WEAK A2 SPACES, THE KASTANAS GAME, AND STRATEGICALLY RAMSEY SETS

Abstract

We introduce the notion of a weak A2 space (or wA2-space), which generalises spaces satisfying Todorčević’s axioms A1–A4 and countable vector spaces. We show that in any Polish weak A2 space, analytic sets are Kastanas Ramsey, and discuss the relationship between Kastanas Ramsey sets and sets in the projective hierarchy. We also show that in all spaces satisfying A1–A4, every subset of $\mathcal {R}$ is Kastanas Ramsey iff Ramsey, generalising the recent result by [2]. Finally, we show that in the setting of Gowers wA2-spaces, Kastanas Ramsey sets and strategically Ramsey sets coincide, providing a connection between the recent studies on topological Ramsey spaces and countable vector spaces.

1 Introduction

In this article we show the notion of a weak A2 space provides a direct connection between the study of the abstract Kastanas game on closed triples satisfying Todorčević’s axioms A1–A4 in [2], and the study of strategically Ramsey subsets of a countable vector space in [12]. We show that the set-theoretic properties of strategically Ramsey subsets of countable vector spaces and Ramsey subsets of topological Ramsey spaces are consequences of the properties of Kastanas Ramsey sets in wA2-spaces.

It was shown in [2] that if is a closed triple satisfying Todorčević’s axioms A1–A4, and it is selective, then a subset of $\mathcal {R}$ is Kastanas Ramsey iff it is Ramsey. On the other hand, Rosendal showed that all analytic subsets of a countable vector space are strategically Ramsey, along with other set-theoretic behaviour of strategically Ramsey sets. This was made further abstract in [11], where de Rancourt introduced the notion of a Gowers space, and showed that all analytic subsets of a Gowers space are strategically Ramsey.

This article presents three main theorems. Our first theorem generalises the result given in [2] to all spaces satisfying A1–A4.

Theorem 1.1. Suppose that is a closed triple satisfying A1–A4. Then a set $\mathcal {X} \subseteq \mathcal {R}$ is Kastanas Ramsey iff it is Ramsey.

Observing that the abstract Kastanas game may be similarly studied on wA2-spaces, we present a few set-theoretic properties of the set of Kastanas Ramsey subsets of a wA2-space $\mathcal {R}$ . If $\mathcal {AR}$ is countable, then $\mathcal {R}$ is a Polish space under the usual metrisable topology, so we may consider the projective hierarchy of subsets of $\mathcal {R}$ .

Theorem 1.2. Suppose that is a wA2-space, and that $\mathcal {AR}$ is countable (so $\mathcal {R}$ is a Polish space under the metrisable topology). Then every analytic set is Kastanas Ramsey.

We will also show that for a large class of wA2-spaces, Theorem 1.2 is consistently optimal (see Theorem 4.8 and Corollary 4.9).

In our last section, we study the corresponding version of the Kastanas game on Gowers space. ¹ We relate the two concepts by introducing the notion of a Gowers wA2-space (in which countable vector spaces are an example of).

Theorem 1.3. Let be a Gowers wA2-space. Then the Kastanas game on $\mathcal {R}$ (as a Gowers space) and the Kastanas game on $\mathcal {R}$ (as a wA2-space) are equivalent. Furthermore, a subset of $\mathcal {R}$ is Kastanas Ramsey iff it is strategically Ramsey.

For the precise statements, see Theorem 5.18 and Proposition 5.13.

2 Weak A2 spaces

In this section, we provide a recap of the axioms of topological Ramsey spaces presented by Todorčević in [15], which would preface the setup of a wA2-space. We follow up by discussing various examples of wA2-spaces, and an overview of Ramsey subsets of wA2-spaces, motivating the need to study an alternative variant of an infinite-dimensional Ramsey property.

2.1 Axioms

We recap the four axioms presented by Todorčević in [15], which are sufficient conditions for a triple to be a topological Ramsey space. Here, $\mathcal {R}$ is a non-empty set, be a quasi-order on $\mathcal {R}$ , and is a function. We also define a sequence of maps $r_n : \mathcal {R} \to \mathcal {AR}$ by $r_n(A) := r(A,n)$ for all $A \in \mathcal {R}$ . Let $\mathcal {AR}_n \subseteq \mathcal {AR}$ be the image of $r_n$ (i.e., $a \in \mathcal {AR}_n$ iff $a = r_n(A)$ for some $A \in \mathcal {R}$ ).

The four axioms are as follows:

1. (A1)

   1. (1) $r_0(A) = \emptyset $ for all $A \in \mathcal {AR}$ .

   2. (2) $A \neq B$ implies $r_n(A) \neq r_n(B)$ for some n.

   3. (3) $r_n(A) = r_m(B)$ implies $n = m$ and $r_k(A) = r_k(B)$ for all $k < n$ .

   (A1) For each $a \in \mathcal {AR}$ , let denote the unique n in which $a \in \mathcal {AR}_n$ . By Axiom A1(3), this n is well-defined.

2. (A2) There is a quasi-ordering on $\mathcal {AR}$ such that:

   1. (1) is finite for all $b \in \mathcal {AR}$ .

   2. (2) iff .

   3. (3) .

3. (A3) We may define the Ellentuck neighbourhoods as follows: For any $A \in \mathcal {R}$ , $a \in \mathcal {AR,}$ and $n \in \mathbb {N}$ , we let

   
   Then the depth function defined by, for $B \in \mathcal {R}$ and $a \in \mathcal {AR}$ ,
   
   satisfies the following:

   1. (1) If , then for all , $[a,A] \neq \emptyset $ .

   2. (2) If and $[a,A] \neq \emptyset $ , then there exists such that $\emptyset \neq [a,A'] \subseteq [a,A]$ .

   For each $A \in \mathcal {R}$ , we let

   
   If $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , we also define
   

4. (A4) If and if , then there exists such that or .

We introduce a useful weakening of Axiom A2, which we shall call weak A2, or just wA2.

Axiom wA2. There is a quasi-ordering on $\mathcal {AR}$ such that:

• (w1) is countable for all $b \in \mathcal {AR}$ .

• (2) iff .

• (3) .

Note that by axiom A1, we may identify each element $A \in \mathcal {R}$ as a sequence of elements of $\mathcal {AR}$ , via the map . Therefore, we may identify $\mathcal {R}$ as a subset of $\mathcal {AR}^{\mathbb {N}}$ .

Definition 2.1. A triple is said to be

1. (1) a closed triple if $\mathcal {R}$ is a metrically closed subset of $\mathcal {AR}^{\mathbb {N}}$ ;

2. (2) a wA2-space if is a closed triple satisfying Axioms A1, wA2, and A3;

3. (3) an A2-space if it is a wA2-space satisfying A2.

Given a wA2-space , we shall focus on the following two topologies on $\mathcal {R}$ :

1. (1) The metrisable topology—we may equip $\mathcal {R}$ with the first difference metric, where for $A,B \in \mathcal {R}$ , $d(A,B) = \frac {1}{2^n}$ , where n is the least integer such that $r_n(A) \neq r_n(B)$ . If $\mathcal {AR}$ is countable, then under this metrisable topology, $\mathcal {R}$ is a Polish space.

2. (2) The Ellentuck topology generated by open sets of the form $[a,A]$ for $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ .

In this article, unless stated otherwise all topological properties of $\mathcal {R}$ will be with respect to the metrisable topology.

2.2 Examples

We discuss several examples of wA2-spaces. All examples below, except for countable vector spaces and the singleton space (Examples 2.8 and 2.9), may be found in [15]. Discussions of countable vector spaces and strategically Ramsey sets may be found in [11–14].

Example 2.2 (Natural numbers/Ellentuck space ).

Let $\mathcal {R} = [\mathbb {N}]^\infty $ . For each n and $A \in [\mathbb {N}]^\infty $ , let $r_n(A)$ be the finite set containing the n least elements of A (so $\mathcal {AR} = [\mathbb {N}]^{<\infty }$ ). Let denote the subset relation $\subseteq $ . It is easy to check that $([\mathbb {N}]^\infty ,\subseteq ,r)$ is a closed triple satisfying A1–A4.

Example 2.3 (Infinite block sequences/Gowers’ space $\mathbf {FIN}_k^{[\infty ]}$ ).

For each , let $\mathbf {FIN}_k$ be the set of all functions $x : \mathbb {N} \to \mathbb {N}$ with finite support (i.e., the set is finite), such that and . For $x,y \in \mathbf {FIN}_k$ , we also define the following:

1. (1) (Tetris operation) $T(x)(n) := \max \{x(n) - 1,0\} \in \mathbf {FIN}_{k-1}$ .

2. (2) .

3. (3) If $x < y$ , define $(x + y)(n) := \max \{x(n),y(n)\} \in \mathbf {FIN}_k$ .

It is easy to see that $<$ is transitive. We let $\mathcal {R} = \mathbf {FIN}_k^{[\infty ]}$ be the set of all infinite $<$ -increasing sequences, and for each , $r_N(A) := (x_n)_{n<N}$ . We have that $\mathcal {AR} = \mathbf {FIN}_k^{[<\infty ]}$ , the set of all finite $<$ -increasing sequences.

Given $a = (x_n)_{n<N} \in \mathbf {FIN}_k^{[<\infty ]}$ , we let

For two $a,b \in \mathbf {FIN}_k^{[<\infty ]}$ , we write iff . Then is a closed triple satisfying A1–A4.

$\mathbf {FIN}_k^{[\infty ]}$ was first introduced by Gowers in [5, 6] to resolve several long-standing problems in Banach space theory. The current formulation of $\mathbf {FIN}_k^{[\infty ]}$ may be found in [15]. The fact that satisfies A4 follows from Gowers’ $\mathbf {FIN}_k$ theorem ([5, Theorem 1] or [15, Theorem 2.2]).

Example 2.4 (Infinite block sequences $\mathbf {FIN}_{\pm k}^{[\infty ]}$ ).

Consider a setup similar to Example 2.3. Instead, we let $\mathbf {FIN}_{\pm k}$ be the set of all functions $x : \mathbb {N} \to \mathbb {Z}$ with finite support, such that , and ( or ). The tetris operation is now modified to:

$$ \begin{align*} T(x)(n) := \begin{cases} x(n) - 1, &\text{if } x(n)> 0, \\ x(n) + 1, &\text{if } x(n) < 0, \\ 0, &\text{otherwise}. \end{cases} \end{align*} $$

The rest of the setup is similar. We let $\mathcal {R} = \mathbf {FIN}_{\pm k}^{[\infty ]}$ be the set of all infinite $<$ -increasing sequences, so $\mathcal {AR} = \mathbf {FIN}_{\pm k}^{[<\infty ]}$ is the set of all finite $<$ -increasing sequences.

Given $a = (x_n)_{n<N} \in \mathbf {FIN}_{\pm k}^{[<\infty ]}$ , we let

Then $\mathbf {FIN}_{\pm k}^{[\infty ]}$ is an A2-space. However, $\mathbf {FIN}_{\pm k}^{[\infty ]}$ does not satisfy A4—for each $x \in \mathbf {FIN}_{\pm k}$ , let

Now let

$$ \begin{align*} Y := \{x \in \mathbf{FIN}_{\pm k} : x(n_x) = k\}. \end{align*} $$

Then, for all $A \in \mathbf {FIN}_{\pm k}^{[\infty ]}$ , and .

$\mathbf {FIN}_{\pm k}^{[\infty ]}$ was first introduced by Gowers in [5], and the current formulation of $\mathbf {FIN}_{\pm k}^{[\infty ]}$ may be found in [15].

Example 2.5 (Hales–Jewett space $W_{Lv}^{[\infty ]}$ ).

Let $L = \bigcup _{n=0}^\infty L_n$ be a countable increasing union of finite alphabets with variable $v \notin L$ . Let $W_{Lv}$ denote the set of all variable-words over $L \cup \{v\}$ , i.e., all finite strings of elements of $L \cup \{v\}$ in which v occurs at least once. For each $x \in W_{Lv}$ and , we let denote the word in which all v’s occurring in x are replaced with a.

Given $x_0,\dots ,x_n \in W_{Lv}$ , we write $(x_i)_{i<n} < x_n$ iff $\sum _{i<n} |x_i| < |x_n|$ . A sequence $(x_n)_{n<N}$ is rapidly increasing if $(x_i)_{i<n} < x_n$ for all $n < N$ . Let $\mathcal {R} = W_{Lv}^{[\infty ]}$ be the set of all infinite rapidly increasing sequences, and for each , $r_N(A) := (x_n)_{n<N}$ . We have that $\mathcal {AR} = W_{Lv}^{[<\infty ]}$ , the set of all finite rapidly increasing sequences.

Given $a = (x_n)_{n<N} \in W_{Lv}^{[<\infty ]}$ , we let

For two $a,b \in W_{Lv}^{[<\infty ]}$ , we write iff , and for all $c \sqsubseteq b$ and $c \neq b$ . Then is a closed triple satisfying A1–A4.

$W_{Lv}^{[\infty ]}$ was first introduced by Hales–Jewett in [7], and the current formulation of $W_{Lv}^{[\infty ]}$ may be found in [15].

Example 2.6 (Strong subtrees $\S _\infty $ ).

Let be a tree (i.e., a subset that is closed under initial segments). We introduce some terminologies.

1. (1) A node $s \in T$ splits in T if there exist $m \neq n$ such that $s^\frown m \in T$ and $s^\frown n \in T$ . For $n \geq 1$ , we let be the set of all $s \in T$ such that s splits in T, and there are such that $s\mathord {\upharpoonright } k_i$ splits for . We then let .

2. (2) A node $s \in T$ is terminal in T if $s^\frown n \notin T$ for all n.

3. (3) The height of a non-empty tree T is defined by:

   

4. (4) T is perfect if for all $s \in T$ , either there exists some $t \sqsupseteq s$ such that t splits in T, or s is terminal in T.

Let $\mathcal {R} = \S _\infty $ be the set of strong subtrees . That is, T is a perfect subtree which satisfies the following conditions:

1. (1) For all $s,t \in A$ such that , s splits in T iff t splits in A.

2. (2) If $s \in A$ and there exists some $t \in A$ such that , then there exists some $u \in A$ such that $s \sqsubseteq u$ and .

For each $A \in \S _\infty $ and , we define

We observe that if $A \in \S _\infty $ , then $r_n(A)$ is a strong subtree of height n, so $\mathcal {AR} = \S _{<\infty }$ is the set of all strong subtrees of finite height. For $a,b \in \S _{<\infty }$ , we write iff $a \subseteq b$ , and for all nodes $s \in a,$ which are terminal in a, s splits in b or is terminal b. Then is a closed triple satisfying A1–A4.

The space $\S _\infty $ is also known as the Milliken space of strong subtrees, and was first introduced by Milliken in [10] to generalise Silver’s partition theorem for infinite trees. The theorem asserting that satisfies A4 (also proven in [10]) is the strong subtree variant of the Halpern–Läuchli theorem. The current formulation may be found in [15].

Example 2.7 (Carlson–Simpson Space $\mathcal {E}^\infty $ ).

Let $\mathcal {R} = \mathcal {E}_\infty $ denote the Carlson–Simpson space, i.e., the collection of equivalence relations A on $\mathbb {N}$ in which $\mathbb {N}/A$ is infinite. For each $x \in \mathbb {N}$ , let $[x]_A$ be the equivalence class of A containing x. We then let be the increasing enumeration of the set of all minimal representatives of the equivalence classes of A. Note that $p_0(A) = 0$ always. For each $A \in \mathcal {E}_\infty $ , we let $r_n(A) := A\mathord {\upharpoonright } p_n(A)$ , i.e., the restriction of the equivalence relation A to the domain . We denote $\mathcal {AE}_\infty := \mathcal {AR}$ . For $a,b \in \mathcal {AE}_\infty $ , we write iff and a is coarser than b. We remark that for all $a \in \mathcal {AE}_\infty $ , is the number of equivalence classes in a. Then is a closed triple satisfying A1–A4.

$\mathcal {E}^\infty $ was first introduced by Carlson–Simpson in [3], as part of their development of topological Ramsey theory. The current formulation may be found in [15].

Example 2.8 (Countable vector spaces $E^{[\infty ]}$ ).

Let $\mathbb {F}$ be a countable field, and let E be an $\mathbb {F}$ -vector space of dimension $\aleph _0$ , with a distinguished Hamel basis . Given $x \in E$ , if , we write . We then define a partial order $<$ on $E \setminus \{0\}$ by:

We let $\mathcal {R} := E^{[\infty ]}$ denote the set of all infinite $<$ -increasing sequences of non-zero vectors in E, and for each , define $r_N(A) := (x_n)_{n<N}$ . We have that $\mathcal {AR} = E^{[<\infty ]}$ is the set of finite $<$ -increasing sequences of non-zero vectors. For two $a := (x_n)_{n<N}, b := (y_m)_{m<M} \in E^{[<\infty ]}$ , we write iff for all $n < N$ . Then is a wA2-space. Furthermore:

1. (1) is an A2-space iff $\mathbb {F}$ is finite—if $\mathbb {F}$ is finite, then for all $a = (x_n)_{n<N},b = (y_m)_{m<M} \in E^{[<\infty ]}$ , iff $x_n$ is an $\mathbb {F}$ -linear combination of $y_0,\dots ,y_{M-1}$ , of which there are only finitely many of. If $\mathbb {F}$ is infinite, then for all , so A2 fails.

2. (2) satisfies A4 iff $|\mathbb {F}| = 2$ . If $|\mathbb {F}| = 2$ , then A4 follows from Hindman’s theorem ([15, Theorem 2.41]). If $|\mathbb {F}|> 2$ , then define the set:

   $$ \begin{align*} Y := \{x \in E : x = e_n + y \text{ for some } n \text{ and } e_n < y\}. \end{align*} $$
   It is not difficult to show that and for all $A \in E^{[<\infty ]}$ .

This Ramsey-theoretic framework of countable vector spaces was first introduced by Rosendal in [12, 13]. He studied the set-theoretic properties of strategically Ramsey sets in this framework, a notion motivated by the Ramsey-theoretic methods employed by Gowers in [6]. Smythe studied the local Ramsey theory of this framework in [14], extending some results by Rosendal to $\mathcal {H}$ -strategically Ramsey sets, where $\mathcal {H}$ is a family satisfying some combinatorial properties.

Example 2.9 (Singleton space).

Let $\mathcal {R} = \{(0,0,\dots )\}$ , the singleton containing the zero sequence. We define $r_n(A) := (0,\dots ,0)$ of length n, and to be the equality relation. Then is a closed triple satisfying A1–A4. Then is a closed triple satisfying A1–A4. The singleton space serves as a pathological example of a topological Ramsey space.

2.3 Ramsey sets

The definition of a Ramsey subset of a topological Ramsey space may be extended to any wA2-spaces.

Definition 2.10. Let be a wA2-space. A set $\mathcal {X} \subseteq \mathcal {R}$ is Ramsey if for all $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , there exists some $B \in [a,A]$ such that $[a,B] \subseteq \mathcal {X}$ or $[a,B] \subseteq \mathcal {X}^c$ .

By the abstract Ellentuck theorem [15, Theorem 5.4], if is a closed triple satisfying A1–A4, a subset of $\mathcal {R}$ is Ramsey iff it is Baire relative to the Ellentuck topology. Since the Ellentuck topology refines the metrisable topology, every subset of $\mathcal {R}$ which is Baire relative to the metrisable topology is Ramsey. We show that A4 is a necessary condition.

Proposition 2.11. Let be an A2-space. The following are equivalent:

1. (1) satisfies A4.

2. (2) Every clopen subset of $\mathcal {R}$ is Ramsey.

Proof. (1) $\implies $ (2) follows from the abstract Ellentuck theorem. For the converse, let $A \in \mathcal {R}$ , $a \in \mathcal {AR}\mathord {\upharpoonright } A,$ and . Define

Since $\mathcal {X}$ is clopen, it is Ramsey. Therefore, there exists some $B \in [a,A]$ such that $[a,B] \subseteq \mathcal {X}$ or $[a,B] \subseteq \mathcal {X}^c$ . If $[a,B] \subseteq \mathcal {X}$ , then , and if $[a,B] \subseteq \mathcal {X}^c$ , then . By A3, we may let such that $[a,B'] \subseteq [a,B]$ , so $B'$ witnesses that A4 holds for .

Furthermore, Ramsey subsets of a wA2-space need not be closed under countable intersections (and are hence not closed under countable unions).

Example 2.12 (Countable vector space $E^{[\infty ]}$ ).

Let $\mathbb {F}$ be a countable field such that $|\mathbb {F}|> 2$ , and let E be an $\mathbb {F}$ -vector space of dimension $\aleph _0$ . Let $Y \subseteq E$ be the set defined in Example 2.8 such that for all $A \in E^{[\infty ]}$ , and . We define $Y_n \subseteq E$ for each n such that $Y_n^c$ is finite for all n, and . This is possible as E is countable.

For each n, we let

Note that .

Claim. $\mathcal {X}_n$ is Ramsey for all n.

Proof. Let $A \in E^{[\infty ]}$ and $a \in E^{[<\infty ]}\mathord {\upharpoonright } A$ . If $a = (x_0,\dots ,x_{n-1}) \neq \emptyset $ , then $[a,A] \subseteq \mathcal {X}_n$ or $\mathcal {X}_n^c$ , depending if $x_0 \in Y_n$ or $x_0 \in Y_n^c$ . Otherwise, let be such that $x < B$ for all $x \in Y_n^c$ , which is possible as $Y_n^c$ is finite. Then $[\emptyset ,B] \subseteq \mathcal {X}_n^c$ .

However, $\mathcal {X}$ is not Ramsey—for all $A \in E^{[\infty ]}$ , there exist and such that $x_0 \in Y$ and $y_0 \in Y^c$ , so $[\emptyset ,A] \not \subseteq \mathcal {X}$ and $[\emptyset ,A] \not \subseteq \mathcal {X}^c$ .

These observations show that Ramsey sets in wA2-spaces are not as well-behaved as Ramsey sets in topological Ramsey spaces, prompting us to consider an alternative notion of Ramsey sets in wA2-spaces—one example being the notion of Kastanas Ramsey.

3 The Kastanas game in wA2-spaces

We shall introduce the abstract Kastanas game in wA2-spaces, and study the set-theoretic properties of Kastanas Ramsey sets.

3.1 The abstract Kastanas game

Definition 3.1 [2, Definition 5.1].

Let be a wA2-space. Let $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ . The Kastanas game played below $[a,A]$ , denoted as $K[a,A]$ , is defined as a game played by Players I and II in the following form:

The outcome of this game is (i.e., the unique element $B \in \mathcal {R}$ such that $r_n(B) = a_n$ for all n). We say that I (resp., II) has a strategy in $K[a,A]$ to reach $\mathcal {X} \subseteq \mathcal {R}$ if it has a strategy in $K[a,A]$ to ensure that the outcome is in $\mathcal {X}$ .

Note that we do not require to satisfy either A2 or A4 for the game to make sense. In particular, we may consider the abstract Kastanas game in countable vector spaces.

Definition 3.2. Let be a wA2-space. A set $\mathcal {X} \subseteq \mathcal {R}$ is Kastanas Ramsey if for all $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , there exists some $B \in [a,A]$ such that one of the following holds:

1. (1) I has a strategy in $K[a,B]$ to reach $\mathcal {X}^c$ .

2. (2) II has a strategy in $K[a,B]$ to reach $\mathcal {X}$ .

The seemingly unintuitive decision to define Kastanas Ramsey sets such that I plays into the set $\mathcal {X}^c$ , instead of $\mathcal {X}$ , allows us to describe the relationship between Kastanas Ramsey sets and strategically Ramsey sets, projections and projective sets more easily.

We conclude the section with some definitions and notations which are useful in studying the Kastanas game.

Definition 3.3. Let be a wA2-space, and let $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ . Consider the Kastanas game $K[a,A]$ .

1. (1) A (partial) state is a tuple containing the plays made by both players in a partial play of the game $K[a,A]$ . For instance, a state ending on the $n^{\text {th}}$ turn of I would be

   $$ \begin{align*} s = (A_0,a_1,B_0,A_1,\dots,A_{n-1}). \end{align*} $$

   The rank of s would be the turn number in which the last play was made (so, in the example above, ).

   A state for I (resp., for II ) is a state as defined above, except only the plays made by I (resp., by II) are listed in the tuple. For instance a state for I would be

   $$ \begin{align*} s_{\textbf{I}} = (A_0,A_1,\dots,A_{n-1}), \end{align*} $$
   and a state for II would be
   $$ \begin{align*} s_{\textbf{II}} = (a_1,B_0,\dots,a_n,B_{n-1}). \end{align*} $$

2. (2) If $s = (A_0,a_1,B_0,\dots ,A_n)$ is a state of rank n, then the realisation of s, denoted as $a(s)$ , is the element of $\mathcal {AR}$ last played by II, i.e., $a_{n-1}$ . We also say that s realises $a(s)$ . If , then $a(s) := a$ .

   If $\sigma $ is a strategy for I (resp., II) in $K[a,A]$ and s is a state for I (resp., II) following $\sigma $ , then $a(s)$ is understood to mean the element $a(s')$ (i.e., $a_n$ ), where $s'$ is the state, following $\sigma $ , such that s is the play made by I (resp., II) in $s'$ .

3. (3) A total state s is an infinite sequence of plays made by both players in a total play of the game $K[a,A]$ . Thus, a total state s would be of the form:

   $$ \begin{align*} s = (A_0,a_1,B_0,A_1,a_2,B_1,\dots). \end{align*} $$

   The realisation of s would be the element , i.e., the unique element $A(s) \in \mathcal {R}$ such that for all n.

4. (4) If s is a state (for I or II, resp.), and n is such that either or s is a total state, then write the restriction of s to rank n, denoted $s\mathord {\upharpoonright } n$ , as the partial state (for I or II, resp.) following s up to turn n of s.

Definition 3.4. If $s = (A_0,a_1,B_0,\dots ,A_n)$ is a state of rank n ending with a play by I, then . If $s = (A_0,a_1,B_0,\dots ,A_n,a_{n+1},B_n)$ is a state of rank n ending with a play by II, then . We also define .

3.2 Basic properties

Let be a wA2-space. We let $\boldsymbol {\mathcal {KR}}$ denote the set of all Kastanas Ramsey subsets of $\mathcal {R}$ , and let be the set of all subsets of $\mathcal {R}$ whose complement is Kastanas Ramsey. In this section, we study various set-theoretic properties of $\boldsymbol {\mathcal {KR}}$ . We state some positive results that are analogous to those of strategically Ramsey sets presented in [12]. The proofs are inspired by those shown in the same article.

Lemma 3.5. Let be a wA2-space. Let $\mathcal {X}_n \subseteq \mathcal {R}$ for each n, and let . For any $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , there exists some $B \in [a,A]$ such that one of the following must hold:

1. (1) I has a strategy in $K[a,B]$ to reach $\mathcal {X}$ .

2. (2) II has a strategy $\tau $ in $K[a,B]$ such that the following holds: For any total state s following $\tau $ , there exists some n such that if $a(s\mathord {\upharpoonright } n) = a_n$ and , then I has no strategy in $K[a_n,B_n]$ to reach $\mathcal {X}_n$ .

See also [12, Lemma 4].

Proof. Let be a countable family of subsets of $\mathcal {R}$ , and let . Fix any $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , and assume that (2) fails for all $B \in [a,A]$ . In particular, applying $n = 0$ to the negation of (2), I has a strategy in $K[a,B]$ to reach $\mathcal {X}_0$ for all $B \in [a,A]$ .

If $B \in [a,A]$ and $b \in \mathcal {AR}\mathord {\upharpoonright }[a,B]$ , say that “(2) holds for $(b,B)$ ” if II has a strategy $\tau $ in $K[b,B]$ such that, for any total state s following $\tau $ , there exists some such that if and , then I has no strategy in $K[a_n,B_n]$ to reach $\mathcal {X}_n$ . Note that this would also imply that I has no strategy in $K[a_n,C]$ to reach $\mathcal {X}_n$ for all $C \in [a_n,B_n]$ .

Claim. For all $B \in [a,A]$ and $b \in \mathcal {AR}\mathord {\upharpoonright }[a,B]$ , (2) holds for $(b,B)$ iff for all $A' \in [b,B]$ , there exists some and $B' \in [b',A']$ such that (2) holds for $(b',B')$ .

Proof. $\underline{\implies} $ : Let $\tau $ be the strategy in $K[b,B]$ witnessing that (2) holds for $(b,B)$ . Given any $A' \in [b,B]$ , consider the play where I begins with A, and player II responds with and $B' \in [b',A']$ according to $\tau $ . The restriction of the strategy $\tau $ to $K[b',B']$ is a strategy witnessing that (2) holds for $(b',B')$ .

$\underline{\impliedby}$ : Suppose that for all $A' \in [b,B]$ , there exists some , $B' \in [b',A']$ and strategy $\tau _{A'}$ witnessing that (2) holds for $(b',B')$ . Define the strategy $\tau $ in $K[b,B]$ such that if I begins with $A'$ , then II responds with $b'$ and $B'$ , then continue according to $\tau _{A'}$ . This gives a strategy witnessing that (2) holds for $(b,B)$ .

Let denote the set of all such that for all $B \in [a',A]$ , (2) does not hold for $(a',B)$ . Since we assumed that (2) fails for all $B \in [a,A]$ , by the previous claim there exists some $A_\emptyset \in [a,A]$ such that . As stated in the first paragraph, I has a strategy $\sigma _\emptyset $ in $K[a,A_\emptyset ]$ to reach $\mathcal {X}_0$ . To finish the proof, we shall construct a strategy $\sigma $ for I in $K[a,A_\emptyset ]$ to reach $\mathcal {X}$ .

For the rest of this proof, all states are assumed to be for II. Let $\sigma (\emptyset ) := \sigma _\emptyset (\emptyset )$ . Now suppose for each state s of $K[a,A_\emptyset ]$ following $\sigma $ of rank n, we define the following:

1. (1) If $n> 0$ and $s' = s\mathord {\upharpoonright }(n - 1)$ , then $A_s \in [a(s),A_{s'}]$ .

2. (2) $\sigma _s$ is a strategy for I in $K[a(s),A_s]$ to reach $\mathcal {X}_n$ .

3. (3) If , then $t_s^i$ is a state of $K[a(s\mathord {\upharpoonright } i),A_{s\mathord {\upharpoonright } i}]$ following $\sigma _{s\mathord {\upharpoonright } i}$ of rank $n - i$ , and $a(t_s^i) = a(s\mathord {\upharpoonright } i)$ .

4. (4) $\sigma _{s\mathord {\upharpoonright }(i+1)}(t_s^{i+1}) \in [a_s,\sigma _{s\mathord {\upharpoonright } i}(t_s^i)]$ , and $\sigma (s) \in [a(s),\sigma _s(t_s^n)]$ .

5. (5) For all and $B \in [b,A_s]$ , (2) does not hold in $(b,B)$ .

Now let s be a state of $K[a,A_\emptyset ]$ following $\sigma $ of rank $n + 1$ . We may write $s = (s\mathord {\upharpoonright } n)^\frown (a(s),B_s)$ . Since $B_s \in [a(s),\sigma (s\mathord {\upharpoonright } n)] \subseteq [a(s),\sigma _\emptyset (t_{s\mathord {\upharpoonright } n}^0)]$ , we may define $t_s^0 := {t_{s\mathord {\upharpoonright } n}^0}^\frown (a(s),B_s)$ , which is a legal state in $K[a,A_\emptyset ]$ following $\sigma _\emptyset $ . We have $\sigma _\emptyset (t_s^0) \in [a(s),B_s] \subseteq [a(s),\sigma _{s\mathord {\upharpoonright } 1}(t_{s\mathord {\upharpoonright } n}^1)]$ , so we may define $t_s^1 := {t_{s\mathord {\upharpoonright } n}^1}^\frown (a(s),\sigma _\emptyset (t_s^0))$ . This again, gives us a legal state in $K[a(s\mathord {\upharpoonright } 1),A_{s\mathord {\upharpoonright } 1}]$ following $\sigma _{s\mathord {\upharpoonright } 1}$ . We may repeat this process to give us states $t_s^i$ for . We let $A_s' := \sigma _{s\mathord {\upharpoonright } n}(t_s^n) \in [a(s\mathord {\upharpoonright } n),A_{s\mathord {\upharpoonright } n}]$ .

Let be the set of all such that there exists some $B \in [a(s),A_s']$ in which (2) holds for $(a',B)$ . By (5) of the induction hypothesis and the claim, we may obtain $A_s \in [a(s),A_s']$ such that , and I has a winning strategy $\sigma _s$ in $K[a(s),A_s]$ to reach $\mathcal {X}_{n+1}$ . Define $\sigma (s) := \sigma _s(\emptyset )$ . This is indeed a legal move, as

$$ \begin{align*} \sigma(s) \in [a(s),A_s] \subseteq [a(s),A_s'] \subseteq [a(s),A_{s\mathord{\upharpoonright} n}] \subseteq [a(s),B_s]. \end{align*} $$

This completes the induction. We see that $\sigma $ is indeed a strategy for I in $K[a,A_\emptyset ]$ to reach $\mathcal {X}$ —if s is a total state of $K[a,A_\emptyset ]$ following $\sigma $ , then $A(s) = A(t_s^n) \in \mathcal {X}_n$ for all n, so . This completes the proof.

Proposition 3.6. For any wA2-space , $\boldsymbol {\mathcal {KR}}$ is closed under countable unions.

See also [12, Theorem 9].

Proof. Let be a countable family of subsets of $\mathcal {R}$ , and let . Fix any $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ . If there exists some $B \in [a,A]$ such that II has a strategy in $K[a,B]$ to reach $\mathcal {X}$ , then we’re done, so assume otherwise. Consider applying Lemma 3.5 to . We claim that (2) fails for all $(a,B),$ where $B \in [a,A]$ , so by the same lemma, I has a strategy in $K[a,A]$ to reach $\mathcal {X}^c$ . Indeed, otherwise let $\tau $ be a strategy in $K[a,B]$ witnessing that (2) holds for $(a,B)$ . Player II shall follow $\tau $ until they reach some turn n, ending with II playing $(a_n,B_n')$ , such that I has no strategy in $K[a_n,B_n']$ to reach $\mathcal {X}_n^c$ . Since $\mathcal {X}_n$ is Kastanas Ramsey, II may instead play $(a_n,B_n)$ in the last turn, where $B_n \in [a_n,B_n']$ , such that II has a strategy in $K[a_n,B_n]$ to reach $\mathcal {X}_n$ . Afterwards, II follows this strategy to reach $\mathcal {X}_n$ . Since $\mathcal {X}_n \subseteq \mathcal {X}$ , this constitutes a strategy for II in $K[a,B]$ to reach $\mathcal {X}$ , contradicting our assumption.

We now turn our attention to some negative results.

Definition 3.7. Let be a wA2-space, and let .

1. (1) is $(a,A)$ -biasymptotic, where $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , if for all , and .

2. (2) is biasymptotic if is $(a,A)$ -biasymptotic for all $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ .

Thus, the assertion that $\mathcal {R}$ does not satisfy A4 is equivalent to the assertion that there exists an $(a,A)$ -biasymptotic set for some $A \in \mathcal {R}$ and $a \in \mathcal {AR}$ . We illustrate some examples here.

Example 3.8 (Infinite block sequences $\mathbf {FIN}_{\pm k}^{[\infty ]}$ ).

Recall that for each $x \in \mathbf {FIN}_{\pm k}$ , we defined

and

$$ \begin{align*} Y := \{x \in \mathbf{FIN}_{\pm k} : x(n_x) = k\}. \end{align*} $$

Then, for all $A \in \mathbf {FIN}_{\pm k}^{[\infty ]}$ , and . Thus, the set

is a biasymptotic set.

Example 3.9 (Countable vector space $E^{[\infty ]}$ ).

Let $\mathbb {F}$ be a field such that $|F|> 2$ , and let E be an $\mathbb {F}$ -vector space of dimension $\aleph _0$ . We defined the set

$$ \begin{align*} Y := \{x \in E : x = e_n + y \text{ for some } n \text{ and } e_n < y\}. \end{align*} $$

We have that and for all $A \in E^{[<\infty ]}$ . Thus, the set

$$ \begin{align*} \{a \in E^{[<\infty]} : a = (x_0,\dots,x_n) \wedge x_n \in Y\} \end{align*} $$

is biasymptotic.

Proposition 3.10. Let be a wA2-space. If A4 fails, then there exists some which is not Ramsey.

Proof. Let be an $(a,A)$ -biasymptotic set for some $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ . Define

Since is $(a,A)$ -biasymptotic, for any $B \in [a,A]$ , there exists some $C \in [a,B]$ such that , and some $C' \in [a,B]$ such that . Consequently, $[a,B] \cap \mathcal {X} \neq \emptyset $ and $[a,B] \cap \mathcal {X}^c \neq \emptyset $ , so $\mathcal {X}$ is not Ramsey as B is arbitrary. However, $\mathcal {X}$ is a countable union of clopen sets, so it is Borel (under the metrisable topology). By the Borel determinacy for , the game $K[a,A]$ to reach $\mathcal {X}$ or $\mathcal {X}^c$ is always determined, so .

Proposition 3.11. Let be a wA2-space, and assume that it has a biasymptotic set. If $\boldsymbol {\mathcal {KR}} \neq \mathcal {P}(\mathcal {R})$ , then $\boldsymbol {\mathcal {KR}}$ is not closed under complements.

Proof. Fix a biasymptotic set , and let $\mathcal {X} \subseteq \mathcal {R}$ be not Kastanas Ramsey. Define two sets as follows:

Observe that both $\mathcal {X}_0^c$ and $\mathcal {X}_1^c$ are Kastanas Ramsey: Indeed, for any $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , II has a winning strategy in $K[a,A]$ to reach $\mathcal {X}_0^c$ by playing for all n, and II also has a winning strategy in $K[a,A]$ to reach $\mathcal {X}_1^c$ by playing for all n. On the other hand, we have that $\mathcal {X}_0 \cup \mathcal {X}_1 = \mathcal {X}$ , so if both $\mathcal {X}_0$ and $\mathcal {X}_1$ are Kastanas Ramsey, then so is $\mathcal {X}$ by Proposition 3.6, a contradiction. Thus, at least one of $\mathcal {X}_0$ or $\mathcal {X}_1$ witnesses that $\boldsymbol {\mathcal {KR}}$ is not closed under complements.

Proposition 3.12. Let be a wA2-space, and assume that it has a biasymptotic set. If $\boldsymbol {\mathcal {KR}} \neq \mathcal {P}(\mathcal {R})$ , then $\boldsymbol {\mathcal {KR}}$ is not closed under finite intersections.

Proof. Fix a biasymptotic set , and let $\mathcal {X} \subseteq \mathcal {R}$ be a Kastanas Ramsey set in which $\mathcal {X}^c$ is not Kastanas Ramsey. Define two sets as follows:

By the same argument as in Proposition 3.11, $\mathcal {X}_0^c$ and $\mathcal {X}_1^c$ are Kastanas Ramsey. However, $\mathcal {X}^c = \mathcal {X}_0^c \cap \mathcal {X}_1^c$ is not.

3.3 Kastanas Ramsey sets in topological Ramsey spaces

We shall now give a proof of Theorem 1.1, which is split into a proof of two different propositions. The first proposition is as follows.

Proposition 3.13 [2, Proposition 4.2].

Let be an A2-space. For every $\mathcal {X} \subseteq \mathcal {R}$ , $A \in \mathcal {R,}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , I has a strategy in $K[a,A]$ to reach $\mathcal {X}$ iff $[a,B] \subseteq \mathcal {X}$ for some $B \in [a,A]$ .

We remark that the proof in [2] assumes that satisfies the following property: If $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , and $b \sqsubseteq a$ but $b \neq a$ , then . While it is not true that all spaces satisfying A1–A4 would also satisfy such a property, the gap may be fixed with a careful enumeration of elements of $\mathcal {AR}$ . We omit the details.

The second proposition is as follows.

Proposition 3.14. Suppose that satisfies A1–A4. For every $\mathcal {X} \subseteq \mathcal {R}$ , $A \in \mathcal {R,}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , if II has a strategy in $K[a,A]$ to reach $\mathcal {X}$ , then I has a strategy in $K[a,A]$ to reach $\mathcal {X}$ .

A proof of Proposition 3.14 for selective topological Ramsey spaces ([2, Definition 5.4]) was provided in [2]. We shall use the idea presented in [4] to instead prove Lemma 5.5 of [2] using a semiselectivity argument.

Definition 3.15. Let $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ .

1. (1) A family of subsets $\vec {\mathcal {D}} = \{\mathcal {D}_b\}_{b \in \mathcal {AR}\mathord {\upharpoonright }[a,A]}$ is dense open below $[a,A]$ if for all $b \in \mathcal {AR}\mathord {\upharpoonright }[a,A]$ , $\mathcal {D}_b$ is a -downward closed subset of $[b,A]$ , and for all $B \in [b,A]$ , there exists some $C \in [b,B]$ such that $C \in \mathcal {D}_b$ .

2. (2) Let $\vec {\mathcal {D}} = \{\mathcal {D}_b\}_{b \in \mathcal {AR}\mathord {\upharpoonright }[a,A]}$ be dense open below $[a,A]$ . We say that $B \in [a,A]$ diagonalises $\vec {\mathcal {D}}$ if for all $b \in \mathcal {AR}\mathord {\upharpoonright }[a,B]$ , there exists some $A_b \in \mathcal {D}_b$ such that $[b,B] \subseteq [b,A_b]$ .

Lemma 3.16. If is an A2-space, then every family of subsets $\vec {\mathcal {D}} = \{\mathcal {D}_b\}_{b \in \mathcal {AR}\mathord {\upharpoonright }[a,A]}$ , which is dense open below $[a,A]$ has a diagonalisation.

In other words, Lemma 3.16 asserts that $\mathcal {R}$ is a “semiselective coideal.”

Proof. Fix some $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ . Suppose that $\vec {\mathcal {D}} = \{\mathcal {D}_b\}_{b \in \mathcal {AR}\mathord {\upharpoonright }[a,A]}$ is dense open below $[a,A]$ . We shall define a fusion sequence in $[a,A]$ , with , such that $A_{n+1} \in [a_{n+1},A_n]$ : Let $A_0 := A$ , and suppose that $A_n$ has been defined. Let $\{b_i : i < N\}$ enumerate the set of all $b \in \mathcal {AR}\mathord {\upharpoonright } A_n$ such that $a \sqsubseteq b$ and . Let $A_{n+1}^0 := A_n$ . If $A_{n+1}^i \in [a_{n+1},A_n]$ has been defined, let $B_{n+1} \in \mathcal {D}_{b_i}$ be such that $B_{n+1} \in [b_i,A_{n+1}^i]$ , which exists as $\mathcal {D}_{b_i}$ is dense open in $[b_i,A]$ . By A3, we then let $A_{n+1}^{i+1} \in [a_{n+1},A_n^i]$ such that $[b_i,A_{n+1}^{i+1}] \subseteq [b_i,B_{n+1}]$ . We complete the induction by letting $A_{n+1} := A_{n+1}^N$ . Let B be the limit of the fusion sequence , and we have that B diagonalises $\vec {\mathcal {D}}$ .

We are now ready to prove Proposition 3.14.

Lemma 3.17. Let be a closed triple satisfying A1–A4. Suppose that and $g : [a,A] \to [a,A]$ are two functions such that for all :

1. (1) .

2. (2) $g(B) \in [f(B),B]$ .

We also say that these two functions $f,g$ are suitable in $[a,A]$ . Then there exists some $E_{f,g} \in [a,A]$ such that for all , there exists some $B \in [a,A]$ such that $f(B) = b$ and $[b,E_{f,g}] \subseteq [b,g(B)]$ .

Proof. For each , we define

$$ \begin{align*} \mathcal{D}_{b,0} &:= \{D \in [b,A] : \exists B \in [a,A] \text{ s.t. } f(B) = b \wedge D \in [b,g(B)]\}, \\ \mathcal{D}_{b,1} &:= \{D \in [b,A] : \forall C \in [a,A], \, g(C) \in [b,D] \to g(C) \notin \mathcal{D}_{b,0}\}. \end{align*} $$

Let $\mathcal {D}_b := \mathcal {D}_{b,0} \cup \mathcal {D}_{b,1}$ . Observe that $\mathcal {D}_b$ is dense open in $[b,A]$ : Clearly both $\mathcal {D}_{b,0}$ and $\mathcal {D}_{b,1}$ are open. If $D \in [b,A]$ and $D \notin \mathcal {D}_{b,1}$ , then there exists some $C \in [a,A]$ such that $g(C) \in [b,D]$ . Then and $g(C) \in \mathcal {D}_{b,0}$ , so $\mathcal {D}_b$ is dense.

By Lemma 3.16, there exists some $D \in [a,A]$ diagonalising $(\mathcal {D}_b)_{b \in \mathcal {AR}\mathord {\upharpoonright }[a,A]}$ . Now let

By A4, there exists some $E_{f,g} \in [a,D]$ such that or . However, we see that the latter case is not possible: In this case, we let $b := f(E_{f,g})$ . Then , so let $B \in \mathcal {D}_{b,1}$ such that $[b,D] \subseteq [b,B]$ . Then $g(E_{f,g}) \in [b,E_{f,g}] \subseteq [b,D] \subseteq [b,B]$ , so $g(E_{f,g}) \notin \mathcal {D}_{b,0}$ . But $g(E_{f,g}) \in [b,B]$ , so this implies that $f(E_{f,g}) \neq b$ , a contradiction.

We shall show that $E_{f,g}$ works. Let . Then , so there exists some $B \in \mathcal {D}_{b,0}$ such that $f(B) = b$ and $[b,E_{f,g}] \subseteq [b,D] \subseteq [b,g(B)]$ , as desired.

Proof of Proposition 3.14.

Let $\sigma $ be a strategy for II in $K[a,A]$ to reach $\mathcal {X}$ . We shall construct a strategy $\tau $ for I in $K[a,A]$ to reach $\mathcal {X}$ as follows: We shall assign a state s (for II) in $K[a,A]$ following $\tau $ , to a state $t_s$ (for II) in $K[a,A]$ , following $\sigma $ , such that:

1. (1) $a(s) = a(t_s)$ .

2. (2) .

3. (3) If $s' \sqsubseteq s$ , then $t_{s'} \sqsubseteq t_s$ .

We begin by defining $t_\emptyset := \emptyset $ . Now suppose that s is a state (for II) in $K[a,A]$ following $\tau $ so far. We define the functions $f_s,g_s$ by stipulating that for all , $(f_s(B),g_s(B)) := \sigma ({t_s}^\frown B)$ . Observe that $f_s,g_s$ are suitable in (when restricted to ), so by Lemma 3.17 there exists some such that for all , there exists some such that $f_s(B_{s,b}) = b$ and $[b,E_s] \subseteq [b,g(B_{s,b})]$ . Thus, we define , and for all and $C \in [b,E_s]$ , define

$$ \begin{align*} t_{s^\frown(b,C)} := {t_s}^\frown(b,g(B_{s,b})). \end{align*} $$

Clearly, (1) and (3) of the induction hypothesis are satisfied. (2) is also satisfied, as

This completes the induction. Since every total state following $\tau $ corresponds to a total state following $\sigma $ with the same outcome, $\tau $ is a strategy for I to reach $\mathcal {X}$ .

Combined with Proposition 3.6, we get the following.

Corollary 3.18. Let be a closed triple satisfying A1–A4. Then the set of (Kastanas) Ramsey subsets of $\mathcal {R}$ forms a $\sigma $ -algebra.

4 Kastanas Ramsey sets and the projective hierarchy

Given a wA2-space , if $\mathcal {AR}$ is countable then the metrisable topology is Polish, allowing us to discuss the projective hierarchy on $\mathcal {R}$ . We discuss some relationships between Kastanas Ramsey sets and sets in the projective hierarchy.

4.1 Projective hierarchy

In this section, we prove a general relationship between Kastanas Ramsey sets and sets in the projective hierarchy, which, in particular, gives a proof of Theorem 1.2.

Given a wA2-space , we shall construct another wA2-space as follows:

1. (1) .

2. (2) Given , let $r_n(A,x) := (r_n(A),u\mathord {\upharpoonright } n)$ . Thus, if , then .

3. (3) We define a $\preceq _{\mathrm {fin}}$ on by stipulating that $(a,p) \preceq _{\mathrm {fin}} (b,q)$ iff .

4. (4) Given , we write

   $$ \begin{align*} (A,u) \preceq (B,v) \iff \forall n \, \exists m[r_n(A,u) \preceq_{\mathrm{fin}} r_m(B,v)]. \end{align*} $$

We remark that $\preceq $ is never a partial order. For instance, if $a \in \mathcal {AR}_2$ , then $(a,(0,1)) \preceq _{\mathrm {fin}} (a,(1,0))$ and $(a,(1,0)) \preceq _{\mathrm {fin}} (a,(0,1))$ , but $(a,(0,1)) \neq (a,(1,0))$ .

Lemma 4.1. Let be a wA2-space (resp., A2-space). Then the closed triple defined above is a wA2-space (resp., A2-space) which has a biasymptotic set.

Proof. It is easy to verify that satisfies A1, wA2 (resp., A2), and A3. A biasymptotic set would be

Let denote the projection to the first coordinate, which is a surjective map which respects (i.e., if $(A,p) \preceq (B,q)$ then ). We also use $\vec {0}$ to denote the infinite tuple of zeroes .

Lemma 4.2. Let be a wA2-space. Let be a subset. Let $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ . If II has a strategy in $K[(a,p),(A,\vec {0})]$ to reach $\mathcal {C}$ for some , then II has a strategy in $K[a,A]$ to reach $\pi _0[\mathcal {C}]$ .

Proof. If $\sigma $ is a strategy for II in $K[(a,p),(A,\vec {0})]$ to reach $\mathcal {C}$ , then the strategy $\tau $ for II in $K[a,A]$ defined by $\tau (A_0,\dots ,A_{n-1}) := (b,B)$ , where $\sigma ((A_0,\vec {0}),\dots ,(A_{n-1},\vec {0})) = ((b,p),(B,u))$ for some $p,u$ , is a strategy to reach $\pi _0[\mathcal {C}]$ .

Lemma 4.3. Let be a wA2-space. Let be a subset. Let $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ . If for all and $B \in [a,A]$ , there exists some $C \in [a,B]$ such that I has a strategy in $K[(a,p),(C,\vec {0})]$ to reach $\mathcal {C}^c$ , then I has a strategy in $K[a,A]$ to reach $\pi _0[\mathcal {C}]^c$ .

Proof. We shall construct a strategy $\tau $ for I in $K[a,A]$ to reach $\pi _0[\mathcal {C}]^c$ . Let enumerate the set . We define a decreasing sequence as follows: Let $\sigma _{p_0}$ be the strategy for I in $K[(a,p_0),(A,\vec {0})]$ to reach $\mathcal {C}^c$ , and let $C_0 := \sigma _{p_0}(\emptyset )$ . We also define $B_{p_0} := A$ . Suppose that $A_k^0$ has been defined. By the hypothesis, there exists some $B_{p_{k+1}} \in [a,C_k]$ such that I has a strategy in $K[(a,p_{k+1}),(B_{p_{k+1}},\vec {0})]$ to reach $\mathcal {C}^c$ . Now let $C_{k+1} := \sigma _{p_{k+1}}(\emptyset )$ . Having constructed the above sequence, we now define .

Note that we may assume that for all partial states t of $K[(a,p),(B_p,\vec {0})]$ for II, $\sigma _p(t) = (B,\vec {0})$ for some $B \in \mathcal {R}$ . Now let s be a partial state of $K[a,A]$ for II following $\tau $ so far, and , and assume that $\tau (s)$ has been defined. Suppose for the induction hypothesis that we have a set such that:

1. (1) $t_{s,q}$ is a partial state of following with $a(t_{s,q}) = (a(s),q)$ .

2. (2) If , then $\tau (s) \in [a(s),A_{s,q}]$ .

Note that for the base case, for each we let $t_{\emptyset ,a}$ be the empty state of the game $K[(a,p),(B_p,\vec {0})]$ . For each and $B_n \in [a_{n+1},\tau (s)]$ , Let enumerate the set , and for each k we let and (which differs from the enumeration in the first paragraph, but it doesn’t matter). We define a decreasing sequence as follows: Let $D_0 := \sigma _{p_0}({t_{s,q_0'}}^\frown ((a_{n+1},q_0),(B_n,\vec {0})))$ . Assuming that $D_k$ has been defined, we let $D_{k+1} := \sigma _{p_k}({t_{s,q_k'}}^\frown ((a_{n+1},q_k),(D_k,\vec {0})))$ . Note that by the induction hypothesis, all the partial states listed above are legal. We conclude the construction of $\tau $ by asserting that .

We shall now show that $\tau $ is a strategy for I in $K[a,A]$ to reach $\pi _0[\mathcal {C}]^c$ . If not, then there exists some complete play $K[a,A]$ following $\tau $ such that II plays $(a_1,B_0,a_2,B_1,\dots )$ , and . In particular, we have that $(C,x) \in \mathcal {C}$ for some . For each n, let and let $p := q_0$ . By our construction of $\tau $ , there exists a complete play of the game $K[(a,p),(B_p,\vec {0})]$ following $\sigma _p$ such that II plays $((a_1,q_1),(B_0,\vec {0}),(a_2,q_2),(B_1,\vec {0}),\dots )$ . But since $\sigma _p$ is a strategy for I in $K[(a,p),(A,\vec {0})]$ to reach $\mathcal {C}^c$ , we have that $(C,x) \in \mathcal {C}^c$ , a contradiction.

These two lemmas lead us to the following results.

Theorem 4.4. Let be a wA2-space. If is Kastanas Ramsey, then $\pi _0[\mathcal {C}] \subseteq \mathcal {R}$ is Kastanas Ramsey.

Theorem 4.5. Let be a wA2-space, and assume that $\mathcal {AR}$ is countable.

1. (1) Every analytic subset of $\mathcal {R}$ is Kastanas Ramsey.

2. (2) If every coanalytic subset of is Kastanas Ramsey, then every $\boldsymbol {\Sigma }_2^1$ subset of $\mathcal {R}$ is Kastanas Ramsey. More generally, for every $n \geq 1$ , if every $\boldsymbol {\Pi }_n^1$ subset of is Kastanas Ramsey, then every $\boldsymbol {\Sigma }_{n+1}^1$ subset of $\mathcal {R}$ is Kastanas Ramsey.

See also [1, Theorem IV.4.14]. We remark that one may alternatively prove that every analytic subset of $\mathcal {R}$ is Kastanas Ramsey using Lemma 3.5, and follow an argument similar to the proof of Theorem 5 of [12].

This also allows us to extend Corollary 3.18.

Corollary 4.6. Suppose that is a closed triple satisfying A1–A4 and assume that $\mathcal {AR}$ is countable. If $\mathcal {X} \subseteq \mathcal {R}$ is in the smallest algebra of subsets of $\mathcal {R}$ containing all analytic sets, then $\mathcal {X}$ is Ramsey.

Proof. The set of Ramsey subsets of $\mathcal {R}$ is closed under complements by definition. By Proposition 3.6, the set of Ramsey subsets of $\mathcal {R}$ is closed under countable intersections, so it forms a $\sigma $ -algebra. By Theorems 1.1 and 1.2, every analytic subset of $\mathcal {R}$ is contained in this $\sigma $ -algebra.

We remark that the abstract Rosendal theorem in [11] uses a similar approach to prove that every analytic subset of of a Gowers space is strategically Ramsey, where given a Gowers space, de Rancourt constructed a second Gowers space which equips a binary sequence along with elements of X.

4.2 $\Sigma _2^1$ well ordering

We dedicate this section to showing that Theorem 1.2 is consistently optimal for a family of sufficiently well-behaved wA2-space.

Definition 4.7. Let be a wA2-space. We say that $\mathcal {R}$ is deep if for all $A \in \mathcal {R}$ , $a \in \mathcal {AR}\mathord {\upharpoonright } A$ and , there exists some $B \in [a,A]$ such that for all , .

We shall see later in the proof of Corollary 4.18 that all examples of wA2-spaces introduced in Section 2.2, except for the singleton space (Example 2.9), are deep. The singleton space is not deep as deepness implies that for all $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright }[a,A]$ , is infinite. We do not know if there are any “natural” examples of wA2-spaces which are not deep.

The main theorem of this section is as follows.

Theorem 4.8. Let be a deep wA2-space, and assume that $\mathcal {AR}$ is countable. Suppose that there exists a $\Sigma _2^1$ -good well-ordering of the reals. Then there exists a $\boldsymbol {\Sigma }_2^1$ subset of $\mathcal {R}$ which is not Kastanas Ramsey.

See also Theorem IV.7.4 of [1]. Observing that if is deep, then so is , we obtain the following corollary.

Corollary 4.9. Let be a deep wA2-space, and assume that $\mathcal {AR}$ is countable. Then there exists a coanalytic subset of which is not Kastanas Ramsey.

We now furnish a proof of Theorem 4.8. We shall now introduce two related games, which serve as a “reduction” of the Kastanas game for each player.

Definition 4.10. Let be a wA2-space. Let $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ . The fusion game played below $[a,A]$ , denoted as $Z[a,A]$ , is defined as a game played by Players I and II in the following form:

The outcome of this game is . We say that I (resp., II) has a strategy in $Z[a,A]$ to reach $\mathcal {X} \subseteq \mathcal {R}$ if it has a strategy in $Z[a,A]$ to ensure that the outcome is in $\mathcal {X}$ .

The following lemma, which roughly states that $Z[a,A]$ is a “reduction” of $K[a,A]$ for I, is obvious.

Lemma 4.11. Let be a wA2-space. For any $A \in \mathcal {R}$ and $a \in \mathcal {AR}$ , if I has a strategy in $K[a,A]$ to reach $\mathcal {X}$ , then I has a strategy in $Z[a,A]$ to reach $\mathcal {X}$ .

Similar to the game in Definition IV.7.2 of [1], it is possible to modify the fusion game to ensure that the set of all partial states is countable.

Definition 4.12. Let be a wA2-space. Let $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ . The game $Z^*[a,A]$ is defined as a game played by Players I and II in the following form:

1. (1) Player I begin by playing some .

2. (2) Player II may choose to either respond with some , or not respond, in which case I plays some .

3. (3) Repeat (2) until II chooses to respond with some for some k. Then I responds by playing some .

4. (4) Again, Player II may choose to either respond with some , or not respond, in which case I plays some .

5. (5) Repeat.

The outcome of this game is . We say that I has a strategy in $Z^*[a,A]$ to reach $\mathcal {X} \subseteq \mathcal {R}$ if either (i.e., II stops playing after some finite stage), or I has a strategy in $Z[a,A]$ to ensure that the outcome is in $\mathcal {X}$ . II has a strategy in $Z^*[a,A]$ to reach $\mathcal {X} \subseteq \mathcal {R}$ if .

It is also easy to see that this gives us another reduction for I.

Lemma 4.13. Let be a wA2-space. For any $A \in \mathcal {R}$ and $a \in \mathcal {AR}$ , if I has a strategy in $Z[a,A]$ to reach $\mathcal {X}$ , then I has a strategy in $Z^*[a,A]$ to reach $\mathcal {X}$ .

We shall now introduce the reduction of $K[a,A]$ for II.

Definition 4.14. Let be a wA2-space. The subasymptotic game played below $[a,A]$ , denoted as $Y[a,A]$ , is defined as a game played by Players I and II in the following form:

The outcome of this game is . We say that I (resp., II) has a strategy in $Y[a,A]$ to reach $\mathcal {X} \subseteq \mathcal {R}$ if it has a strategy in $Y[a,A]$ to ensure that the outcome is in $\mathcal {X}$ .

Lemma 4.15. Let be a deep wA2-space. For any $A \in \mathcal {R}$ and $a \in \mathcal {AR}$ , if II has a strategy in $K[a,A]$ to reach $\mathcal {X}$ , then II has a strategy in $Y[a,A]$ to reach $\mathcal {X}$ .

Proof. We first note that since $\mathcal {R}$ is deep, it is always possible for II to respond with a legal move in $Y[a,A]$ . The deepness of $\mathcal {R}$ also allows us to view $Y[a,A]$ as a “special case” of $K[a,A]$ : In $K[a,A]$ , if II responded with $(a_k,B_{k-1})$ , and I wants to restrict the next response by II such that for some , then I can respond to $(a_k,B_{k-1})$ by playing any $A_k \in [a_k,B_{k-1}]$ such that for all , . Therefore, a strategy for II in $K[a,A]$ to reach $\mathcal {X}$ may be passed to a strategy for II in $Y[a,A]$ to reach $\mathcal {X}$ .

Definition 4.16. Let be a wA2-space. A set $\mathcal {X} \subseteq \mathcal {R}$ is pre-Kastanas Ramsey if for all $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , there exists some $B \in [a,A]$ such that one of the following holds:

1. (1) I has a strategy in $Z^*[a,B]$ to reach $\mathcal {X}^c$ .

2. (2) II has a strategy in $Y[a,B]$ to reach $\mathcal {X}$ .

It is clear from the definition that every Kastanas Ramsey set is pre-Kastanas Ramsey.

Proof of Theorem 4.8.

It suffices to construct a $\boldsymbol {\Sigma }_2^1$ subset of $\mathcal {R}$ which is not pre-Kastanas Ramsey. Define the following two sets:

$$ \begin{align*} S_{\textbf{I}} &:= \{(a,A,\sigma) : a \in \mathcal{AR}\mathord{\upharpoonright} A \wedge \sigma \text{ is a strategy for } \textbf{I} \text{ in } Z^*[a,A]\}, \\ S_{\textbf{II}} &:= \{(a,A,\sigma) : a \in \mathcal{AR}\mathord{\upharpoonright} A \wedge \sigma \text{ is a strategy for } \textbf{II} \text{ in } Y[a,A]\}. \end{align*} $$

Note that a strategy in a game is a function from the set of partial states of the game to a play of the game. Since $\mathcal {AR}$ is countable, the sets of partial states of $Z^*[a,A]$ and of $Y[a,A]$ are also countable, and in both games all players play from a countable set. Therefore, we may naturally embed both sets $S_{\textbf {I}}$ and $S_{\textbf {II}}$ to the reals, giving us a $\Sigma _2^1$ -well ordering $\prec _s$ of $S_{\textbf {I}}$ and $S_{\textbf {II}}$ . Note that $\prec _s$ is of order-type , so every triple in $S_{\textbf {I}} \cup S_{\textbf {II}}$ has countably many $\prec _s$ -predecessors.

Given a triple $(a,A,\sigma ) \in S_{\textbf {I}} \cup S_{\textbf {II}}$ , we build $B_{a,A,\sigma } \in \mathcal {R}$ by an increasing sequence $a = b_0 \sqsubseteq b_1 \sqsubseteq \cdots $ with , and let . We consider two cases.

1. (1) If $(a,A,\sigma ) \in S_{\textbf {I}}$ , then we shall construct some $B_{a,A,\sigma } \in \mathcal {R}$ such that $B_{a,A,\sigma }$ is the outcome of some full play in $Z^*[a,A]$ , with I following $\sigma $ , and that $B_{a,A,\sigma } \neq B_{a',A',\sigma '}$ for all $(a',A',\sigma ') \prec _s (a,A,\sigma )$ . We do this as follows: We first enumerate the $\prec _s$ -predecessors by . Following a play in $Z^*[a,A]$ where I follows $\sigma $ , suppose that I started the $n^{\text {th}}$ turn with $B_n \in [b_n,A]$ . If or , then pick any . Otherwise, since $\mathcal {R}$ is deep we may pick some such that . This construction ensures that $B_{a,A,\sigma } \neq B_{a_n,A_n,\sigma _n}$ for all .

2. (2) If $(a,A,\sigma ) \in S_{\textbf {II}}$ , then we shall construct some $B_{a,A,\sigma } \in \mathcal {R}$ such that $B_{a,A,\sigma }$ is the outcome of some full play in $Y[a,A]$ , with II following $\sigma $ , and that $B_{a,A,\sigma } \neq B_{a',A',\sigma '}$ for all $(a',A',\sigma ') \prec _s (a,A,\sigma )$ . We do this as follows: Again, we enumerate the $\prec _s$ -predecessors by . Following a play in $Y[a,A]$ where II follows $\sigma $ , suppose that the sequence $b_n$ has been played so far. If , we then ask that I respond with , so that for any $b_{n+1}$ that II respond with next, . Otherwise, I may respond with any . This construction ensures that $B_{a,A,\sigma } \neq B_{a_n,A_n,\sigma _n}$ for all .

Now let

$$ \begin{align*} \mathcal{X} := \{B_{a,A,\sigma} : (a,A,\sigma) \in S_{\textbf{I}}\}. \end{align*} $$

$\mathcal {X}$ is $\boldsymbol {\Sigma }_2^1$ , as the well-ordering $\prec _s$ is $\Sigma _2^1$ and the construction of $\mathcal {X}$ is natural from $\prec _s$ . We then see that $\mathcal {X}$ is not pre-Kastanas Ramsey: Let $A \in \mathcal {R}$ and $a \in \mathcal {AR}$ .

1. (1) If $\sigma $ is a strategy for I in $Z^*[a,A]$ , then $B_{a,A,\sigma }$ is the outcome of a run following $\sigma $ such that $B_{a,A,\sigma } \notin \mathcal {X}^c$ , so $\sigma $ is not a winning strategy for I.

2. (2) If $\sigma $ is a strategy for II in $Y[a,A]$ , then $B_{a,A,\sigma }$ is the outcome of a run following $\sigma $ such that $B_{a,A,\sigma } \notin \mathcal {X}$ , so $\sigma $ is not a winning strategy for II.

This completes the proof.

We remark that the set $\{B_{a,A,\sigma } : (a,A,\sigma ) \in S_{\textbf {I}}\}$ would similarly produce a $\boldsymbol {\Pi }_2^1$ subset of $\mathcal {R}$ that is not pre-Kastanas Ramsey. Since such a well-ordering exists in Gödel’s constructible universe, we may conclude the following.

Corollary 4.17 ( $\mathsf {V=L}$ ).

Let be a deep wA2-space, and assume that $\mathcal {AR}$ is countable. Then there exists a $\boldsymbol {\Sigma }_2^1$ subset of $\mathcal {R}$ which is not Kastanas Ramsey.

Corollary 4.18 ( $\mathsf {V=L}$ ).

The following wA2-spaces have a $\boldsymbol {\Sigma }_2^1$ subset which is not Kastanas Ramsey:

1. (1) $([\mathbb {N}]^\infty ,\subseteq ,r)$ .

2. (2) , the topological Ramsey space of infinite block sequences.

3. (3) , a variant of the space of infinite block sequences.

4. (4) , the Hales–Jewett space.

5. (5) $(\mathcal {S}^\infty ,\subseteq ,r)$ , the topological Ramsey space of strong subtrees.

6. (6) , the Carlson–Simpson space.

7. (7) , the space of infinite-dimensional block subspaces of a countable vector space.

Proof. By Corollary 4.17, it suffices to show that every wA2-space above is deep.

1. (1) Let $A = \{n_0,n_1,\dots \} \in [\mathbb {N}]^\infty $ , and let $a \subseteq A$ be finite. For all N such that $n_N> \max (a)$ , we have that . Therefore, and .

2. (2) Let $A = (x_0,x_1,\dots ) \in \mathbf {FIN}_k^{[\infty ]}$ , and let $a \in \mathbf {FIN}_k^{[<\infty ]}\mathord {\upharpoonright } A$ . For all N such that $x_N> a$ , we have that . Therefore, and .

3. (3) The proof is identical to that of .

4. (4) Let $A = (x_0,x_1,\dots ) \in W_{Lv}^{[\infty ]}$ , and let $a = (y_i)_{i<n} \in W_{Lv}^{[<\infty ]}\mathord {\upharpoonright } A$ . Since A is rapidly increasing, for N large enough we have that $\sum _{i<n} |y_i| < |x_N|$ . Therefore, and .

5. (5) Let $A \subseteq T$ be a strong subtree. Given any $a \in \S _{<\infty }\mathord {\upharpoonright } A$ , we let $S_a$ be the set of terminal nodes in a. Since a is a strong subtree, for some N. We observe that

   

   Fix any $a \in \S _{<\infty }\mathord {\upharpoonright } A$ and be large enough. For each $s \in S_a$ , let be such that $s \sqsubseteq t_{s,0}$ , $s \sqsubseteq t_{s,1}$ and $t_{s,0} \neq t_{s,1}$ . We let

   $$ \begin{align*} b := \{u \in A : u \sqsubseteq t_{s,i} \text{ for some } s \in S_a \text{ and } i \in \{0,1\}\}. \end{align*} $$

   Observe that a is an initial segment of b, and every terminal node in a splits in b. Thus, and .

6. (6) Let $A \in \mathcal {E}_\infty $ , and let $a \in \mathcal {AE}_\infty \mathord {\upharpoonright } A$ . Let be the increasing enumeration of the minimal representatives of A. Given , we define the equivalence relation b on $\{0,1,\dots ,p_N(A)-1\}$ as follows: Given $i,j \in \mathbb {N}$ , we define

   

   We shall show that and .

   Given $i,j < p_N(A)$ such that $(i,j) \in A$ , if or then $(i,j) \in b$ . Otherwise, $(i,j) \in b$ as well. Therefore, b is an equivalence relation on which is coarser than A, so . To see that $a \sqsubseteq b$ —if and $(i,j) \in b$ , then $(i,j) \in A$ , so $(i,j) \in a$ as a is coarser than A. Finally, the equivalence classes in b are either of the form $[i]_b$ for some (of which $(i,j) \notin a$ implies that $[i]_b \neq [j]_b$ ), or $[i]_b$ for any (of which $[i]_b = [j]_b$ for all ). Therefore, b has many equivalence classes, i.e., .

7. (7) The proof is identical to that of .

5 Strategically Ramsey sets and Gowers spaces

5.1 Gowers spaces

de Rancourt first introduced Gowers spaces in [11] as a common abstraction to the topological Ramsey space $[\mathbb {N}]^\infty $ (i.e., the Ellentuck space or Mathias–Silver space) and countable vector spaces (i.e., Rosendal space). We recall the definition.

Definition 5.1 [11, Definition 2.1].

A Gowers space is a quintuple , where $P \neq \emptyset $ is the set of subspaces, $X \neq \emptyset $ is at most countable (the set of points), are two quasi-orders on P, and is a binary relation, satisfying the following properties:

1. (1) For all $p,q \in P$ , if , then .

2. (2) For all $p,q \in P$ , if , then there exists some $r \in P$ such that , and .

3. (3) For every -decreasing sequence of P, there exists some $p^* \in P$ such that for all .

4. (4) For all $p \in P$ and , there exists some $x \in X$ such that .

5. (5) For all and $p,q \in P$ , if and , then .

Given $p,q \in P$ , we also write iff and .

de Rancourt proceeded to introduce various games in this abstract setting. We hereby provide a summary of the games we’re interested in. Note that we have employed some changes in the names/notations of the game.

Definition 5.2 [11, Definition 2.2].

For each $p \in P$ , the adversarial Gowers game $AG(p)$ is defined as a game played by Players I and II in the following form:

such that $x_n,y_n \in X$ and $p_n,q_n \in P$ for all n, and that the following additional condition must be fulfilled for all :

1. (1) .

2. (2) .

3. (3) and .

The outcome of this game is $(x_0,y_0,x_1,y_1,\dots )$ . We say that I (resp., II) has a strategy in $K(p)$ to reach if it has a strategy in $K(p)$ to ensure that the outcome is in $\mathcal {X}$ .

Definition 5.3 [11, Definition 2.2].

For each $p \in P$ , the adversarial Gowers game for I $AG_{\textbf {I}}(p)$ (resp., for II $AG_{\textbf {II}}(p)$ ) is the game $AG(p)$ with the following additional restrictions:

1. (1) For $AG_{\textbf {I}}(p)$ , I can only play $q_n$ such that .

2. (2) For $AG_{\textbf {I}}(p)$ , II can only play $p_n$ such that .

Definition 5.4 [11, Definition 2.5].

For each $p \in P$ , the de Rancourt game $R(p)$ is the game $AG(p)$ with the following additional restriction:

1. (1) For all , and .

Definition 5.5 [11, Definition 3.1].

For each $p \in P$ , the Gowers game $G(p)$ is defined as a game played by Players I and II in the following form:

such that $x_n \in X$ and $p_n \in P$ for all n, and that the following additional condition must be fulfilled for all :

1. (1) .

2. (2) .

The outcome of this game is $(x_0,x_1,\dots )$ . We say that I (resp., II) has a strategy in $K(p)$ to reach if it has a strategy in $K(p)$ to ensure that the outcome is in  $\mathcal {X}$ .

Definition 5.6 [11, Definition 3.1].

For each $p \in P$ , the asymptotic game $F(p)$ is the game $G(p)$ with the following additional restriction:

1. (1) For all , .

We now introduce several variants of game-theoretic Ramsey properties.

Definition 5.7 [11, Definition 2.3].

A set is adversarially Ramsey if for all $p \in P$ , there exists some such that one of the following holds:

1. (1) I has a strategy in $AG_{\textbf {I}}(q)$ to reach $\mathcal {X}$ .

2. (2) II has a strategy in $AG_{\textbf {II}}(q)$ to reach $\mathcal {X}^c$ .

Definition 5.8 [11, Definition 3.2].

A set is strategically Ramsey if for all $p \in P$ , there exists some such that one of the following holds:

1. (1) I has a strategy in $F(q)$ to reach $\mathcal {X}^c$ .

2. (2) II has a strategy in $G(q)$ to reach $\mathcal {X}$ .

Definition 5.9. A set is de Rancourt Ramsey if for all $p \in P$ , there exists some such that one of the following holds:

1. (1) I has a strategy in $R(q)$ to reach $\mathcal {X}$ .

2. (2) II has a strategy in $R(q)$ to reach $\mathcal {X}^c$ .

Proposition 5.10. Let $p \in P$ and .

1. (1) I has a strategy in $R(q)$ to reach $\mathcal {X}$ for some iff there exists some such that I has a strategy in $AG_{\mathbf {I}}(q)$ to reach $\mathcal {X}$ .

2. (2) II has a strategy in $R(q)$ to reach $\mathcal {X}$ for some iff there exists some such that II has a strategy in $AG_{\mathbf {II}}(q)$ to reach $\mathcal {X}$ .

Proof. The forward direction for both statements has been proven in Proposition 2.6 of [11], so we only prove the converse for (1) (the proof for the converse for (2) is almost verbatim). Suppose $\sigma $ is a strategy for I in $AG_{\textbf {I}}(p)$ to reach $\mathcal {X}$ , and we define a strategy $\tau $ for I in $R(p)$ . For each state s for II of $R(p)$ following $\tau $ , we shall correspond it to a state $t_s$ for II of $AG_{\textbf {I}}(p)$ realising $a(s)$ . Start by letting $t_{(p_0)} := (p_0)$ for any . Now let s be a state of $R(p)$ for II following $\tau $ so far, with $s = {s'}^\frown (y_n,p_{n+1})$ . Suppose by the induction hypothesis that there exists a corresponding state $t_s$ of the game $AG_{\textbf {I}}(p)$ such that:

1. (1) $a(s') = a(t_{s'})$ ;

2. (2) $\sigma (t_{s'}) = (x_n,q_n')$ for some .

Now let $t_s := {t_{s'}}^\frown (y_n,p_{n+1})$ , and suppose $\sigma (t_s) = (x_{n+1},q_{n+1}')$ for some . Since , by Property (2) of Definition 5.1 there exists some such that . Then $\tau (s) := (x_{n+1},p_{n+1})$ is a legal continuation. This completes the inductive definition of $\tau $ , which is a winning strategy as every complete play following $\tau $ corresponds to a complete play following $\sigma $ realising the same sequence.

5.2 The Kastanas game

We now introduce (our version of) the Kastanas game for Gowers spaces.

Definition 5.11 [11, Definition 2.5].

For each $p \in P$ , the Kastanas game $K(p)$ is defined as a game played by Players I and II in the following form:

such that $x_n \in X$ and $p_n,q_n \in P$ for all n, and that the following additional condition must be fulfilled for all :

1. (1) .

2. (2) and .

The outcome of this game is $(x_0,x_1,\dots )$ . We say that I (resp., II) has a strategy in $K(p)$ to reach if it has a strategy in $K(p)$ to ensure that the outcome is in  $\mathcal {X}$ .

Definition 5.12. A set is Kastanas Ramsey if for all $p \in P$ , there exists some such that one of the following holds:

1. (1) I has a strategy in $K(q)$ to reach $\mathcal {X}^c$ .

2. (2) II has a strategy in $K(q)$ to reach $\mathcal {X}$ .

Proposition 5.13. A subset is Kastanas Ramsey iff $\mathcal {X}$ is strategically Ramsey [11, Definition 3.2].

We shall prove this proposition as a corollary of Proposition 5.10.

Proof. We let be a Gowers space, and assume WLOG that $0 \notin X$ . We then define a relation such that for all :

1. (1) If is odd, then for all $x \in X \cup \{0\}$ and $p \in P$ , $s^\frown x \blacktriangleleft P$ iff $x = 0$ .

2. (2) If is even, then for all $x \in X \cup \{0\}$ and $p \in P$ , $s^\frown x \blacktriangleleft P$ iff $x \neq 0$ and .

It is easy to verify that is a Gowers space. We now define an injective function by

$$ \begin{align*} f((x_0,\dots,x_{n-1})) := (x_0,0,x_1,0\dots,x_{n-1},0) \end{align*} $$

and naturally extend f to . Note that f is injective. For each $p \in P$ , we also define the functions $g,h$ by

$$ \begin{align*} g(p_0,0,p_1,0,p_2,\dots) &:= (p_0,p_1,p_2,\dots), \\ h(x_0,q_0,x_1,q_1,\dots) &:= (x_0,x_1,\dots). \end{align*} $$

We may now observe that:

1. (1) $\sigma $ is a strategy for I in $K(p)$ to reach $\mathcal {X}$ iff:

   $$ \begin{align*} s \mapsto \begin{cases} \sigma(s), &\text{if } s = \emptyset, \\ (0,\sigma(s)), &\text{if } s \neq \emptyset \\ \end{cases} \end{align*} $$
   is a strategy for II in $R(p)$ to reach $f[\mathcal {X}]$ .

2. (2) $\sigma $ is a strategy for II in $K(p)$ to reach $\mathcal {X}$ iff $\sigma \circ g$ is a strategy for I in $R(p)$ to reach $f[\mathcal {X}]$ .

3. (3) $\sigma $ is a strategy for I in $F(p)$ to reach $\mathcal {X}$ iff:

   $$ \begin{align*} s \mapsto \begin{cases} (\sigma \circ h)(s), &\text{if } s = \emptyset, \\ (0,(\sigma \circ h)(s)), &\text{if } s \neq \emptyset \\ \end{cases} \end{align*} $$
   is a strategy for II in $AG_{\textbf {II}}(p)$ to reach $f[\mathcal {X}]$ .

4. (4) $\sigma $ is a strategy for II in $G(p)$ to reach $\mathcal {X}$ iff $s \mapsto ((\sigma \circ g)(s),p)$ is a strategy for I in $AG_{\textbf {I}}(p)$ to reach $f[\mathcal {X}]$ .

Therefore, the proposition follows from Proposition 5.10.

5.3 Gowers wA2-spaces

We shall now reformulate the above result in the context of wA2-spaces. In [9], Mijares introduced the notion of an almost reduction for spaces satisfying A1–A4, which may be applied to wA2-spaces. We introduce a variant of this almost reduction, restricted to a fixed initial segment.

Notation 5.14. Let be a wA2-space. Given $A,B \in \mathcal {R}$ and $a \in \mathcal {AR}$ , we write iff there exists some $b \in \mathcal {AR}\mathord {\upharpoonright }[a,A]$ such that $[b,A] \subseteq [b,B]$ .

Note that need not be a transitive relation—counterexample would be the topological Ramsey space of strong subtrees (which satisfies A1–A4). Note also that, by A1, we may identify each element $a \in \mathcal {AR}$ with the sequence .

Definition 5.15. Let be a wA2-space. We say that $\mathcal {R}$ is Gowers if there exists a relation such that the following properties hold:

1. (G1-5) For all $A \in \mathcal {R}$ and $a \in \mathcal {AR}\mathord {\upharpoonright } A$ , is a Gowers space (when identifying elements of $\mathcal {AR}$ with ).

2. (G6) Let $A,B \in \mathcal {R}$ . Let $a \in \mathcal {AR}\mathord {\upharpoonright } A \cap \mathcal {AR}\mathord {\upharpoonright } B$ .

   1. (1) $[a,A] \subseteq [a,B]$ iff for all .

   2. (2) If there exists some N such that for all $n \geq N$ , then .

3. (G7) For all $A \in \mathcal {R}$ , $a \in \mathcal {AR}\mathord {\upharpoonright } A,$ and , there exists some such that for all $b \in \mathcal {AR}\mathord {\upharpoonright }[a,A]$ , if then .

Example 5.16 (Natural numbers/Ellentuck space ).

We show that $([\mathbb {N}]^\infty ,\subseteq ,r)$ is a Gowers wA2-space.

1. (G1-5) Let $A \in [\mathbb {N}]^\infty $ and $a \in [\mathbb {N}]^{<\infty }\mathord {\upharpoonright } A$ . Note for all $B,C \in [a,A]$ , iff $C \setminus N \subseteq B$ for some $N \geq \max (a)$ . Given $b = a \cup \{x_{|a|},\dots ,x_n\} \in [\mathbb {N}]^{<\infty }\mathord {\upharpoonright }[a,A]$ and $B \in [a,A]$ , we define iff $x_n \in B$ .

   1. (1) Clearly $C \subseteq B$ implies that .

   2. (2) If , then there exists some n such that $D := a \cup (C \setminus n) \subseteq B$ . Then $D \subseteq C$ , $D \subseteq B,$ and as $(C \setminus a) \setminus (D \setminus a) \subseteq n$ .

   3. (3) Let be a $\subseteq $ -decreasing sequence in $[a,A]$ , and let $C \subseteq B_0 \setminus a$ be such that $C \subseteq ^* B_n$ for all n. Then $a \cup C \in [a,A]$ and for all n.

   4. (4) Given $B \in [a,A]$ and $b \in [\mathbb {N}]^{<\infty }\mathord {\upharpoonright }[a,A]$ , for any $x \in B$ such that $\max (b) < x$ .

   5. (5) If $b = a \cup \{x_0,\dots ,x_n\} \in [\mathbb {N}]^{<\infty }\mathord {\upharpoonright }[a,A]$ , and $C \subseteq B$ , then $x_n \in C \subseteq B$ , so .

2. (G6) Let $A,B \in [\mathbb {N}]^\infty $ and $b \in [\mathbb {N}]^{<\infty }\mathord {\upharpoonright }[a,A] \cap [\mathbb {N}]^{<\infty }\mathord {\upharpoonright }[a,B]$ . We write $A = \{x_0,x_1,\dots \}$ and (i.e., $\max (a) = x_{m-1}$ ).

   1. (1) We have that:

      

   2. (2) If for all $n \geq N> m$ , then we have that $A \setminus (a \cup \{x_{|a|},\dots ,x_N\}) \subseteq B$ , so .

3. (G7) Let $A \in [\mathbb {N}]^\infty $ , $a \in [\mathbb {N}]^{<\infty }\mathord {\upharpoonright } A,$ and $B \subseteq A$ . Let , and let $C := r_m(A) \cup (B \setminus \max (a))$ . Then for all $b \in [\mathbb {N}]^{<\infty }[a,A]$ , if $b = a \cup \{x_{|a|},\dots ,x_n\}$ and , then $x_n> \max (a) = \max (r_m(A))$ and $x_n \in C \subseteq B$ , so .

Example 5.17 (Countable vector space $E^{[\infty ]}$ ).

We show that is a Gowers wA2-space. Given some , we denote $A/N := (x_n)_{n \geq N}$ .

1. (G1-5) Let $A \in E^{[\infty ]}$ and $a \in E^{[<\infty ]}\mathord {\upharpoonright } A$ . Note for all $B,C \in [a,A]$ , iff for some . Given $b = a^\frown (x_{|a|},\dots ,x_n) \in E^{[<\infty ]}\mathord {\upharpoonright }[a,A]$ and $B \in [a,A]$ , we define iff .

   1. (1) Clearly implies that .

   2. (2) If , then there exists some such that . Then , and as .

   3. (3) Let be a -decreasing sequence in $[a,A]$ , and let be such that for all n ( may be constructed by picking ). Then $a^\frown C \in [a,A]$ and for all n.

   4. (4) Given $B \in [a,A]$ and $b \in E^{[<\infty ]}\mathord {\upharpoonright }[a,A]$ , for any $x \in B$ such that $\max (b) < x$ .

   5. (5) If $b = a^\frown (x_{|a|},\dots ,x_n) \in E^{[<\infty ]}$ , and , then , so .

2. (G6) Let $A,B \in E^{[\infty ]}$ and $b \in E^{[\infty ]}\mathord {\upharpoonright }[a,A] \cap E^{[\infty ]}\mathord {\upharpoonright }[a,B]$ . We write $A = (x_0,x_1,\dots )$ and .

   1. (1) We have that:

      

   2. (2) If for all $n \geq N> m$ , then we have that , so .

3. (G7) Let $A \in E^{[\infty ]}$ , $a \in E^{[<\infty ]}\mathord {\upharpoonright } A,$ and $B \subseteq A$ . Let , and let $C := r_m(A)^\frown (B/N)$ , where . Then for all $b \in E^{[\infty ]}[a,A]$ , if $b = a \cup \{x_{|a|},\dots ,x_n\}$ and , then and , so .

Theorem 5.18. Let be a Gowers wA2-space, and let $\mathcal {X} \subseteq \mathcal {R}$ . Let $A \in \mathcal {R}$ and $a \in \mathcal {AR}$ . The following are equivalent:

1. (1) I (resp., II) has a strategy in $K[a,A]$ to reach $\mathcal {X}$ .

2. (2) I (resp., II) has a strategy in $K(A)$ (as a game of the Gowers space ) to reach $\mathcal {X} \cap [a,A]$ .

Proof. (1)⟹(2), Player I: Let $\sigma $ be a strategy for I in $K[a,A]$ to reach $\mathcal {X}$ . We define a strategy $\tau $ for I in $K(A)$ as follows: Let s be a state for II in $K(A)$ following $\tau $ so far, and suppose that $s = {s'}^\frown (a_n,B_{n-1})$ , and we have defined a state $t_{s'}$ for II in $K[a,A]$ such that $a(s) = a(t_{s'})$ (i.e., they realise the same finite sequence so far), and . Note that for the base case, we define $t_\emptyset := \emptyset $ and $\tau (\emptyset ) := \sigma (\emptyset )$ .

Since and $a_n \sqsubseteq \sigma (t_{s'})$ , by G7 there exists some $C_{n-1} \in [a_n,\sigma (t_{s'})]$ such that for all $b \in \mathcal {AR}\mathord {\upharpoonright }[a_n,C_n]$ , if then . We may thus define the legal continuation $t_s := {t_{s'}}^\frown (a_n,C_{n-1})$ . Then by G6, , so by G2 (i.e., Property (2) of Definition 5.1) we may define to be such that . This completes the inductive definition of $\tau $ , and it is a winning strategy for I as every complete play s of $K(A)$ following $\tau $ induces a complete play $t_s$ of $K[a,A]$ following $\sigma $ , with the same outcome.

(1)⟹(2), Player II: Let $\sigma $ be a strategy for II in $K[a,A]$ to reach $\mathcal {X}$ . We define a strategy $\tau $ for II in $K(A)$ as follows: Let s be a state for I in $K(A)$ following $\tau $ so far, and suppose that $s = {s'}^\frown (\tau (s'),A_n)$ , and we have defined a state $t_{s'}$ for I in $K[a,A]$ such that $a(s) = a(t_{s'})$ and . Note that for the base case, we define $t_{(A_0)} := (A_0)$ and $\tau ((A_0)) := \sigma ((A_0))$ .

We write $\tau (s') = (a_n,B_{n-1})$ and $\sigma (t_{s'}) = (a_n,C_{n-1})$ . Since , by G7 there exists some $A_n' \in [a_n,B_{n-1}]$ such that for all $b \in \mathcal {AR}\mathord {\upharpoonright }[a_n,B_{n-1}]$ , if then . We may thus define the legal continuation $t_s := {t_{s'}}^\frown (A_n')$ . By G7, if $\sigma (t_{s'}) = (a_{n+1},C_n)$ , then we may define $\tau (s) = (a_{n+1},B_n)$ , where and . This completes the inductive definition of $\tau $ , and it is a winning strategy for II as every complete play s of $K(A)$ following $\tau $ induces a complete play $t_s$ of $K[a,A]$ following $\sigma $ , with the same outcome.

The proof of (2) $\implies $ (1) for both players is similar but simpler, mostly using G7 to make the necessary changes to the strategy. For instance, suppose that $\sigma $ is a strategy for I in $K(A)$ to reach $\mathcal {X} \cap [a,A]$ . Suppose that in the game $K[a,A]$ , II responded with $(a_n,B_{n-1})$ , and $\sigma $ then responds with some . By G7, we instead ask I to respond with $A_n \in [a_n,B_{n-1}]$ such that for all , if then . Then this modification gives I a strategy in $K[a,A]$ to reach $\mathcal {X}$ .

Alternatively, one may define the corresponding de Rancourt game for wA2-spaces, then define a corresponding notion of de Rancourt Ramsey, and prove using similar methods that the corresponding notion of de Rancourt Ramsey is equivalent to that for Gowers spaces. Then Theorem 5.18 may be deduced using the maps $g,h$ defined in the proof of Proposition 5.13.

Consequently, we have the following immediate corollary.

Corollary 5.19. Let be a Gowers wA2-space, and let $\mathcal {X} \subseteq \mathcal {R}$ . The following are equivalent:

1. (1) $\mathcal {X}$ is Kastanas Ramsey (as in Definition 3.2).

2. (2) For all $A \in \mathcal {R}$ and $a \in \mathcal {AR}$ , $\mathcal {X} \cap [a,A]$ is a Kastanas Ramsey subset of $[a,A]$ (as in Definition 3.2).

3. (3) For all $A \in \mathcal {R}$ and $a \in \mathcal {AR}$ , $\mathcal {X} \cap [a,A]$ is a Kastanas Ramsey subset of $[a,A]$ (as in Definition 5.12 for the Gowers space ).

Since a countable vector space $E^{[\infty ]}$ is a Gowers wA2-space with countable $\mathcal {AR}$ , we may conclude all the following classical facts of strategically Ramsey sets.

Corollary 5.20. Let be the wA2-space of infinite-dimensional block subspaces of a countable vector space.

1. (1) $\mathcal {X} \subseteq E^{[\infty ]}$ is Kastanas Ramsey (as in Definition 3.2) iff $\mathcal {X}$ is strategically Ramsey (as in Definition 1 of [12]).

2. (2) Every analytic subset of $E^{[\infty ]}$ is strategically Ramsey.

3. (3) The set of strategically Ramsey subsets of $E^{[\infty ]}$ is closed under countable unions, but not under complement and finite intersection.

Proof.

1. (1) Combine Proposition 5.13 and Corollary 5.19.

2. (2) Combine Proposition 5.13 and Theorem 1.2.

3. (3) By Example 3.9, there is a biasymptotic subset of $E^{[<\infty ]}$ .

5.4 Coanalytic sets

Combined with Proposition 5.13 and Theorem IV.7.5 of [1], we get a positive answer to Question 2 in the context of a countable vector space. This section shows that this is, in fact, a consequence of Corollary 4.9 and a suitable choice of coding.

Let E be a vector space over a countable field with a dedicated Schauder basis . Let $Y \subseteq E$ be a biasymptotic set (i.e., for all $A \in E^{[\infty ]}$ , and ). We first define $\delta : E \to 2$ by stipulating that:

$$ \begin{align*} \delta(x) := \begin{cases} 1, &\text{if } x \in Y, \\ 0, &\text{if } x \notin Y. \end{cases} \end{align*} $$

Fix some $a \in E^{[<\infty ]}$ and . Let $E^{[<\infty ]}(a)$ denote the set of all $b \in E^{[\infty ]}$ such that $a \sqsubseteq b$ . We now define a map by stipulating that:

We may then extend this function to a continuous map . By Corollary 5.20(1), we may replace Kastanas Ramsey sets with strategically Ramsey sets in our discussion.

Lemma 5.21. Let . Let $A \in E^{[\infty ]}$ , $a \in E^{[<\infty ]}\mathord {\upharpoonright } A,$ and . Suppose that I has a strategy in $F[a,A]$ to reach . Then I has a strategy in $F[(a,p),(A,\vec {0})]$ to reach $\mathcal {C}$ .

Proof. Let $\sigma $ be a strategy for I in $F[a,A]$ to reach . We define a strategy $\tau $ for I in $F[(a,p),(A,\vec {0})]$ as follows: We first let $\tau (\emptyset ) := (\sigma (\emptyset ),\vec {0})$ , and $t_\emptyset := \emptyset $ . Now suppose, for the induction hypothesis, that for all states s of $F[(a,p),(A,\vec {0})]$ for II of rank n, there exists a state $t_s$ of $F[a,A]$ for II of rank $2n$ such that and $\tau (s) = (\sigma (t_s),\vec {0})$ . Let s be a state for II of rank $n + 1$ , and suppose that , where $\varepsilon \in \{0,1\}$ . We let $t_s := {t_{s\mathord {\upharpoonright } n}}^\frown (x_n,y_n)$ , where $y_n$ is any element of E such that $y_n \in Y$ iff $\varepsilon = 1$ . Observe that . This finishes the construction of the strategy $\tau $ . Now let s be a complete play in $F[(a,p),(A,\vec {0})]$ following $\tau $ . Since $\sigma $ is a strategy that reaches , , so

as desired.

Lemma 5.22. Let . Let $A \in E^{[\infty ]}$ , $a \in E^{[<\infty ]}\mathord {\upharpoonright } A,$ and . Suppose that II has a strategy in $G[a,A]$ to reach . Then II has a strategy in $G[(a,p),(A,\vec {0})]$ to reach $\mathcal {C}$ .

Proof. Let $\sigma $ be a strategy for II in $G[a,A]$ to reach . We define a strategy $\tau $ for II in $G[(a,p),(A,\vec {0})]$ as follows: For any , we let $t_{(B)} := (B)$ . Now let s be a state of $G[(a,p),(A,\vec {0})]$ for I, and suppose that we have defined a corresponding state $t_s$ of $G[a,A]$ of rank $2n$ such that , and . Let $x_n,y_n$ be such that $x_n = \sigma (t_s)$ and $y_n = \sigma ({t_s}^\frown (A))$ . We then let

$$ \begin{align*} \tau(s) := \begin{cases} (x_n,1), &\text{if } y_n \in Y, \\ (x_n,0), &\text{if } y_n \notin Y. \end{cases} \end{align*} $$

Now let $t_{s^\frown {(B)}} := {t_s}^\frown (A,B)$ . This finishes the construction of the strategy $\tau $ , and by a similar reasoning to the last paragraph of Lemma 5.21, $\tau $ is a strategy for II in $G[(a,p),(A,\vec {0})]$ to reach $\mathcal {C}$ .

We thus obtain the following variant of Theorem 4.5.

Theorem 5.23 [1, Theorem IV.4.14].

If every coanalytic subset of $E^{[\infty ]}$ is strategically Ramsey, then every $\boldsymbol {\Sigma }_2^1$ subset of $E^{[\infty ]}$ is strategically Ramsey. More generally, for every $n \geq 1$ , if every $\boldsymbol {\Pi }_n^1$ subset of $E^{[\infty ]}$ is strategically Ramsey, then every $\boldsymbol {\Sigma }_{n+1}^1$ subset of $E^{[\infty ]}$ is strategically Ramsey.

Proof. Suppose on the contrary that there exists a $\boldsymbol {\Sigma }_{n+1}^1$ non-strategically Ramsey subset of $E^{[\infty ]}$ . By Theorem 4.5, there exists a $\boldsymbol {\Pi }_n^1$ subset which is not strategically Ramsey. Therefore, there exists some $A \in E^{[\infty ]}$ and such that for all $(B,\vec {0}) \in [(a,p),(A,\vec {0})]$ , neither I has a strategy in $F[(a,p),(B,\vec {0})]$ to reach $\mathcal {C}^c$ , nor II has a strategy in $G[(a,p),(B,\vec {0})]$ to reach $\mathcal {C}$ . By Lemmas 5.21 and 5.22, this implies that for all $B \in [a,A]$ , neither I has a strategy in $F[a,B]$ to reach , nor II has a strategy in $G[a,B]$ to reach . Therefore, is a $\boldsymbol {\Pi }_n^1$ set (as is continuous) which is not strategically Ramsey.

Corollary 5.24. Suppose that there exists a $\Sigma _2^1$ -good well-ordering of the reals. Then there exists a coanalytic subset of $E^{[\infty ]}$ which is not strategically Ramsey.

Proof. By Theorem 4.8, there exists a coanalytic which is not strategically Ramsey. Now apply Theorem 5.23.

We remark that we have also essentially proved Corollary IV.4.13 of [1]: If $\mathcal {X} \subseteq E^{[\infty ]}$ is analytic, then $\mathcal {X} = \pi _0[\mathcal {C}]$ for some $G_\delta $ subset . Then is also a $G_\delta $ subset of $E^{[\infty ]}$ for all $a,p$ , as is always continuous and $E^{[\infty ]}(a)$ is a clopen subset of $E^{[\infty ]}$ . Therefore, proving that every $G_\delta $ subset of $E^{[\infty ]}$ is strategically Ramsey would imply that every analytic subset of $E^{[\infty ]}$ is strategically Ramsey.

6 Further remarks and open questions

Todorčević proved in [15] that if is a closed triple satisfying A1–A4, then every Suslin-measurable subset of $\mathcal {R}$ is Ramsey. This is strictly stronger than Corollary 4.6, which is our best conclusion from our general results regarding Kastanas Ramsey sets. However, by Proposition 3.12, Kastanas Ramsey sets need not be closed under the Suslin operation in general.

Question 1. Can we prove a general result about Kastanas Ramsey subsets of a wA2-space which, when restricted to the setting of topological Ramsey space, directly implies that Ramsey subsets are closed under the Suslin operation?

There are various set-theoretic properties shared by Ramsey subsets of topological Ramsey spaces and strategically Ramsey subsets of countable vector spaces, but it is not apparent to us if one can provide general results (in the context of wA2-spaces) which encompass them. One such property concerns the statement “Every set is Kastanas Ramsey.” It is a classic result that in Solovay’s model, every subset of a Polish space has the property of Baire. Since the Ellentuck topology refines the Polish topology, we have the following.

Theorem 6.1. Let be a closed triple satisfying A1–A4 such that $\mathcal {AR}$ is countable. Let $\kappa $ be an inaccessible cardinal, and let $\mathcal {G}$ be -generic. Then in the model , every subset of $\mathcal {R}$ is Ramsey.

On the other hand, we have the following property of strategically Ramsey sets.

Theorem 6.2 (Lopez-Abad [8]).

Let $\kappa $ be a supercompact cardinal, and let $\mathcal {G}$ be -generic. Then in the model , every subset of $E^{[\infty ]}$ is strategically Ramsey.

This leads to the following conjecture.

Conjecture. Let be a wA2-space, and assume that $\mathcal {AR}$ is countable. It is consistent with sufficiently large cardinal assumptions that every subset of $\mathcal {R}$ is Kastanas Ramsey.

We conclude with a question that naturally extends Corollary 4.9.

Question 2. Let be a sufficiently well-behaved wA2-space. Assume that it has a biasymptotic set, and that $\mathcal {AR}$ is countable. If there exists a $\Sigma _2^1$ -good well-ordering of the reals, then must there exist a coanalytic subset of $\mathcal {R}$ which is not Kastanas Ramsey?