Ideal games and Ramsey sets

It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal $\mathcal{H}$, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a co-ideal $\mathcal{H}$ is semiselective if and only if Matet's game-theoretic characterization of the $\mathcal{H}$-Ramsey property holds. This lifts Kastanas's characterization of the classical Ramsey property to its optimal setting, from the point of view of the local Ramsey theory and gives a game-theoretic counterpart to a theorem of Farah \cite{far}, asserting that a co-ideal $\mathcal{H}$ is semiselective if and only if the family of $\mathcal{H}$-Ramsey subsets of $\N^{[\infty]}$ coincides with the family of those sets having the abstract $\mathcal{H}$-Baire property. Finally, we show that under suitable assumptions, for every semiselective co-ideal $\mathcal H$ all sets of real numbers are $\mathcal H$-Ramsey.


Introduction
Let N be the set of nonnegative integers. Given an infinite set A ⊆ N, the symbol A [∞] (resp. A [<∞] ) represents the collection of the infinite (resp. finite) subsets of A. Let A [n] denote the set of all the subsets of A with n elements. If a ∈ N [<∞] is an initial segment of A ∈ N [∞] then we write a ⊏ A. Also, let A/a := {n ∈ A : max(a) < n}, and write A/n to mean A/{n}. } is a basis for Ellentuck's topology, also known as exponential topology. In [1], Ellentuck gave a characterization of Ramseyness in terms of the Baire property relative to this topology (see Theorem 2.1 below).
Let (P, ≤) be a poset, a subset D ⊆ P is dense in P if for every p ∈ P , there is q ∈ D with q ≤ p. D ⊆ P is open if p ∈ D and q ≤ p imply q ∈ D. P is σ-distributive if the intersection of countably many dense open subsets of P is dense. P is σ-closed if every decreasing sequence of elements of P has a lower bound. The complement I = ℘(N) \ H is the dual ideal of H. In this case, as usual, we write H = I + . We will suppose that coideals differ from ℘(N). Also, we say that a nonempty family F ⊆ H is H-disjoint if for every A, B ∈ F , A ∩ B ∈ H. We say that F is a maximal H-disjoint family if it is H-disjoint and it is not properly contained in any other H-disjoint family as a subset.
A subset X of N [∞] is Ramsey if for every [a, A] = ∅ with A ∈ N [∞] there exists B ∈ [a, A] such that [a, B] ⊆ X or [a, B] ∩ X = ∅. Some authors have used the term "completely Ramsey" to express this property, reserving the term "Ramsey" for a weaker property. Galvin and Prikry [3] showed that all Borel subsets of N [∞] are Ramsey, and Silver [11] extended this to all analytic sets. Mathias in [9] showed that if the existence of an inaccessible cardinal is consistent with ZF C then it is consistent, with ZF + DC, that every subset of N [∞] is Ramsey. Mathias introduced the concept of a selective coideal (or a happy family), which has turned out to be of wide interest. Ellentuck [1] characterized the Ramsey sets as those having the Baire property with respect to the exponential topology of N [∞] .
A game theoretical characterization of Ramseyness was given by Kastanas in [5], using games in the style of Banach-Mazur with respect to Ellentuck's topology.
In this work we will deal with a game-theoretic characterization of the following property: H-Ramseyness is also called local Ramsey property.
Mathias considered sets that are H-Ramsey with respect to a selective coideal H, and generalized Silver's result to this context. Matet [8] used games to characterize sets which are Ramsey with respect to a selective coideal H. These games coincide with the games of Kastanas if H is N [∞] and with the games of Louveau [7] if H is a Ramsey ultrafilter.
Given a coideal H ⊂ N [∞] , let In general, this is not a basis for a topology on N [<∞] , but the following abstract version of the Baire property and related concepts will be useful: if it is the union of countably many Exp(H)-nowhere dense sets.
Given a decreasing sequence A 0 ⊇ A 1 ⊇ A 2 ⊇ · · · of infinite subsets of N, a set B is a diagonalization of the sequence (or B diagonalizes the sequence) if and only if B/n ⊆ A n for each n ∈ B. A coideal H is selective if and only if every decreasing sequence in H has a diagonalization in H.
A coideal H has the Q + -property, if for every A ∈ H and every partition (F n ) n of A into finite sets, there is S ∈ H such that S ⊆ A and |S ∩ F n | ≤ 1 for every n ∈ N.  Since σ-closedness implies σ-distributivity, then semiselectivity follows from selectivity, but the converse does not hold (see [2] for an example).
In section 2 we list results of Ellentuck, Mathias and Farah that characterize topologically the Ramsey property and the local Ramsey property. In section 3 we define a family of games, and present the main result, which states that a coideal H is semiselective if and only if the H-Ramsey sets are exactly those for which the associated games are determined. This generalizes results of Kastanas [5] and Matet [8]. The proof is given in section 4. In section 5 we relate semiselectivity of coideals with the Fréchet-Urysohn property, and show that in Solovay's model every semiselective coideal has the Fréchet-Urysohn property.
We thank A. Blass and J. Bagaria for helping us to correct some deficiencies in previous versions of the article.
The following are the main results concerning the characterization of the Ramsey property and the local Ramsey property in topological terms.  In the next section we state results by Kastanas [5] and Matet [8] (Theorems 3.1 and 3.2 below) which are the game-theoretic counterparts of theorems 2.1 and 2.2, respectively; and we also present our main result (Theorem 3.3 below), which is the game-theoretic counterpart of Theorem 2.3.

Characterizing Ramseyness with games.
The following is a relativized version of a game due to Kastanas [5], employed to obtain a characterization of the family of completely Ramsey sets (i.e. H-Ramsey for H = N [∞] ). The same game was used by Matet in [8] to obtain the analog result when H is selective.
we define a two-player game G H (a, A, X ) as follows: player I chooses an element A 0 ∈ H ↾ A; II answers by playing n 0 ∈ A 0 such that a ⊆ n 0 , and B 0 ∈ H∩(A 0 /n 0 ) [∞] ; then I chooses A 1 ∈ H∩B [∞] 0 ; II answers by playing n 1 ∈ A 1 and B 1 ∈ H ∩ (A 1 /n 1 ) [∞] ; and so on. Player I wins if and only if a ∪ {n j : j ∈ N} ∈ X .
A strategy for a player is a rule that tells him (or her) what to play based on the previous moves. A strategy is a winning strategy for player I if player I wins the game whenever she (or he) follows the strategy, no matter what player II plays. Analogously, it can be defined what is a winning strategy for player II. The precise definitions of strategy for two players games can be found in [6,10].
Let s = {s 0 , . . . , s k } be a nonempty finite subset of N, written in its increasing order, and − → B = {B 0 , . . . , B k } be a sequence of elements of H. We say that the pair (s, − → B ) is a legal position for player II if (s 0 , B 0 ), . . . , (s k , B k ) is a sequence of possible consecutive moves of II in the game G H (a, A, X ), respecting the rules. In this case, if σ is a winning strategy for player I in the game, we say that σ(s, − → B ) is a realizable move of player I according to σ. Notice that if r ∈ B k /s k and C ∈ H ↾ B k /s k then (s 0 , B 0 ), . . . , (s k , B k ), (r, C) is also a sequence of possible consecutive moves of II in the game. We will sometimes use the notation (s, − → B , r, C), and say that (s, − → B , r, C) is a legal position for player II and σ(s, − → B , r, C) is a realizable move of player I according to σ.
We say that the game G H (a, A, X ) is determined if one of the players has a winning strategy.
So Theorem 3.3 is a game-theoretic counterpart to Theorem 2.3 in the previous section, in the sense that it gives us a game-theoretic characterization of semiselectivity. Obviously, it also gives us a characterization of H-Ramseyness, for semiselective H, which generalizes the main results of Kastanas in [5] and Matet in [8] (Theorems 3.1 and 3.2 above).
It is known that every analytic set is H-Ramsey for H semiselective (see Theorem 2.2 in [2] or Lemma 7.18 in [15]). We extend this result to the projective hierarchy. Please see [6] or [10] for the definitions of projective set and of projective determinacy. Proof. Let X be a projective subset of N [∞] . Fix A ∈ H, a ∈ N [<∞] . By the projective determinacy over the reals, the game G H (a, A, X ) is determined. Then, Theorem 3.3 implies that X is H-Ramsey.

Proof of the main result
Throughout the rest of this section, fix a semiselective coideal H. Before proving Theorem 3.3, in Propositions 4.1 and 4.7 below we will deal with winning strategies of players in a game G H (a, A, X ). Proof. Suppose σ is a winning strategy for I. We will suppose that a = ∅ and A = N without loss of generality.
Let A 0 = σ(∅) be the first move of I using σ. We will define a tree T of finite subsets of A 0 ; and for each s ∈ T we will also define a family M s ⊆ A

Proof. Fix s ∈ T and A ∈ H ↾
is a legal position for player II. Then, using the maximality of M s , chooseF ∈ M s such that   Let us prove that E 1 ∈ H: Suppose E 1 ∈ H. By the definitions, (∀n ∈ E 1 )Ê/n ∈ U n . Let n 0 = min(E 1 ) and fix s 0 ⊂Ê such that max(s 0 ) = n 0 and satisfying, in particular, the following: Notice that |s 0 | > 1, by the construction of the M s 's. Now, let m = max(s 0 \ {n 0 }). Then m ∈ E 0 and thereforeÊ/m ∈ U m ⊆ U s 0 \{n 0 } . So there exists F ∈ M s 0 \{n 0 } such thatÊ/m ⊆ F . Since m < n 0 then n 0 ∈ F . But F ∩ E 1 /n 0 = E 1 /n 0 ∈ H. A contradiction.
Hence, E 1 ∈ H and therefore E 0 ∈ H. Then E := E 0 is as required.   (k 0 , B 0 ). Thus, by the choice of E and applying Claim 4.4 iteratively, we prove that {k i } i≥0 is generated in a run of the game in which player I has used his winning strategy σ. Therefore The converse is trivial. This completes the proof of Proposition 4.1.
Now we turn to the case when player II has a winning strategy. The proof of the following is similar to the proof of Proposition 4.3 in [8]. First we show a result we will need in the sequel, it should be compared with lemma 4.2 in [8]. The set D n = U n ∪ V n is dense open in H ↾ B. Choose E ∈ H ↾ B such that for each n ∈ E, E/n ∈ D n . Let E 0 = {n ∈ E : E/n ∈ U n } and E 1 = {n ∈ E : E/n ∈ V n }. Now, suppose E 1 ∈ H. Then, for each n ∈ E 1 , E 1 /n ∈ V n . Let n 1 = f (E 1 ). So n 1 ∈ E 1 by the definition of f . But, by the definition of g, g(E 1 ) ⊆ E 1 /n 1 and so E 1 /n 1 ∈ V n 1 ; a contradiction. Therefore, E 1 ∈ H. Hence E 0 ∈ H, since H is a coideal. The set E f,g := E 0 is as required. Proof. Let τ be a winning strategy for II in G H (a, A, X ) and let A ′ ∈ H ↾ A be given. We are going to define a winning strategy σ for I, in G H (a, A ′ , N [∞] \ X ), in such a way that we will get the required result by means of Proposition 4.1. So, in a play of the game G H (a, A ′ , N [∞] \ X ), with II's successive moves being (n j , B j ), j ∈ N, define A j ∈ H and E f j ,g j as in Lemma 4.6, for f j and g j such that (1) For allÂ ∈ H ↾ A ′ , (f 0 (Â), g 0 (Â)) = τ (Â); (2) For allÂ ∈ H ↾ B j ∩ g j (A j ), Now, let σ(∅)=E f 0 ,g 0 and σ((n 0 , B 0 ), · · · , (n j , B j ))=E f j+1 ,g j+1 .
Conversely, let A 0 be the first move of I in the game. Then there exists E ∈ H ↾ A 0 such that [a, E] ∩ X = ∅. We define a winning strategy for player II by letting her (or him) play (min E, E \ {min E}) at the first turn, and arbitrarily from there on.
We are ready now for the following: X a .
Fix A ∈ H and a ∈ N [<∞] with [a, A] = ∅, and define a winning strategy σ for player I in G H (a, A, X ), as follows: let σ(∅) be any element of D 0 such that σ(∅) ⊆ A. At stage k, if II's successive moves in the game are (n j , B j ), j ≤ k, let σ ((n 0 , B 0 ), . . . , (n k , B k )) be any element of D k+1 such that σ ((n 0 , B 0 ), . . . , (n k , B k )) ⊆ B k . Notice that a ∪ {n 0 , n 1 , n 2 , . . . } ∈ X a . So the game G H (a, A, X ) is determined for every A ∈ H and a ∈ N [<∞] with [a, A] = ∅. Then, by our assumptions, X is H-Ramsey. So given A ∈ H, there exists B ∈ H ↾ A such that B [∞] ⊆ X or B [∞] ∩ X = ∅. The second alternative does not hold, so X ∩ H is dense in (H, ⊆). Hence, H is semiselective.
We say that an coideal H ⊆ N [∞] has the Fréchet-Urysohn property if The following characterization of the Fréchet-Urysohn property is taken from [12,14]. It provides a method to construct ideals with that property. Given A ⊆ N [∞] , define the orthogonal of A as A ⊥ := {A ∈ N [∞] : (∀B ∈ A) (|A ∩ B| < ∞)}. Notice that A ⊥ is an ideal. The following result is probably known but we include its proof for the sake of completeness.  The previous result can be extended from suitable axioms. Farah [2] shows that if there is a supercompact cardinal, then every semiselective coideal in L(R) has the Fréchet-Urysohn property.
As we show below, it is also an easy consequence of results of [2] and [9] that in Solovay's model every semiselective coideal has the Fréchet-Urysohn property.
Notice that every real in V [G] has a name in V λ , and names for subsets of H or countable sequences of subsets of H are contained in V λ . Also, the forcing Col(ω, < λ) is a subset of V λ . Therefore the same statement is valid in the structure (V λ , ∈,Ḣ, Col(ω, < λ)). This statement is Π 1 1 over this structure, and since λ is Π 1 1 -indescribable, there is κ < λ such that in (V κ , ∈,Ḣ ∩ V κ , Col(ω, < λ) ∩ V κ ) it holds ∀Ḋ∀τ (p Col(ω,<κ) (Ḋ is a name for a sequence of dense open subsets ofḢ ∩ V κ We can get κ inaccessible, since there is a Π 1 1 formula expressing that λ is inaccessible. Also, κ is such that p and the names for the real parameters in the definition of A and for A belong to V κ . If we let G κ = G ∩ Col(ω, < κ), then G κ ⊆ Col(ω, < κ), and is generic over V . Also, And since every subset (or sequence of subsets) of H ∩ V [G κ ] which belongs to V [G κ ] has a name contained in V κ , we have that, in V [G κ ], H∩V κ is semiselective, and in consequence it has both the Prikry and the Mathias properties. Now the proof can be finished as in [9]. . Therefore, [a, x \ a] is contained in A or is disjoint from A.
As in [9], we obtain the following.
Corollary 5.8. If ZF C is consistent with the existence of a weakly compact cardinal, then so is ZF + DC and "every set of reals is H-Ramsey for every semiselective coideal H".