ -formula $\varphi (\overline {x})$ such that for all $\overline {a}$ from $\mathcal {A}$ , $\mathcal {A} \vDash \varphi (\overline {a})$ if and only if $\overline {a} \in X$ , and $0$ -definable if $\varphi (\overline {x})$ may be chosen without parameters from $\mathcal {A}$ . A set $X \subseteq {|\mathcal {A}|}^n$ is quantifier-free definable if it is $0$ -definable by way of a quantifier-free formula $\varphi (\overline {x})$ . A structure $\mathcal {A}$ is $\omega $ -homogeneous if any partial isomorphism between finite subsets of $\mathcal {A}$ can be extended to an automorphism of $\mathcal {A}$ . For the basics of formulas, structures, and Fraïssé theory the reader is referred to [Reference Hodges12, Reference Marker20].

For structures $\mathcal {A}, \mathcal {B}$ , $\mathcal {A} \subseteq \mathcal {B}$ always denotes that $\mathcal {A}$ is a substructure of $\mathcal {B}$ , in which case they are structures in the same signature. The age of a structure $\mathcal {A}$ , $\textrm {age}(\mathcal {A})$ is the set of all finitely generated substructures of $\mathcal {A}$ , modulo $L({\mathcal {A}})$ -isomorphism. We will use Roman letters $A, B$ for finitely generated substructures of a given structure $\mathcal {A}$ . For structures $\mathcal {A}, \mathcal {A}'$ , $\mathcal {A} \cong \mathcal {A}'$ means that the structures are isomorphic (and thus, they are in the same signature). To emphasize the shared signature, we might write $\mathcal {A} \cong _L \mathcal {A}'$ , where $L=L({\mathcal {A}})=L(\mathcal {A}')$ . We say that a structure $\mathcal {A}$ is rigid if the only automorphism of $\mathcal {A}$ is the identity map.

Fix a signature L, an L-structure $\mathcal {A}$ , and an integer $n \geq 1$ .

1. (1) Given a set $\Delta $ of L-formulas, a $\Delta $ -n-type (over $\emptyset $ in $\mathcal {A}$ ) is a set of n-ary formulas from $\Delta $ that is consistent with $\textrm {Th}(\mathcal {A})$ . A complete $\Delta $ -n-type $\pi $ is a $\Delta $ -n-type such that for every n-ary formula $\varphi $ from $\Delta $ , either $\varphi \in \pi $ or $\neg \varphi \in \pi $ . If we drop the use of n, we mean $\Delta $ -n-type for some n.

2. (2) In the case that $\Delta $ is the set of all quantifier-free L-formulas, we call a $\Delta $ -n-type in $\mathcal {A}$ a quantifier-free n-type in $\mathcal {A}$ . In the case that $\Delta $ is the set of all L-formulas, we call a $\Delta $ -n-type in $\mathcal {A}$ an n-type in $\mathcal {A}$ . Such types are described as “complete” if they are complete $\Delta $ -n-types for the appropriate $\Delta $ .

3. (3) An n-type p over $\emptyset $ in $\mathcal {A}$ is realized (in $\mathcal {A}$ ) if there exists $\overline {a} \in {|\mathcal {A}|}^n$ such that $\mathcal {A} \vDash \varphi (\overline {a})$ for all $\varphi \in p$ .

4. (4) A structure $\mathcal {A}$ is $\kappa $ -saturated if for all subsets $S \subseteq |\mathcal {A}|$ such that $|S| < \kappa $ , $\mathcal {A}$ realizes all types in $\mathcal {A}$ over S. We say that $\mathcal {A}$ is saturated, if it is $||\mathcal {A}||$ -saturated.

5. (5) We define $S^{\mathcal {M}}_n(\emptyset )$ to be the space of all complete n-types over $\emptyset $ in $\mathcal {A}$ with the usual Stone topology, with basic open sets $[\psi ] := \{p \in S^{\mathcal {M}}_n(\emptyset ) \mid \psi \in p\}$ .

6. (6) Given a tuple $\overline {a} \in {|\mathcal {A}|}^n$ , $\textrm {tp}_\Delta ^{\mathcal {A}}(\overline {a})$ is the complete $\Delta $ -n-type of $\overline {a}$ in $\mathcal {A}$ . For $\textrm {tp}_\Delta ^{\mathcal {A}}(\overline {a})$ , we write $\textrm {qftp}^{\mathcal {A}}(\overline {a})$ when $\Delta $ is the set of all quantifier-free formulas, and we write $\textrm {tp}^{\mathcal {A}}(\overline {a})$ when $\Delta $ is the set of all formulas.

7. (7) For same-length tuples $\overline {a}, \overline {a}'$ from $\mathcal {A}$ , we will use $\overline {a} \sim _{\mathcal {A}} \overline {a}'$ to mean that $\textrm {qftp}^{\mathcal {A}}(\overline {a})=\textrm {qftp}^{\mathcal {A}}(\overline {a}')$ , $\overline {a} \equiv _\Delta ^{\mathcal {A}} \overline {a}'$ to mean that $\textrm {tp}_\Delta ^{\mathcal {A}}(\overline {a})=\textrm {tp}_\Delta ^{\mathcal {A}}(\overline {a}')$ , and $\overline {a} \equiv ^{\mathcal {A}} \overline {a}'$ to mean that $\textrm {tp}^{\mathcal {A}}(\overline {a})=\textrm {tp}^{\mathcal {A}}(\overline {a}')$ .

2.1 Structural Ramsey theory and topological dynamics

We start this section with some standard definitions from structural Ramsey theory (see [Reference Kechris, Pestov and Todorčević16, Introduction of part (D)] and [Reference Nešetřil24, Reference Nguyen Van Thé28]). Given L-structures $A, B$ we define ${B \choose A}$ to be the set of all substructures $A' \subseteq B$ such that $A' \cong A$ and $\textrm {Emb}(A,B)$ to be the set of all L-embeddings $f:A \rightarrow B$ . Given an L-embedding $h: B \rightarrow C$ , we define $h \circ \textrm {Emb}(A,B) = \{h \circ f : f \in \textrm {Emb}(A,B) \} \subseteq \textrm {Emb}(A,C)$ .

Following [Reference Masulović22], we define two types of Erdős–Rado partition arrow.

Definition 2.3. Given a signature L, L-structures $A, B, \mathcal {M}$ , and integers $k, d \geq 1$ , the notation

$$ \begin{align*}\mathcal{M} \longrightarrow (B)^A_{r,d}\end{align*} $$

denotes that for all r-colorings $c: {\mathcal {M} \choose A} \rightarrow r$ , there exists $B' \subseteq \mathcal {M}$ , $B' \cong B$ , such that $\left |c({B' \choose A})\right | \leq d$ .

We say that the structure $B'$ above is $\leq d$ -chromatic (for the coloring c on copies of A).

If $d=1$ , it will be dropped in the notation, and we will write

$$ \begin{align*}\mathcal{M} \longrightarrow (B)^A_{r}.\end{align*} $$

Moreover,

denotes that for all r-colorings $c: \textrm {Emb}(A,\mathcal {M}) \rightarrow r$ , there exists $h \in \textrm {Emb}(B,\mathcal {M})$ such that $|c(h \circ \textrm {Emb}(A,B))| \leq d$ .

If $d=1$ , it will be dropped in the notation, and we will write

Definition 2.4. Let $\mathcal {K}$ be a class of finitely generated L-structures, for some signature L, and let $A, B \in \mathcal {K}$ . We say that $(A, B)$ is a Ramsey duo for $\mathcal {K}$ if for all integers $r \geq 2$ there exists $C \in \mathcal {K}$ such that

$$ \begin{align*}C \longrightarrow (B)^A_{r} .\end{align*} $$

We say that $B'$ is a copy of B homogeneous for c (on copies of A).

Definition 2.5. We say that $\mathcal {K}$ has the Ramsey property (RP) if for all $A, B \in \mathcal {K}$ , $(A, B)$ is a Ramsey duo for $\mathcal {K}$ .

Example 2.6. The following classes have RP:

1. (1) All finite sets in $L = \emptyset $ [Reference Ramsey30].

2. (2) All finite linear orders in $L= \{<\}$ [Reference Ramsey30].

3. (3) All finite simple graphs with no loops with an ordering on the vertices in $L = \{R,<\}$ [Reference Abramson and Harrington1, Reference Nešetřil and Rödl26].

4. (4) All finite n-regular hypergraphs with linear orders in $L=\{R,<\}$ , where R is n-ary [Reference Abramson and Harrington1, Reference Nešetřil and Rödl26].

5. (5) Convexly ordered finite equivalence relations in $L= \{E,<\}$ (known, see discussion after Corollary 6.8 in [Reference Kechris, Pestov and Todorčević16]).

6. (6) Finite Boolean algebras in $L=\{\vee ,\wedge ,\neg ,\mathbf {0},\mathbf {1}\}$ [Reference Graham and Rothschild11].

Example 2.7. The following classes do not have RP:

1. (1) All finite simple graphs with no loops [Reference Nešetřil and Rödl25].

2. (2) Finite equivalence relations with any ordering on points in $L = \{E,<\}$ (Theorem 6.4 in [Reference Kechris, Pestov and Todorčević16]).

3. (3) Partial orders with any linear ordering on points in $L = \{<,\prec \}$ [Reference Sokić35].

As we can see in examples above, the class of finite graphs does not have the Ramsey property, but its expansion by linear orders does. This phenomenon leads to the notion of a Ramsey degree.

Definition 2.8. Let $\mathcal {K}$ be a class of L-structures and let $A\in \mathcal {K}$ . We say that A has finite Ramsey degree in $\mathcal {K}$ if there is an integer $d \geq 1$ such that for every $B\in \mathcal {K}$ and every $r\geq 2$ , there is $C\in \mathcal {K}$ such that $C \longrightarrow (B)^A_{r,d}$ .

We define $d(A,\mathcal {K})$ to be the least such integer d, if it exists, and otherwise define $d(A,\mathcal {K})=\infty $ . If $d(A,\mathcal {K})$ is finite, it is the Ramsey degree of A in $\mathcal {K}$ .

Observation 2.9. A class $\mathcal {K}$ has RP if and only if $d(A,\mathcal {K})=1$ for all $A \in \mathcal {K}$ .

We can make a related definition for an infinite structure $\mathcal {M}$ .

Definition 2.10. Given an L-structure $\mathcal {M}$ and a finitely generated substructure $A \subseteq \mathcal {M}$ , we say that A has finite small Ramsey degree in $\mathcal {M}$ if for some integer $d \geq 1$ , for all finitely generated structures $B \subseteq \mathcal {M}$ , for all integers $r \geq 2$ ,

$$ \begin{align*}\mathcal{M} \longrightarrow (B)^A_{r,d}.\end{align*} $$

If A has finite small Ramsey degree in $\mathcal {M}$ , then we define $d(A,\mathcal {M})$ to be the least integer d such that for all finitely generated structures $B \subseteq \mathcal {M}$ , for all integers $r \geq 2$ , $\mathcal {M} \longrightarrow (B)^A_{r,d}$ .

If A does not have finite small Ramsey degree in $\mathcal {M}$ , we define $d(A,\mathcal {M}) = \infty $ .

We say that $(A,B)$ is a Ramsey duo for $\mathcal {M}$ if for all integers $r \geq 2$ ,

$$ \begin{align*}\mathcal{M} \rightarrow (B)^A_{r}.\end{align*} $$

Proposition 2.11. Fix a signature L and a locally finite L-structure M. Let $\mathcal {K}:=\textrm {age}(\mathcal {M})$ . Then, for any finite substructures $A, B \subseteq \mathcal {M}$ , $(A,B)$ is a Ramsey duo for $\mathcal {K}$ if and only if $(A,B)$ is a Ramsey duo for $\mathcal {M}$ .

A proof for Proposition 2.11 is straightforward and provided in the Appendix.

Given the ability to do calculations in a countably infinite structure, the following is common usage:

Definition 2.12. We say that $\mathcal {A}$ has RP if $\textrm {age}(\mathcal {A})$ has RP.

Striking connections between dynamics of an automorphism group of an $\omega $ -homogeneous structure $\mathcal {A}$ and Ramsey degrees of $\textrm {age}(\mathcal {A})$ were established in [Reference Kechris, Pestov and Todorčević16]. The work of these authors and others shows that this relationship is best explained in terms of the Ramsey properties of embeddings rather than substructures.

Definition 2.13. Let $\mathcal {K}$ be a class of L-structures and let $A\in \mathcal {K}$ . We say that A has finite Ramsey degree for embeddings in $\mathcal {K}$ if there is an integer $d \geq 1$ such that for every $B\in \mathcal {K}$ and every $r\geq 2$ there is $C\in \mathcal {K}$ such that for every coloring $c:\textrm {Emb}(A,C)\rightarrow \{0,1,\ldots ,r-1\},$ there is $h\in \textrm {Emb}(B,C)$ such that c on $h\circ \textrm {Emb}(A,B)$ takes at most d colors.

We define $d_e(A,\mathcal {K})$ to be the least such integer d, if it exists, and otherwise define $d_e(A,\mathcal {K})=\infty $ . If $d_e(A,\mathcal {K})$ is finite, it is the Ramsey degree for embeddings of A in $\mathcal {K}$ .

If $d_e(A,\mathcal {K})=1$ for all $A \in \mathcal {K}$ , then we say that $\mathcal {K}$ has the Ramsey property for embeddings.

Definition 2.14. Given an L-structure $\mathcal {M}$ and a finitely generated substructure $A \subseteq \mathcal {M}$ , say that A has finite small Ramsey degree for embeddings in $\mathcal {M}$ if for some integer $d \geq 1$ , for all finitely generated structures $B \subseteq \mathcal {M}$ , for all integers $r \geq 2$ ,

We define $d_e(A,\mathcal {M})$ to be the least such integer d, if it exists, and otherwise define $d_e(A,\mathcal {M})=\infty $ . If $d_e(A,\mathcal {M})$ is finite, it is the Ramsey degree for embeddings of A in  $\mathcal {M}$ .

It is well known that the Ramsey property for a class $\mathcal {K}$ that is an age of finite structures can be understood as a property of an infinite structure $\mathcal {M}$ with ${\textrm {age}(\mathcal {M}) = \mathcal {K}}$ . This is proved for a countable age of finite structures in Proposition 3 of [Reference Nguyen Van Thé28] using ultrafilters, and the generalization to Ramsey degrees is proved in the more general setting of a category $\mathbb {C}$ with a distinguished subcategory $\mathbb {C}_{\textrm {fin}}$ satisfying certain assumptions in Lemma 3.4 of [Reference Masulović22]. In the Appendix we provide a model-theoretic argument for Lemma 2.15 by compactness as an alternative approach.

Lemma 2.15. Fix a signature L and a locally finite L-structure $\mathcal {M}$ . Let ${\mathcal {K}:=\textrm {age}(\mathcal {M})}$ . Then, for any finite substructure $A \subseteq \mathcal {M}$ ,

1. (1) $d_e(A,\mathcal {K}) = d_e(A,\mathcal {M})$ , and

2. (2) $d(A,\mathcal {K}) = d(A,\mathcal {M})$ .

The relationship between Ramsey degrees for structures and Ramsey degrees for embeddings has a long history, dating back to work in [Reference Abramson and Harrington1, Reference Fouché10, Reference Nešetřil and Rödl26]. This history is cataloged in the introduction to Section 10 of [Reference Kechris, Pestov and Todorčević16], in which the case of $\mathcal {K}$ having an ordered expansion with RP is worked out in detail. A more general result in a category theory context is offered by Proposition 3.1 of [Reference Masulović22].

Proposition 2.16 [Reference Kechris, Pestov and Todorčević16, Reference Masulović22].

Given a signature L and a class $\mathcal {K}$ of finite L-structures and $A \in \mathcal {K}$ , if $d_e(A,\mathcal {K})$ is finite, then so is $d(A,\mathcal {K})$ and

$$ \begin{align*}d_e(A,\mathcal{K}) = |\textrm{Aut}(A)| \cdot d(A,\mathcal{K}).\end{align*} $$

Below we state the famous Kechris–Pestov–Todorčević correspondence from [Reference Kechris, Pestov and Todorčević16] between the Ramsey property of $\textrm {age}(\mathcal {A})$ for a (countable) $\omega $ -homogeneous structure $\mathcal {A}$ and the fixed point on compacta property of $\textrm {Aut}(\mathcal {A}).$ We start by introducing the necessary notions from topological dynamics.

Let G be a topological group and X a compact Hausdorff space. A continuous function $\alpha :G\times X\to X$ is a G-flow if:

1. (1) $\alpha (e,x)=x$ for any $x\in X$ and e the neutral element of G,

2. (2) $\alpha (gh,x)=\alpha (g,\alpha (h,x))$ for every $g,h\in G$ and $x\in X$ .

In, other words, a flow is a continuous group action, where continuity is considered with respect to the product topology. We typically write $gx$ in place of $\alpha (g,x).$ A G-flow on X is minimal if X does not contain a non-empty proper closed G-invariant subset. A homomorphism between G-flows X and Y is a G-equivariant continuous map $\phi :X\to Y,$ i.e., for every $g\in G$ and $x\in X,$ we have $\phi (gx)=g\phi (x).$ If $\phi $ is onto, we say that Y is a quotient of X and if $\phi $ is bijective, it is called an isomorphism. Ellis showed that up to isomorphism, for every topological group G, there is a unique universal minimal flow, $M(G)$ , that is, a minimal G-flow which has every minimal G-flow as a quotient (see [Reference Ellis8] for discrete and [Reference Ellis9] for arbitrary groups). We call G extremely amenable if every G-flow X has a fixed point—a point $x_0\in X$ such that $gx_0=x_0$ for every $g\in G.$ It immediately follows that G is extremely amenable if and only if $M(G)$ is a single point (and thus every minimal G-flow is a single point).

Theorem 2.17 ([Reference Kechris, Pestov and Todorčević16] for countable structures; [Reference Bartošová3] for uncountable structures).

Let $\mathcal {A}$ be an $\omega $ -homogeneous structure. The following are equivalent.

1. (1) The group $\textrm {Aut}(\mathcal {A})$ is extremely amenable.

2. (2) The class $\textrm {age}(\mathcal {A})$ satisfies the Ramsey property and consists of rigid elements—structures whose automorphism group is trivial.

In fact a consequence of Proposition 2.16 was observed as Proposition 2.3 in [Reference Mašulović and Scow23] stated in a more general category theory context, namely that a class $\mathcal {K}$ of finite structures has the Ramsey property for embeddings if and only if $\mathcal {K}$ has the Ramsey property for substructures and every member of $\mathcal {K}$ is rigid. Thus item (2) in Theorem 2.17 can be replaced by

1. (2)’ $\textrm {age}(\mathcal {A})$ satisfies the Ramsey property for embeddings.

In [Reference Kechris, Pestov and Todorčević16], the authors computed a number of universal minimal flows of automorphism groups of countable $\omega $ -homogeneous structures whose ages have finite Ramsey degrees. In fact, Zucker later proved in [Reference Zucker37] that finite Ramsey degrees are equivalent to the universal minimal flow being metrizable, as stated precisely below.

Theorem 2.18 [Reference Kechris, Pestov and Todorčević16, Reference Zucker37].

Let $\mathcal {A}$ be a countable $\omega $ -homogeneous structure. The following are equivalent.

1. (1) The universal minimal flow $M(\textrm {Aut}(\mathcal {A}))$ is metrizable.

2. (2) The class $\textrm {age}(\mathcal {A})$ has finite Ramsey degrees for embeddings.

2.3 The modeling property

The first use of Definition 2.22 was in [Reference Ehrenfeucht and Mostowski7] to construct models with many automorphisms. The presentation of Definition 2.22 that we adopt in this paper can be found in [Reference Shelah34].

Definition 2.22 [Reference Ehrenfeucht and Mostowski7].

Given a structure $\mathcal {M}$ , a set $\Delta $ of $L(\mathcal {M})$ -formulas, and an integer $n \geq 1$ , a $\Delta $ -n-indiscernible sequence is a sequence of finite l-tuples $\overline {a}_i$ from $\mathcal {M}$ for some integer $l \geq 1$ indexed by some linear order $\mathcal {I} = (|\mathcal {I}|,<)$ such that for all increasing n-tuples $\overline {\imath },\overline {\jmath }$ from $\mathcal {I}$ ,

$$ \begin{align*}\overline{a}_{\overline{\imath}} \equiv_\Delta^{\mathcal{M}} \overline{a}_{\overline{\jmath}} .\end{align*} $$

In the study of classification theory in model theory there has been significant use of a generalization of this notion named “ $\mathcal {I}$ -indexed indiscernible sets” in [Reference Shelah34] which we will define as follows.

Definition 2.23 [Reference Shelah34].

Fix a structure $\mathcal {I}$ , an integer $l \geq 1$ , and l-tuples $\overline {a}_i$ from some structure $\mathcal {M}$ , for all $i \in \mathcal {I}$ . We say that $(\overline {a}_i \mid i \in \mathcal {I})$ is an $\mathcal {I}$ -indexed indiscernible set if for any integer $n \geq 1$ , for all n-tuples $\overline {\imath },\overline {\jmath }$ from $\mathcal {I}$ ,

$$ \begin{align*}\overline{\imath} \sim_{\mathcal{I}} \overline{\jmath} \Rightarrow \overline{a}_{\overline{\imath}} \equiv^{\mathcal{M}} \overline{a}_{\overline{\jmath}} .\end{align*} $$

We say that $(\overline {a}_i \mid i \in \mathcal {I})$ is an $\mathcal {I}$ -indexed indiscernible sequence if $\mathcal {I}$ is an ordered structure, and additionally a generalized indiscernible sequence when $\mathcal {I}$ is clear from context.

We review some definitions and basic results from [Reference Scow33], where the second author gave a more complete proof that the modeling property is a model-theoretic analogue of the Ramsey property.

Definition 2.24 [Reference Scow32].

Given an integer $l \geq 1$ , an $L'$ -structure $\mathcal {I}$ , an L-structure $\mathcal {M}$ and an $\mathcal {I}$ -indexed set of l-tuples from $\mathcal {M}$ , $X = (\overline {a}_i \mid i \in \mathcal {I})$ , we define the EM-type of X ( $\textrm {EMtp}(X)$ ) to be a syntactic type in variables $(\overline {x}_i \mid i \in \mathcal {I})$ , where $|\overline {x}_i| = l$ for each $i \in \mathcal {I}$ , as follows:

$$ \begin{align*}&\textrm{EMtp}(X) = \{ \psi(\overline{x}_{i_0},\ldots,\overline{x}_{i_{n-1}}) \mid \psi \\ &\quad\in L, \overline{\imath} \in {|\mathcal{I}|}^n \textrm{~and~} (\forall \overline{\jmath} \in {|\mathcal{I}|}^n)( \overline{\jmath} \sim_{\mathcal{I}} \overline{\imath} \Rightarrow \mathcal{M} \vDash \psi(\overline{a}_{j_0},\ldots,\overline{a}_{j_{n-1}})) \}.\end{align*} $$

Proposition 2.25 (Proposition 2 of [Reference Scow32]).

Given an $L'$ -structure $\mathcal {I}$ and an L-structure $\mathcal {M}$ , fix sets of l-tuples from $\mathcal {M}$ indexed by $\mathcal {I}$ , $X = (\overline {a}_i \mid i \in \mathcal {I})$ and $Y = (\overline {b}_i \mid i \in \mathcal {I})$ . We have that $Y \vDash \textrm {EMtp}(X)$ if and only if for any integer $n \geq 1$ , for all complete quantifier-free n-types $\eta $ in $\mathcal {I}$ and all $n \cdot l$ -ary formulas $\varphi \in L$ , if we have the rule

$$ \begin{align*}(\forall \overline{\jmath}) ( \mathcal{I} \vDash \eta(\overline{\jmath}) \Rightarrow \mathcal{M} \vDash \varphi(\overline{a}_{\overline{\jmath}})),\end{align*} $$

then we have the rule

$$ \begin{align*}(\forall \overline{\jmath}) ( \mathcal{I} \vDash \eta(\overline{\jmath}) \Rightarrow \mathcal{M} \vDash \varphi(\overline{b}_{\overline{\jmath}})).\end{align*} $$

Definition 2.26 [Reference Scow32].

Fix sequences of parameters $X = (\overline {a}_i \mid i \in \mathcal {A})$ , $Y = (\overline {b}_i \mid i \in \mathcal {A})$ , where $\overline {a}_i, \overline {b}_i$ are from some L-structure $\mathcal {M}$ .

We say Y is locally based on X if for any finite set of L-formulas, $\Delta $ , and for any finite tuple $(j_0, \ldots , j_{n-1})$ from $\mathcal {A}$ , there exists a tuple $(i_0, \ldots , i_{n-1})$ from $\mathcal {A}$ such that

$$ \begin{align*}\overline{\jmath} \sim_{\mathcal{A}} \overline{\imath}\end{align*} $$

and

$$ \begin{align*}\overline{b}_{\overline{\jmath}} \equiv^{\mathcal{M}}_\Delta \overline{a}_{\overline{\imath}},\end{align*} $$

where Y and X are understood from context, this property will be referred to as local basedness.

We give a proof sketch to illustrate the idea behind Proposition 2.27.

Proposition 2.27 (Proposition 2 in [Reference Scow32]).

Fix sequences of parameters $X = (\overline {a}_i \mid i \in \mathcal {I})$ , $Y = (\overline {b}_i \mid i \in \mathcal {I})$ , where $\overline {a}_i, \overline {b}_i$ are l-tuples from some L-structure $\mathcal {M}$ , for some integer $l \geq 1$ . We have that Y is locally based on X if and only if $Y \vDash \textrm {EMtp}(X)$ .

Proof Suppose $Y \vDash \textrm {EMtp}(X)$ . To show local basedness, fix a finite set of L-formulas $\Delta $ and a finite tuple $\overline {\jmath }$ from $\mathcal {I}$ with complete quantifier-free type $\eta $ in $\mathcal {I}$ . Let $\varphi $ be a conjunction of all the formulas in the finite $\Delta $ -type of $\overline {b}_{\overline {\jmath }}$ . Suppose, for contradiction, there is no $\overline {\imath } \sim _{\mathcal {I}} \overline {\jmath }$ such that $\overline {a}_{\overline {\imath }} \equiv ^{\mathcal {M}}_\Delta \overline {b}_{\jmath }$ . Then

$$ \begin{align*}(\forall \overline{\imath}) ( \mathcal{I} \vDash \eta(\overline{\imath}) \Rightarrow \mathcal{M} \vDash \neg \varphi(\overline{a}_{\overline{\imath}})).\end{align*} $$

Since $Y \vDash \textrm {EMtp}(X)$ , we must have that

$$ \begin{align*}(\forall \overline{\jmath}) ( \mathcal{I} \vDash \eta(\overline{\jmath}) \Rightarrow \mathcal{M} \vDash \neg \varphi(\overline{b}_{\overline{\jmath}})),\end{align*} $$

which contradicts the $\Delta $ -type of $\overline {b}_{\overline {\jmath }}$ .

Suppose Y is locally based on X. To show $Y \vDash \textrm {EMtp}(X)$ , consider a rule from $\textrm {EMtp}(X)$ :

$$ \begin{align*}(\forall \overline{\imath}) ( \mathcal{I} \vDash \eta(\overline{\imath}) \Rightarrow \mathcal{M} \vDash \varphi(\overline{a}_{\overline{\imath}})).\end{align*} $$

Fix any $\overline {\jmath }$ from $\mathcal {I}$ such that $\mathcal {I} \vDash \eta (\overline {\jmath })$ . By local basedness, there is $\overline {\imath } \sim _{\mathcal {I}} \overline {\jmath }$ such that $ \overline {b}_{\overline {\jmath }} \equiv _{\{\varphi \}} \overline {a}_{\overline {\imath }}$ . By the rule for $\eta $ , $\mathcal {M} \vDash \varphi (\overline {a}_{\overline {\imath }})$ . Thus we have that $\mathcal {M} \vDash \varphi (\overline {b}_{\overline {\jmath }})$ , as well. And so we have proved the rule

$$ \begin{align*}(\forall \overline{\jmath}) ( \mathcal{I} \vDash \eta(\overline{\jmath}) \Rightarrow \mathcal{M} \vDash \varphi(\overline{b}_{\overline{\jmath}})).\end{align*} $$

Since this is true for any rule, $Y \vDash \textrm {EMtp}(X)$ .

Definition 2.28 [Reference Scow32].

Given a structure $\mathcal {I}$ , we say that $\mathcal {I}$ -indexed indiscernible sets have the modeling property if for any integer $l \geq 1$ , any $||\mathcal {I}||^+$ -saturated structure $\mathcal {M}$ , and any $\mathcal {I}$ -indexed set of l-tuples from $\mathcal {M}$

$$ \begin{align*}X = (\overline{a}_i \mid i \in \mathcal{I}) ,\end{align*} $$

there exists an $\mathcal {I}$ -indexed indiscernible set of l-tuples from $\mathcal {M}$

$$ \begin{align*}Y = (\overline{b}_i \mid i \in \mathcal{I}),\end{align*} $$

such that $Y \vDash \textrm {EMtp}(X)$ (equivalently, Y is locally based on X, by Proposition 2.27).

Remark 2.29. In fact, it suffices to require that $\mathcal {M}$ in Definition 2.28 be $||\mathcal {I}||$ -saturated, since the type describing Y has $||\mathcal {I}||$ variables and no parameters from $\mathcal {M}$ , as noted in Remark 2.2. Previously, in [Reference Scow32], the bound was required to be $||\mathcal {I}||^+$ not $||\mathcal {I}||$ .

Theorem 2.30 (Dictionary Theorem [Reference Scow31–Reference Scow33]).

Suppose that $\mathcal {I}$ is a locally finite ordered structure. Then $\mathcal {I}$ -indexed indiscernible sequences have the modeling property if and only if $\textrm {age}(\mathcal {I})$ has RP.

Theorem 2.30 fails when we drop order:

Example 2.31. Let $\mathcal {I}=(\mathbb {N},=)$ and note that $\textrm {age}(\mathcal {I})$ has RP by Example 2.6. If we take an $\mathcal {I}$ -indexed set in $\mathcal {M}:=(\mathbb {N},<)$ , $X = (i \mid i \in \mathcal {I})$ , then there is no $\mathcal {I}$ -indexed indiscernible set in any extension $\mathcal {M}' \succeq \mathcal {M}$ locally based on X. Such a set would need to have $\textrm {tp}^{\mathcal {M}'}(i,j)=\textrm {tp}^{\mathcal {M}'}(j,i)$ for $i\neq j\in \mathbb {N}$ , which is not possible.

Example 2.31 illustrates why rigidity has been important in applications of structural Ramsey theory to generalized indiscernible sequences in model theory. Consider an injection $f: \mathcal {I} \rightarrow \mathcal {M}$ and parameters $(a_i : i \in \mathcal {I})$ such that $a_i = f(i)$ , for all $i \in \mathcal {I}$ . Given a finite substructure $A \subseteq \mathcal {I}$ of size n, the injection f induces a (possibly infinite) coloring on tuples $(a_0,\ldots ,a_{n})$ such that $\textrm {ran} (\overline {a}) \cong A$ , where $\textrm {tp}^{\mathcal {M}} (f(\overline {a}))$ is the color of $\overline {a}$ . Thus, if finitely generated substructures of $\mathcal {M}$ are rigid, then we will not have an $\mathcal {I}$ -indexed indiscernible set locally based on $(a_i : i \in \mathcal {I})$ if A has a nontrivial automorphism. Note that the modeling property (Definition 2.28) is a universal statement about all structures $\mathcal {M}$ , and so it would immediately fail for $\mathcal {I}$ -indexed indiscernible sets if $\mathcal {I}$ is not rigid. The reason for the modeling property to be a universal property is so that $\mathcal {I}$ can function as a tool in classification theory to compare all theories, even those theories whose models have rigid finitely generated substructures. In fact, theories whose models are linearly ordered by a formula in the language play an important role in classification theory as they are unstable and have the strict-order property.

Theorem 2.30 fails when we drop local finiteness:

Example 2.32. Let $\mathcal {I} = (\mathbb {Z},p,s,<)$ be the structure on $\mathbb {Z}$ with the usual order $<$ and where $p, s$ are unary function symbols interpreted as “predecessor” and “successor”, respectively. The only possible finitely generated substructure of $\mathbb {Z}$ is the whole structure. Since $||\textrm {age}(\mathcal {I})||=1$ , the class trivially has RP. However, we will show that $\mathcal {I}$ -indexed indiscernibles do not have the modeling property, showing the essentialness of the assumption that $\mathcal {I}$ be locally finite in Theorem 2.30. Let $\mathcal {M}$ be the Fraïssé limit of finite convexly ordered equivalence classes in signature $\{E,\prec \}$ . Let $X=(a_i \mid i \in \mathbb {Z})$ be such that all $a_i$ for i odd are in one E-class that we call $\texttt {Odd}$ and all $a_j$ for j even are in a separate E-class that we call $\texttt {Even}$ . Moreover, let $i < j \Rightarrow a_i \prec a_j$ , and $\texttt {Odd}<\texttt {Even}$ in $\mathcal {M}$ . Within this example, we use $\sim $ to denote E-equivalence in the following visualization, where elements are listed in $\prec $ -increasing order in $\mathcal {M}$ :

$$ \begin{align*}\ldots a_1 \sim a_3 \sim a_5 \ldots \nsim \ldots a_2 \sim a_4 \sim a_6 \ldots.\end{align*} $$

The type $\textrm {EMtp}(X)$ requires that whenever $j=s(s(i))$ , $x_i \prec x_j$ and $E(x_i,x_j)$ . However, a decision is not made about $x_i \prec x_j$ when $j=s(i)$ , since sometimes $a_i \prec a_j$ and sometimes $a_j \prec a_i$ , when $j=s(i)$ , though $a_i, a_j$ are always E-inequivalent when $j=s(i)$ . Suppose there is an $\mathcal {I}$ -indexed indiscernible set locally based on X. If this indiscernible set chooses the rule that $b_i \prec b_j$ , whenever $j=s(i)$ , then $\mathcal {M}$ would admit equivalence classes that are not convexly ordered, e.g., with $b_1 \sim b_3$ but

$$ \begin{align*}\ldots b_1 \nsim b_2 \nsim b_3 \ldots\end{align*} $$

a contradiction. The alternative, choosing $b_i \succ b_{s(i)}$ , is incompatible with ${b_i \prec b_{s(s(i))}}$ .

Theorem 2.33 [Reference Scow33].

Let $\mathcal {A}$ and $\mathcal {B}$ be any structures. Suppose that $\mathcal {A}$ is a semi-retract of $\mathcal {B}$ . Furthermore, suppose that $\mathcal {B}$ -indexed indiscernible sets have the modeling property. Then $\mathcal {A}$ -indexed indiscernible sets have the modeling property.

Corollary 2.34 [Reference Scow33].

Let $\mathcal {A}$ and $\mathcal {B}$ be locally finite ordered structures. Suppose that $\mathcal {A}$ is a semi-retract of $\mathcal {B}$ and $\mathcal {B}$ has RP. Then $\mathcal {A}$ has RP.

Proof This follows by Theorems 2.33 and 2.30.

3 Semi-retractions, reducts, and examples

In this section, we consider basic examples of semi-retractions and give a characterization of semi-retractions in Theorem 3.7 under certain assumptions.

Proof The $\Rightarrow $ is from Corollary 2.34. It remains to show $\Leftarrow $ .

Suppose that $\mathcal {A}$ has RP. Let $X = (a_i \mid i \in \mathcal {A})$ enumerate $\mathcal {B}$ where $a_i = i$ . Since $\mathcal {A}$ has RP, by Theorem 2.30, $\mathcal {A}$ -indexed indiscernible sets have the modeling property. Let $Y = (b_i \mid i \in \mathcal {A})$ be an $\mathcal {A}$ -indexed indiscernible set locally based on X. By the saturation assumption, we know that we can witness Y in $\mathcal {B}$ , and not just in some elementary extension of $\mathcal {B}$ (see Remark 2.2). Define $f: \mathcal {A} \rightarrow \mathcal {B}$ to take $i \mapsto b_i$ . This is an injective map, since $a_i \neq a_j$ for all $i \neq j$ and Y is $\mathcal {A}$ -indexed indiscernible locally based on X. It remains to show that

$$ \begin{align*}\mathcal{A} \xrightarrow{f} \mathcal{B} \xrightarrow{id} \mathcal{A}\end{align*} $$

is a semi-retraction.

By Observation 3.2, we already know that the identity map is qftp-respecting. We must show the remaining properties from Definition 2.19: (i) $\overline {\imath }_1 \sim _{\mathcal {A}} \overline {\imath }_2 \Rightarrow f(\overline {\imath }_1) \sim _{\mathcal {B}} f(\overline {\imath }_2)$ and that (ii) $\overline {\jmath } \sim _{\mathcal {A}} id(f(\overline {\jmath })) = f(\overline {\jmath })$ , for all tuples $\overline {\jmath }, \overline {\imath }_1, \overline {\imath }_2 \in {|\mathcal {A}|}^n$ , for all $n<\omega $ . We have (i) as a direct consequence of $\mathcal {A}$ -indexed indiscernibility. We have (ii) from local basedness: for every finite subset of $L({\mathcal {B}})$ -formulas $\Delta $ , for any $\overline {\jmath } \in {|\mathcal {A}|}^n$ , there is $\overline {\imath } \sim _{\mathcal {A}} \overline {\jmath }$ from $\mathcal {A}$ such that $b_{\overline {\jmath }} \equiv ^{\mathcal {B}}_{\Delta } a_{\overline {\imath }}$ . In other words, $f(\overline {\jmath }) = b_{\overline {\jmath }} \equiv ^{\mathcal {B}}_{\Delta } a_{\overline {\imath }} = \overline {\imath }$ . Given an arbitrary $\overline {\jmath }$ , let $\Delta $ contain the formula equivalent to $\textrm {qftp}^{\mathcal {B}}(f(\overline {\jmath }))$ , and fix a corresponding $\overline {\imath } \sim _{\mathcal {A}} \overline {\jmath }$ . Then, $f(\overline {\jmath }) \sim _{\mathcal {B}} \overline {\imath }$ , which implies that $f(\overline {\jmath }) \sim _{\mathcal {A}} \overline {\imath }$ , since $\mathcal {A}$ is a quantifier-free reduct of $\mathcal {B}$ . Since $f(\overline {\jmath }) \sim _{\mathcal {A}} \overline {\imath } \sim _{\mathcal {A}} \overline {\jmath }$ , we have that $f(\overline {\jmath }) \sim _{\mathcal {A}} \overline {\jmath }$ , showing (ii).

Example 3.8. Let $\mathcal {B}:=\mathcal {R}^<$ be the random ordered graph (the Fraïssé limit of finite ordered graphs) and $\mathcal {A}:= \mathcal {B} \upharpoonright \{<\}$ , so $\mathcal {A}$ is isomorphic to the rational linear order. It is easy to see that $\mathcal {A}$ is a quantifier-free reduct of $\mathcal {B}$ , $\mathcal {B}$ is countably saturated, and both are locally finite and ordered. By Theorem 3.7, $\mathcal {A}$ is a semi-retract of $\mathcal {B}$ .

In order to find a semi-retraction $(g,f)$ between $\mathcal {A}$ and $\mathcal {B}$ , one strategy is to take an indiscernible sequence $(g(i) \mid i \in \mathcal {A})$ in $\mathcal {R}^<$ , i.e., such that for all integers $n \geq 0$ , for all $i_0<\dots <i_{n-1}$ and $j_0<\dots <j_{n-1}$ from $\mathcal {A}$ , the map $g(i_k) \mapsto g(j_k)$ for all $k<n$ provides an isomorphism of ordered graphs in $\mathcal {B}$ . Such an indiscernible sequence is guaranteed to exist in $\mathcal {B}$ , but in such a straightforward case, we can discern the existence of such a one directly: let $g(\mathcal {A})$ be a copy of the countably infinite complete graph $K_\omega $ whose vertices form a dense linear order without endpoints under $<$ (guaranteed to exist in $\mathcal {B}$ by its saturation), where $g: \mathcal {A} \rightarrow \mathcal {B}$ is set up to preserve order. Then f can be taken to be merely the identity map, as follows:

$$ \begin{align*}\mathcal{A} \xrightarrow{g} \mathcal{B} \xrightarrow{f=\textrm{id}} \mathcal{A} .\end{align*} $$

This pair $(g,f)$ now witnesses that $\mathcal {A}$ is a semi-retract of $\mathcal {R}$ .

Though both $\mathcal {A},\mathcal {B}$ are well-known to have RP, this fact can be thought of as an instance of transfer by Corollary 5.5.

A similar argument can be made for the case $\mathcal {A} = (\mathbb {Q},<)$ and $\mathcal {B} = (|\mathcal {B}|, E, \prec ),$ where $\mathcal {B}$ is either of the ordered equivalence relation structures described in Examples 2.6 or 2.7.

Next we look at three locally finite ordered structures known to have RP: the Shelah tree $\mathcal {I}_{\textrm {stree}}$ , the strong tree $\mathcal {I}_{\textrm {strtree}}$ , and the convexly ordered equivalence relation $\mathcal {I}_{\textrm {eq}}$ . The structure $\mathcal {I}_{\textrm {eq}}$ has RP (Example 2.6) which fact is related to the usefulness of mutually indiscernible sequences in model theory (see Theorem III.7.12(iii) in [Reference Shelah34] for an early example of mutually indiscernible sequences) even if RP is not explicitly mentioned. An infinitary proof that shows that $\mathcal {I}_{\textrm {stree}}$ has RP is given in the Appendix of [Reference Shelah34] and a finitary proof is given in Lemma 2 in Section 2.2 of Chapter 2 of [Reference Nguyen Van Thé27]. The fact that $\mathcal {I}_{\textrm {strtree}}$ has RP is implicitly used in the proof of Theorem III.7.11 in [Reference Shelah34] and used explicitly in the survey paper [Reference Kim, Kim and Scow17] (see [Reference Scow32] for a more detailed discussion of this history).

We recall the following example of a semi-retraction that transfers RP from $\textrm {age}(\mathcal {I}_{\textrm {strtree}})$ to $\textrm {age}(\mathcal {I}_{\textrm {eq}})$ .

The following example is an application of Theorem 3.7.

4 (Counter)example

This section is inspired by an example of Mašulović from [Reference Mašulović21] where a pre-adjunction (see Section 6.1) is constructed to transfer RP from the category of finite naturally ordered Boolean algebras to the category of finite linearly ordered graphs. Both classes were previously known to have RP (see [Reference Abramson and Harrington1, Reference Nešetřil and Rödl26] for linearly ordered graphs, and [Reference Graham and Rothschild11] for Boolean algebras), but the mechanism of transfer is quite interesting with a number of applications.

Let $\mathcal {R}$ denote the countably infinite Random graph (the Fraïssé limit of finite graphs) and let $\mathcal {H}_n$ denote the countably infinite random n-regular hypergraph (the Fraïssé limit of finite n-regular hypergraphs). Let $\mathcal {B}_{ba}$ denote the countable atomless Boolean algebra (the Fraïssé limit of finite Boolean algebras) with its induced partial order denoted by $<^{\mathcal {B}_{ba}}$ . Below we show that $\mathcal {R}$ and $\mathcal {H}_n$ are semi-retracts of $\mathcal {B}_{ba}$ . By Theorem 5.1, this gives us upper bounds on Ramsey degrees for embeddings of finite graphs, respectively, finite n-regular hypergraphs, in terms of Ramsey degrees for embeddings of finite Boolean algebras, and consequently on Ramsey degrees of substructures by Corollary 5.2 (see Remark 4.2 for a more detailed analysis).

4.1 Semi-retracts of the countable atomless Boolean algebra

Theorem 4.1. The countable random graph $\mathcal {R}$ is a semi-retract of the countable atomless Boolean algebra $\mathcal {B}_{ba}$ .

Proof We define a graph relation R on $\mathcal {B}_{ba}$ as Mašulović does on power set algebras of finite sets in [Reference Mašulović21]: $(a,b)\in R$ if and only if $a \neq b$ and $a\wedge b\neq \mathbf {0}$ . We will build a graph embedding $g:\mathcal {R}\rightarrow (\mathcal {B}_{ba},R)$ that will be qftp-respecting when $\mathcal {B}_{ba}$ is considered as a Boolean algebra. Since $\mathcal {R}$ is universal for countable graphs, there must be a graph embedding $f: (\mathcal {B}_{ba},R)\rightarrow \mathcal {R}$ . Clearly, f will remain qftp-respecting with respect to $\mathcal {B}_{ba}$ in the signature of Boolean algebras. Since g and f are graph embeddings, $f\circ g$ is a graph embedding from $\mathcal {R}$ to itself. Thus $(g,f)$ will be a semi-retraction between $\mathcal {R}$ and $\mathcal {B}_{ba}$ .

Let $V=\{v_n:n\in \omega \}$ be an enumeration of vertices in $\mathcal {R}$ . Recall that an antichain in a Boolean algebra is a collection of non-zero elements such that every two distinct elements have zero meet. Pick an infinite antichain $B=\{b_n:n\in \omega \}$ in $\mathcal {B}_{ba}$ . For any $b_n\in B,$ let $\{b_n^i:i\in \omega \}$ be an antichain $\leq ^{\mathcal {B}_{ba}}$ -below $b_n.$ Define $g:\mathcal {R}\to \mathcal {B}_{ba}$ by

$$\begin{align*}g(v_n)=b_n\vee\bigvee_{i<n}b_i^n(\varepsilon),\end{align*}$$

where

$$\begin{align*}b_i^n(\varepsilon)=\begin{cases} b_i^n,\ \textrm{if } (v_i,v_n)\in E,\\ \mathbf{0},\ \textrm{otherwise}. \end{cases} \end{align*}$$

We will show that g is qftp-respecting: Every quantifier-free type in $\mathcal {R}$ is determined by E on pairs of points, and that needs to be reflected in the image of g. By the definition of g, if $(v_i,v_n)\in E$ and $i<n$ then $g(v_i)\wedge g(v_n)=b_i^n \neq \mathbf {0}$ . If $(v_i,v_n)\notin E,$ then $g(v_i)\wedge g(v_n)=\mathbf {0}$ .

Any Boolean expression with variables $x_0,\ldots , x_{n-1}$ has its unique equivalent full disjunctive normal form, that is, a disjunction of clauses, where each clause consist of conjunction of literals $x_i$ or $\neg x_i$ so that each of the variables $x_i$ appears in every clause. We will show that an n-type of elements of $\mathcal {B}_{ba}$ in the image of g depends only on the type of the graph they encode:

1. (1) For every triple of distinct numbers $i,j,k$ : $g(v_i)\wedge g(v_j)\wedge g(v_k)=\textbf {0}$ .

2. (2) Whenever $i< j,$ we have that $g(v_i)\wedge g(v_j)=\begin {cases} \mathbf {0},\ \ \text {if } (v_i,v_j)\notin E,\\ b_i^j,\ \ \text {if } (v_i,v_j)\in E. \end {cases}$

It follows that every non-zero clause has at most two positive literals. Further,

1. (3) For any $i\neq j$ ,

   $$\begin{align*}g(v_i)\wedge \neg g(v_j)=g(v_i)\wedge \neg(g(v_i)\wedge g(v_j))= \!\begin{cases} g(v_i),\ \text{if } (v_i,v_j)\notin E,\\ g(v_i) \wedge \neg b_i^j, \ \text{if } (v_i,v_j)\in E \ \text{and} \ i<j, \\ g(v_i) \wedge \neg b_j^i, \ \text{if } (v_i,v_j)\in E \ \text{and } i>j. \end{cases} \end{align*}$$

2. (4) For $i\neq j,\ \neg g(v_i)\wedge \neg g(v_j)=\neg (g(v_i)\vee g(v_j))$ by de Morgan’s law.

Let $v_{i_0}, v_{i_2},\ldots , v_{i_{n-1}}$ be n distinct vertices of $\mathcal {R}$ . We will consider Boolean expression in $g(v_{i_0}),\ldots , g(v_{i_{n-1}})$ in full disjunctive normal form. From the analysis above, we have three cases.

1. (i) If a clause contains no positive literal, then it is equal to $\neg (\bigvee _{k=0}^{n-1}g(v_{i_k}))$ , which is a positive element disjoint from any $g(v_{i_k}).$

2. (ii) If a clause has one positive literal $g(v_{i_k})$ and all other $n-1$ negative, it equals to

   $$\begin{align*}g(v_{i_k})\wedge \neg(\bigvee \{g(v_{i_k})\wedge g(v_{i_j}):(v_{i_k},v_{i_j})\in E\}),\end{align*}$$
   which is $\leq ^{\mathcal {B}_{ba}} g(v_{i_k})$ and disjoint from every $g(v_{i_k})\wedge g(v_{i_j})$ for $j\neq k.$

3. (iii) If a clause contains two positive literals $g(v_{i_k}), g(v_{i_l})$ , then it is equal to $g(v_{i_k})\wedge g(v_{i_l}).$

There are no non-trivial equations with quantifier free formulas about an n-tuple in the image of g other than comparison with $\mathbf {0}$ . Since clauses in items (i) and (ii) are never equal to $\mathbf {0},$ the entire type is decided by the clauses as in (iii), which corresponds exactly to the type of the finite graph in the preimage.

Remark 4.2. We say that A is a Ramsey object in $\mathcal {K}$ if for all $B \in \mathcal {K}$ , $(A,B)$ is a Ramsey duo for $\mathcal {K}$ . The class of finite Boolean algebras has RP (as in Example 2.6) while only complete or empty graphs are Ramsey objects in the class of finite graphs, as shown in [Reference Nešetřil and Rödl25]. Therefore, to secure transfer of RP by semi-retractions, the requirement on rigidity of $\mathcal {B}$ in Corollary 5.5 or on relational signature in Theorem 5.9 cannot be completely removed. The reason in the example of Theorem 4.1 is that there can be two distinct copies $\Gamma _1,\Gamma _2$ of the same finite graph in $\mathcal {R}$ such that $f^{-1}(\Gamma _1)$ and $f^{-1}(\Gamma _2)$ have the same quantifier-free type and generate the same Boolean subalgebra A of $\mathcal {B}_{ba}$ (thanks to function symbols in the signature of Boolean algebras). This would not have been possible if $\textrm {age}(\mathcal {B}_{ba})$ were rigid, since a partial isomorphism from $f^{-1}(\Gamma _1)$ to $f^{-1}(\Gamma _2)$ given by having the same quantifier-free type would extend to a non-trivial automorphism of A.

Theorem 5.1 applied to $(g,f)$ shows that the finite Ramsey degree of a finite graph $\Gamma $ in the class of finite graphs with embeddings is bounded above by the Ramsey degree of the Boolean subalgebra of $\mathcal {B}_{ba}$ generated by $g(\Gamma )$ in the class of finite Boolean algebras with embeddings (which is equal to $n!$ , where n is the number of atoms in $\left <g(\Gamma )\right>$ ). It can be derived from results in [Reference Abramson and Harrington1] and [Reference Nešetřil and Rödl26] that the exact Ramsey degree for embeddings of a graph on m vertices is $m!$ . Thus our estimate is optimal only in the case of discrete graphs.

Question 4.3. The example of a semi-retraction on Theorem 4.1 was inspired by an example in [Reference Mašulović21]. However, Mašulović’s example is about the classes of finite ordered graphs and finite naturally ordered Boolean algebras, rather than their unordered versions presented here. Is the random ordered graph (the Fraïssé limit of finite linearly ordered graphs) a semi-retract of the countable atomless Boolean algebra with a generic normal order (the Fraïssé limit of finite naturally ordered Boolean algebras)?

Let $\mathcal {H}_n=(V,E)$ be the countable random n-regular hypergraph. Mašulović commented in his paper [Reference Mašulović21] that the method he used to witness the Ramsey property of linearly ordered graphs by that of finite naturally ordered Boolean algebras did not generalize to higher arity hypergraphs. We were able to overcome this obstacle in the unordered case. As in graphs, one needs to ensure that the g-part of the semi-retraction maps to a portion of the Boolean algebra, where every type is determined only by the hypergraph relation. This can be achieved by a construction similar to that in Theorem 4.1, while making sure that all k-tuples for $k<n$ have the same type.

Theorem 4.4. For any integer $n\geq 2$ , $\mathcal {H}_n$ is a semi-retract of $\mathcal {B}_{ba}$ .

Proof We define an n-ary hypergraph embedding $H_n$ on $\mathcal {B}_{ba}$ by $(b_0,\ldots ,b_{n-1})\in H_n$ iff $\bigwedge _{i<j<n} b_i \neq b_j$ and $\bigwedge _{l=0}^{n-1} b_l\neq \textbf {0}$ . We will define an hypergraph embedding $g:\mathcal {H}_n\to (\mathcal {B}_{ba}, H_n)$ and by universality of $\mathcal {H}_n,$ there is an embedding $f:(\mathcal {B}_{ba},H_n)\to \mathcal {H}_n$ . Since $H_n$ is quantifier-free definable in $\mathcal {B}$ , taking the reduct $\mathcal {B}_{ba}$ of $(\mathcal {B}_{ba}, H_n),\ (g,f)$ will be a semi-retraction between $\mathcal {B}_{ba}$ and $\mathcal {H}_n$ , as in the case of the random graph. However, we need to be more careful when defining g than in the graph case – in $(\mathcal {B}_{ba},H_n)$ there are different $<n$ -types, which is not true in $\mathcal {H}_n$

For every $1\leq k\leq n,$ let $B_k=\{b_{\overline {\imath }}: \overline {\imath }\in \omega ^{[k]}\}$ be an antichain in $\mathcal {B}_{ba}$ , where $\omega ^{[k]}$ denotes all strictly increasing sequences in $\omega $ of length k. We further require that if $\overline {\imath }$ is an initial segment of $\overline {\jmath },$ then $b_{\overline {\jmath }} <^{\mathcal {B}_{ba}} b_{\overline {\imath }}.$ Let $\{v_l:l\in \omega \}$ be an enumeration of vertices of $\mathcal {H}_n$ and let E denote the set of hyperedges. We define $g:\mathcal {H}_n\to (\mathcal {B}_{ba},H_n)$ by

$$\begin{align*}g(v_l)=\bigvee_{\overline{\imath}(|\overline{\imath}|-1)=l, |\overline{\imath}|<n} b_{\overline{\imath}}\vee \bigvee_{\overline{\imath}(|\overline{\imath}|-1)=l,|\overline{\imath}|=n, (v_{i(0)},\ldots,v_{i(n-1)})\in E} b_{\overline{\imath}}. \end{align*}$$

Since each $B_k$ is an antichain, we have that for any $\overline {\imath }\in \omega ^{[k]}$

$$\begin{align*}\bigwedge_{l=0}^{k-1} g(v_{i(l)})\neq\mathbf{0} \ \text{iff } k<n, \text{\ or } k=n \ \text{and}\ (v_{i(0)},\ldots,v_{i(n-1)})\in H_n. \end{align*}$$

We have that g and f are hypergraph embeddings and thus $fg$ is an embedding. As in the proof of Theorem 4.1, one can show that every N-type for $N\geq n$ in $g(\mathcal {H}_n)$ is determined by n-types. Here it is crucial that every two k-tuples for $k<n$ have the same type and therefore n-types are determined solely by the hypergraph relation. We can conclude that g considered as a map $\mathcal {H}_n\to \mathcal {B}_{ba}$ is qftp-respecting.

Remark 4.5. As in 4.2, by Theorem 5.1, the Ramsey degree for embeddings of a finite hypergraph H in the class of finite hypergraphs is bounded above by the Ramsey degree of $\left <g(H)\right>$ in the class of finite Boolean algebras with embeddings.

5 Formula-free approach to transferring the RP

In prior work [Reference Scow33], Corollary 2.34 demonstrated how semi-retractions transfer RP when $\mathcal {A}$ and $\mathcal {B}$ are both ordered and locally finite. In this section we show in Corollary 5.5 that some of the assumptions in Corollary 2.34 may be dropped. Namely, we only need rigidity of the structures in $\textrm {age}(\mathcal {B})$ and we only need local finiteness of $\mathcal {A}$ . In fact, the transfer can be done locally for a pair of finite structures $(A,B)$ in $\mathcal {A}$ , as demonstrated in Corollary 5.4. Moreover, if the signatures of $\mathcal {A}$ and $\mathcal {B}$ are relational, we no longer need rigidity in $\textrm {age}(\mathcal {B})$ as shown in Theorem 5.9.

In Remark 4.2, we noted that the rigidity assumption in Corollary 5.4 or the relational language in Theorem 5.9 cannot be removed. However, if we consider the Ramsey property for embeddings rather than for substructures, which is the natural framework in dynamical applications, we obtain a transfer principle for finite Ramsey degrees for embeddings (Theorem 5.1) that does not assume more about $\mathcal {A}$ and $\mathcal {B}$ other than that $\mathcal {A}$ is a locally finite semi-retract of $\mathcal {B}$ . We include examples that demonstrate the necessity of the assumptions in Corollaries 5.4 and 5.5.

In Proposition 2.16, we recalled the exact formula relating the Ramsey degree for embeddings and the Ramsey degree for substructures in case of a locally finite class. Therefore Theorem 5.1 immediately yields the following corollary.

In Theorem 4.1, we provided an example of the random graph as a semi-retract of the countable atomless Boolean algebra. While the class of finite Boolean algebras is a Ramsey class, the class of finite graphs is not, so we cannot simply say that semi-retractions transfer the Ramsey property for structures. However, the existence of the semi-retraction provides a bound on finite Ramsey degrees of finite graphs as per Corollary 5.2. We point out that the reason that Section 4 does not provide a direct transfer of the Ramsey property is that the signature of Boolean algebras contains function symbols, $\textrm {age}(\mathcal {B}_{ba})$ is not rigid, and the semi-retraction defined in the proof of Theorem 4.1 allows two distinct finite isomorphic graphs to generate the same Boolean algebra via preimages under f. We show in Theorem 5.9 that restricting to relation symbols and in Corollary 5.5 that restricting to $\mathcal {B}$ with age consisting of rigid structure, suffice to transfer the Ramsey property, respectively.

Example 2.31 explained why rigidity is essential for applications of Ramsey theory to indiscernible sequences. Thus we present a few special cases of Theorem 5.1 in the rigid setting.

Proof By Proposition 2.16, $d(A,\mathcal {K}) \leq d_e(A,\mathcal {K})$ . If $\mathcal {K}'$ consists of rigid elements, then $d_e(A',\mathcal {K}') = (1) \cdot d(A',\mathcal {K}')$ . Thus we have the inequalities:

$$ \begin{align*}d(A,\mathcal{K}) \leq d_e(A,\mathcal{K}) \leq d_e(A',\mathcal{K}') = d(A',\mathcal{K}')=d,\end{align*} $$

where the middle inequality follows from Theorem 5.1.

Corollary 5.4 follows easily from the argument for Theorem 5.1: we merely need to be aware of how we fix B in the beginning and set $d=1$ . An alternative argument, not as a corollary of Theorem 5.1, is presented in the Appendix in Section 8.1.

Corollary 5.5 is a special case of Corollary 5.4.

To show the necessity of the assumptions in Corollary 5.5, we give an example of a structure $\mathcal {A}$ that is a semi-retract of $\mathcal {B}$ , where each structure is in a non-relational signature, such that $\mathcal {A}$ fails to be locally finite, the element in $\textrm {age}(\mathcal {B})$ is not rigid, and RP fails to transfer from $\mathcal {B}$ to $\mathcal {A}$ .

To show the necessity of the assumptions in Corollary 5.4, we give an example of a structure $\mathcal {A}$ that is a semi-retract of $\mathcal {B}$ , where $\mathcal {A}$ is in a non-relational signature, such that all structures in $\textrm {age}(\mathcal {B})$ are finite and rigid, and a Ramsey duo for $\mathcal {B}$ fails to transfer to a Ramsey duo for $\mathcal {A}$ , specifically because the pair in $\mathcal {A}$ is a counterexample to local finiteness in $\mathcal {A}$ .

Example 5.7 can be modified slightly to satisfy the assumptions of Corollary 5.5.

It would be possible to prove Theorem 5.9 with no mention of function symbols, but there is a property, the “restricted inverse images under f” property (defined in Definitions 5.10 and 5.13), that could be of independent interest in the functional case. For this reason, we pursue a slightly longer development of the argument than needed here. The reader who would like to pursue the additional details for a proof of Corollary 5.5 using the “restricted inverse images under f” property, is invited to read Section 8.1 within the Appendix.

The argument for Theorem 5.1 is of a category-theoretic nature and so certain details at the level of substructures may not be immediately evident. For example, consider Definition 5.10 and what it would take for a certain finite $B_0 \subseteq \mathcal {B}$ to have this property.

It turns out that if $\mathcal {A}$ is a semi-retract of $\mathcal {B}$ via $(g,f),$ so long as $B_0$ is isomorphic to an image under g of a structure in $\mathcal {A}$ containing A as a substructure, then $B_0$ has restricted inverse images under f for $A.$

To develop the proof, we first state the analogues of Definition 5.10 and Proposition 5.11 in the case that the signatures contain function symbols. Since Definition 5.13 seems technical at first, we motivate it with Example 5.12.

Proof Fix structures $\mathcal {A}, \mathcal {B}$ and a pair of maps $g: \mathcal {A} \rightarrow \mathcal {B}$ and $f: \mathcal {B} \rightarrow \mathcal {A}$ witnessing that $\mathcal {A}$ is a semi-retract of $\mathcal {B}$ . Fix finite tuples $\overline {a},\overline {b}$ enumerating substructures of $\mathcal {A}$ and let $\overline {a}_0:=g(\overline {a}), \overline {b}_0:=g(\overline {b})$ , $\overline {a}_1:=f(\overline {a}_0)$ , and $\overline {b}_1:=f(\overline {b}_0)$ . Let $n:=|\overline {a}|$ .

Fix a tuple $\overline {c}_1$ in $\left < \overline {b}_1 \right>_{\mathcal {A}}$ such that:

(1) $$ \begin{align} \overline{c}_1 \sim_{\mathcal{A}} \overline{a}. \end{align} $$

We will argue that $\overline {c}_0:=f^{-1}(\overline {c}_1)$ is in $\left < \overline {b}_0 \right>_{\mathcal {B}}$ : Since $fg$ is an $L({\mathcal {A}})$ -embedding and we assumed $\overline {b} = \left < \overline {b} \right>_{\mathcal {A}}$ , it must be that $\overline {b}_1 = \left < \overline {b}_1 \right>_{\mathcal {A}}$ . Thus $\overline {c}_1 \subseteq \overline {b}_1$ so there is some $\overline {c} \subseteq \overline {b}$ such that $g(\overline {c}) = \overline {c}_0$ and $f(\overline {c}_0) = \overline {c}_1$ , and therefore $f^{-1}(\overline {c}_1)=g(\overline {c}) \subseteq \overline {b}_0$ .

By the embedding property of semi-retractions,

(2) $$ \begin{align} \overline{c} \sim_{\mathcal{A}} \overline{c}_1. \end{align} $$

The equations (1) and (2) imply that:

(3) $$ \begin{align} \overline{c} \sim_{\mathcal{A}} \overline{a}. \end{align} $$

Thus, by the qftp-respecting property of semi-retractions:

(4) $$ \begin{align} \overline{c}_0 = g(\overline{c}) \sim_{\mathcal{B}} g(\overline{a}) = \overline{a}_0. \end{align} $$

By (4), we may conclude that $\overline {c}_0=f^{-1}(\overline {c}_1) \sim _{\mathcal {B}} \overline {a}_0$ , as desired.

To complete the proof, fix any $\overline {b}_0'$ from $\mathcal {B}$ such that

(5) $$ \begin{align} \overline{b}_0' \sim_{\mathcal{B}} \overline{b}_0. \end{align} $$

Since $\overline {b}_0' \sim _{\mathcal {B}} \overline {b}_0$ , $f(\overline {b}_0') \sim _{\mathcal {A}} f(\overline {b}_0)=\overline {b}_1$ and so $f(\overline {b}_0')$ inherits $\overline {b}_1$ ’s property of enumerating a substructure of $\mathcal {A}$ , i.e., $\left < f(\overline {b}_0') \right>_{\mathcal {A}}=f(\overline {b}_0')$ .

Fix any $\overline {e}_1 \subseteq \overline {b}_1'=\left < f(\overline {b}_0') \right>_{\mathcal {A}}=f(\overline {b}_0')$ such that $\overline {e}_1 \sim _{\mathcal {A}} \overline {a}$ . Then there exists $\overline {e}_0:=f^{-1}(\overline {e}_1) \subseteq \overline {b}_0'$ . The similarity (5) guarantees the existence of some $\overline {e}_0' \subseteq \overline {b}_0$ (on the same coordinates as $\overline {e}_0 \subseteq \overline {b}_0'$ ) such that $\overline {e}_0' \sim _{\mathcal {B}} \overline {e}_0$ . By the qftp-respecting property for semi-retractions, $f(\overline {e}_0') \sim _{\mathcal {A}} f(\overline {e}_0) = \overline {e}_1 \sim _{\mathcal {A}} \overline {a}$ , and we just argued that $\overline {b}_0$ has the restricted inverse images under f property for $\overline {a}$ witnessed by $\overline {a}_0$ , so $\overline {e}_0' \sim _{\mathcal {B}} \overline {a}_0$ , thus $\overline {e}_0 \sim _{\mathcal {B}} \overline {a}_0$ , as desired.

Now we are ready to adapt Proposition 5.14 to the case of relational signatures.

Having pointed out this technical property, we can deduce the relational case with little machinery.