INFINITE DIMENSIONAL PERFECT SET THEOREMS 1. Introduction the Perfect Set Theorem Says That an Analytic Subset of a Polish Space

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 365, Number 1, January 2013, Pages 23–58 S 0002-9947(2012)05468-7 Article electronically published on June 8, 2012

INFINITE DIMENSIONAL PERFECT SET THEOREMS

TAMAS´ MATRAI´

Abstract. What largeness and structural assumptions on A ⊆ [R]ω can guarantee the existence of a non-empty perfect set P ⊆ R such that [P ]ω ⊆ A? Such a set P is called A-homogeneous. We show that even if A is open, in general it is independent of ZFC whether for a cardinal κ, the existence of an A-homogeneous set H ∈ [R]κ implies the existence of a non-empty perfect A-homogeneous set. On the other hand, we prove an infinite dimensional analogue of Mycielski’s Theorem: if A is large in the sense of a suitable Baire category-like notion, then there exists a non-empty perfect A-homogeneous set. We introduce fusion games to prove this and other infinite dimensional perfect set theorems. Finally we apply this theory to show that it is independent of ZFC whether Tukey reductions of the maximal analytic cofinal type can be witnessed by definable Tukey maps.

1. Introduction The Perfect Set Theorem says that an analytic subset of a Polish space is either countable or has a non-empty perfect subset (see e.g. [11, Theorem 29.1, p. 226]). The complexity assumption in this result is consistently optimal: in L there exists 1 an uncountable Π1 set without non-empty perfect subsets (see e.g. [9, Corollary 25.37]). However, one is often obliged to quest for a perfect set which satisfies multidimensional relations. Let N<ωbe fixed and let [X]N denote the set of N element subsets of X.ThenanN dimensional perfect set theorem should address the following problem: what largeness and structural assumptions on A ⊆ [X]N can guarantee the existence of a non-empty perfect set P ⊆ X such that [P ]N ⊆ A? Such a set P is called A-homogeneous. In [12, Theorem 2.2, p. 620], W. Kubi´sprovedthatifX is a Polish space, N A ⊆ [X] is Gδ and there exists an uncountable A-homogeneous set, then there exists a non-empty perfect A-homogeneous set. Obviously, as far as Gδ sets are concerned, this result is the exact multidimensional analogue of the Perfect Set Theorem. The surprising fact is that the complexity assumption in this result is also optimal: the Turing reducibility relation on 2ω (see Definition 2.4) defines ω 2 ω an Fσ set A ⊆ [2 ] such that there exists an A-homogeneous set H ⊆ 2 with |H| = ω1 but there is no non-empty perfect A-homogeneous set. For A analytic, W. Kubi´s and S. Shelah [13] investigated a rank function which decides whether the existence of an A-homogeneous set with a given cardinality implies the existence of a non-empty perfect A-homogeneous set (see also [8], [23]).

Received by the editors October 8, 2009 and, in revised form, September 20, 2010. 2010 Mathematics Subject Classiﬁcation. Primary 03E15. This research was partially supported by the OTKA grants K 61600, K 49786 and K 72655 and by the NSERC grants 129977 and A-7354.

c 2012 American Mathematical Society Reverts to public domain 28 years from publication 23

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They obtain in particular that for every α<ω1, it is consistent with ZFC that ω 2 there exists an Fσ set A ⊆ [2 ] such that there exists an A-homogeneous set of cardinality ℵα but there is no non-empty perfect A-homogeneous set. Nevertheless, the most frequently applied perfect set theorem is a classical result of J. Mycielski, which says that if X is a non-empty perfect Polish space and N AN ⊆ [X] (N<ω) are co-meager relations, then there is a non-empty perfect set N N P ⊆ X such that [P ] ⊆ AN (N<ω). So, in particular, if A ⊆ [X] is co-meager, then there exists a non-empty perfect A-homogeneous set. Obviously, the largeness assumption in this result is not optimal. In the present paper we study the existence of non-empty perfect homogeneous sets for inﬁnite dimensional relations A ⊆ [X]ω. Unlike in the ﬁnite dimensional case, it is not obvious how to topologize [X]ω. Therefore we usually assume that A ⊆ Xω is symmetric, i.e., it is invariant under permutations of coordinates in Xω and a set H ⊆ X is called A-homogeneous if the injective sequences of H, ω ISω(H)={(xn)n<ω ∈ H : xn = xm (n

Theorem 1.1. Let κ be a cardinal and suppose that there exists an Fσ set A ⊆ [2ω]2 such that there exists an A-homogeneous set of cardinality κ but there is no non-empty perfect A-homogeneous set. Then there exists a symmetric open set U ⊆ (2ω)ω such that there exists a U-homogeneous set of cardinality κ but there is no non-empty perfect U-homogeneous set. Thus in the infinite dimensional case, by the above mentioned result of W. Kubi´s and S. Shelah, even for open relations, it is consistent that the existence of a homogeneous set of large cardinality does not imply the existence of a non-empty perfect homogeneous set (see Corollary 2.3). On the other hand, Mycielski’s Theorem has an infinite dimensional analogue. For a Polish space X, consider the σ-ideal M generated by the sets of the form × ω\(n+1) ⊆ n+1 n<ω(Mn X )whereMn X (n<ω) are meager. As we will see, an easy application of Mycielski’s Theorem yields that if A ⊆ Xω satisfies Xω \A ∈ M, then there exists a non-empty perfect A-homogeneous set (see Theorem 4.1). A more involved task is to find sufficient conditions for Xω \ A ∈ M. In Section 4 we provide such sufficient conditions. In particular, we will show the following (see Corollary 4.7.2). Theorem 1.2. Let X be a Polish space, let A ⊆ Xω be co-analytic and suppose that there exists a non-meager A-homogeneous set. Then there exists a non-empty perfect A-homogeneous set. As a corollary, we obtain that in the iterated perfect set model, for every co- analytic set A ⊆ Xω if there exists an A-homogeneous set of cardinality continuum, then there exists a non-empty perfect A-homogeneous set (see Theorem 4.9). More- over, we also obtain that in Cohen extensions the existence of a homogeneous set of sufficiently large cardinality implies the existence of a non-empty perfect homogeneous set (see Theorem 4.10). Thus by combining these results and Theorem 1.1, in Section 4 we will prove the following.

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Theorem 1.3. Let 1 <α<ω1 be an ordinal. Then it is independent of ZFC whether for an open set A ⊆ (2ω)ω,theexistenceofanA-homogeneous set of cardinality ℵα implies the existence of a non-empty perfect A-homogeneous set. The key result toward Theorem 1.2 is proved by using a game which is obtained as a fusion of Banach-Mazur games played on higher and higher dimensional powers of X (see Definition 3.2). In Section 3 we introduce this game and characterize the winning strategies of the players (see Theorem 3.4). It seems that our procedure of taking fusions is applicable to a wide class of games of descriptive set theory. As an illustration, in Section 3.2 we briefly study the fusion of Perfect Set Property games (see Definition 3.13 and Corollary 3.18). Independent of our work, M. Sabok [22] introduced and studied similar games (see also [21]). We discuss the relation between fusion games and some games of [21], [22] and [30] in the introduction of Section 3. Our study of infinite dimensional perfect set theorems was motivated by the problem of whether Tukey reducibilities of the maximal analytic cofinal type Imax can be witnessed by definable Tukey maps. In Section 5 we recall the relevant definitions and show that this problem is independent of ZFC (see Theorem 5.7). Finally we would like to thank Pandelis Dodos, David H. Fremlin, Michael Hruˇsák, Arnold W. Miller, Benjamin D. Miller, Marcin Sabok, Lajos Soukup, Juris Stepr¯ans, Boban Veliˇcković, William A. R. Weiss and Jindˇrich Zapletal for their helpful discussions. We are indebted to Stevo Todorˇcević for his overall support in our research. We gratefully acknowledge the hospitality of the Department of Mathematics of Rutgers University, New Jersey.

2. Open relations The main result of this section is the following slightly generalized version of Theorem 1.1. Recall that for every set H and α ≤ ω, ISα(H)={(xn)n<α ∈ α α H : xn = xm (n

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Proof. Apply Galvin’s Theorem with X = P and B = A ∩ [P ]2. ω We introduce some terminology and notation. Fix a metric d1 on 2 . For every ω n 0 0, let + ω ω B ((xk)k

δ((xk)k 0(n<ω)andA \ Δ2 = Fn. n<ω For every n<ω,set + ω n+3 Un = {B ((xk)k 0; hence + B ((xk)k 0(0

(i) for every i, j < ω, i = j implies qi = qj , (ii) for every 0

Let q0,q1 ∈ Q be arbitrary satisfying q0 = q1.Let0

Then εn > 0byIS2(Q) ∩ A = ∅ and (2.1). To satisfy (i) and (iii), let qn+1 ∈ Q \{qi : i ≤ n} be arbitrary satisfying d1(qn+1,qi) <εi/2(i ≤ n). By the inductive assumption (iii) for n,suchaqn+1 exists, namely any qn+1 ∈ Q \{qi : i ≤ n} suﬃciently close to qn fulﬁlls the requirements. + Suppose (qk)k<ω ∈ U,say(qk)k<ω ∈ Un.Then(qk)k<ω ∈ B ((xk)k

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δ((xk)k

(iv) d1(qki ,xli ) <δ/4(i

We have 0

(2.2) and (v), d2((qka ,qkb ),Fn) <δ/4. So to summarize, ε d ((q ,q ),F ) δ d (q ,q ) < kn+1 ≤ 2 ka kb n < . 1 kn+2 kn+1 2 2 8 Again by (iv), δ δ d (q ,x ) < ,d(q ,x ) < , 1 kn+1 ln+1 4 1 kn+2 ln+2 4 so the triangle inequality yields δ d (x ,x ) ≤ d (x ,q )+d (q ,q )+d (q ,x ) < 5 . 1 ln+1 ln+2 1 ln+1 kn+1 1 kn+1 kn+2 1 kn+2 ln+2 8 This contradicts the deﬁnition of δ.

Finally suppose b = n + 2; as in the previous case, we estimate d1(xln+1 ,xln+2 ). By (iv), δ δ δ (2.3) d (q ,x ) < ,d(q ,x ) < ,d(q ,x ) < . 1 ka la 4 1 kn+1 ln+1 4 1 kn+2 ln+2 4 So by the triangle inequality, ≤ d1(xln+1 ,xln+2 ) d1(xln+1 ,qkn+1 )+d1(qkn+1 ,qkn+2 )+d1(qkn+2 ,xln+2 ) (2.4) δ < + d (q ,q ). 2 1 kn+1 kn+2

We have 0

i.e. d1(qkn+2 ,qkn+1 ) <δ/2. By (2.4) we obtained d1(xln+1 ,xln+2 ) <δ, which again contradicts the deﬁnition of δ.

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Corollary 2.3. For every α<ω1, it is consistent with ZFC that there exists a symmetric open set U ⊆ (2ω)ω such that there exists a U-homogeneous set of cardinality ℵα but there is no non-empty perfect U-homogeneous set. Proof. By [13, Corollary 5.13, p. 159] or [23, Theorem 1.13, p. 15], it is con- ω 2 sistent with ZFC that there exists an Fσ set A ⊆ [2 ] such that there exists an A-homogeneous set of cardinality ℵα but there is no non-empty perfect A- homogeneous set. So the statement follows from Theorem 2.1. As we mentioned in the introduction, the Turing reducibility relation is a ZFC ex- ω 2 ample for an Fσ set T ⊆ [2 ] such that there exists an uncountable T -homogeneous set but there is no non-empty perfect T -homogeneous set. For the sake of complete- ness, we recall (a simplified version of) this relation and prove its above-mentioned properties. Definition 2.4. For every j<ω,set < { ∈ ω × ω j+1 · j } Tj = (x, y) 2 2 : x(i)=y(2 i +2 )(i<ω) > { ∈ ω × ω ∈ <} and Tj = (x, y) 2 2 :(y, x) Tj . Then the Turing reducibility relation ⊆ ω × ω < ∪ > T 2 2 is defined by T = j<ω(Tj Tj ).

Proposition 2.5. The relation T is symmetric and Fσ is such that there exists an uncountable T -homogeneous set but there is no non-empty perfect T -homogeneous set. < > Proof. It is obvious that Tj , Tj (j<ω) are closed sets, so T is symmetric and Fσ. ω ω j+1 j Observe that for given (xj )j<ω ⊆ 2 , the point y ∈ 2 defined by y(2 · i +2 )= ∈ < ⊆ xj (i)(i, j < ω)satisfies(xj,y) Tj T (j<ω). Hence a straightforward transfinite recursion yields an uncountable T -homogeneous set. Finally let P ⊆ 2ω be a non-empty perfect set. Observe that for every j<ω, > > ∩ × Tj is the graph of a function; in particular, Tj (P P )(j<ω) are meager in P × P . By symmetry, this yields that T ∩ (P × P ) is meager in P × P . Hence P cannot be T -homogeneous, as required. From T , by Theorem 2.1, we get a symmetric open set U ⊆ (2ω)ω with analogous properties. In particular, this yields an example of an open set U ⊆ (2ω)ω which is dense even in the box topology; still there is no non-empty perfect U-homogeneous set. We refer to Section 6.1 for a further discussion of alternative topologies.

3. Fusion games In this section we introduce the fusion of infinite sequences of games. The construction can be performed for most of the usual games of descriptive set theory. However, the method of the characterization of the winning strategies in a fusion game depends on the games whose fusion is taken. Therefore, here we study only the fusion game of Banach-Mazur games in detail, which is the most relevant for our perfect set theorems. In addition, in Section 3.2 we briefly discuss the fusion game of Perfect Set Property games. Independent of our work, M. Sabok [22] introduced and studied games which are very similar to fusion games (see also [21]). The approach in these works, which originates from [30], makes explicit the connection between such games and iterated forcing. Fusion games and the corresponding ideal M (see Definition 3.3) are also to be compared to the games and ideals of [30, Definition 5.1.1, p. 225].

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3.1. Banach-Mazur games. Let X be a Choquet space such that there is a metric d on X whose balls are open in X. The diameter of a set A ⊆ X is denoted by diamX (A). Recall that for every A ⊆ X,intheBanach-Mazur game with payoﬀ set A (see e.g. [11, Section 21.D, p. 153]) two players play I : U(0) U(2) ... U(2n) ...

II: U(1) U(3) ... U(2n +1) ... where U(n) ⊆ X (n<ω) are non-empty open sets such that U(n) ⊇ U(n+1)and −n diamX (U(n)) < 2 (n<ω), and player II wins the game if and only if i<ω U(i) ∈ is a singleton and i<ω U(i) A. The following is well known (see e.g. [11, Theorem 8.33, p. 51]). Theorem 3.1. In the Banach-Mazur game with payoﬀ set A, (1) player I has a winning strategy if and only if there is a non-empty open set U ⊆ X such that A ∩ U is meager; (2) player II has a winning strategy if and only if X \ A is meager. We deﬁne the fusion game of Banach-Mazur games. If Y is a set and s, t ∈ Y <ω, |s| denotes the length of s and st stands for the sequence s(0) ... <ω s(|s|−1)t(0) ...t(|t|−1). We write s t if s = t||s|.IfT ⊆ Y is a tree and ω n<ω, set levn(T )={t ∈ T : |t| = n} and [T ]={η ∈ Y : η|n ∈ T (n<ω)}. Also recall that the product space of Choquet spaces is Choquet (see e.g. [11, Exercise 8.13, p. 44]).

Deﬁnition 3.2. For every k<ω,letXk be a Choquet space such that there is ⊆ G ametricdk on Xk whose balls are open in Xk. For every A k<ω Xk, ω(A) denotes the fusion game of the Banach-Mazur games with payoﬀ set A,inwhich players I and II play

I : U0(0) (U0(2),U1(0)) ... (Ui(2(n − i)))i≤n ...

II: U0(1) (U0(3),U1(1)) ... (Ui(2(n − i)+1))i≤n ...

where for every k<ω, Uk(i) ⊆ Xk (i<ω) are non-empty open sets such that ⊇ −i Uk(i) Uk(i + 1) and diamXk (Uk(i)) < 2 (i<ω), and player II wins the game ∈ if and only if for every k<ω, i<ω Uk(i)isasingletonand( i<ω Uk(i))k<ω A. We denote by Gω the tree of partial runs of this game, ordered by end-extension. We set Uω = Uk : n<ω,Uk ⊆ Xk is non-empty open (k

A quasi-strategy of player I is a non-empty pruned tree σ ⊆Gω such that for every n<ω, U ∈ lev2n+1(σ)andu ∈Uω, U u ∈Gω implies U u ∈ σ. Similarly, a quasi-strategy of player II is a non-empty pruned tree σ ⊆Gω such that for every n<ω, U ∈ lev2n(σ)andu ∈Uω, U u ∈Gω implies U u ∈ σ. For every pruned tree σ ⊆Gω,set W (σ)= (xk)k<ω ∈ Xk : ∃((Ui(2(k − i)))i≤k,(Ui(2(k − i)+1))i≤k)k

xk = Uk(i)(k<ω) . i<ω

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For every P ∈{I,II},Σ(P ) denotes the set of all quasi-strategies of player P in G W { ⊆ ∃ ∈ ⊆ } the game ω,andweset (P )= A k<ω Xk : σ Σ(P )(W (σ) A) .

The winning strategies in Gω are characterized by the following Baire Category- like notion. For arbitrary A ⊆ X × Y ,wesetPrX (A)={x ∈ X : ∃y ∈ Y ((x, y) ∈ A)}.

Deﬁnition 3.3. With the notation of Deﬁnition 3.2, for an arbitrary sequence ⊆ Sn k≤n Xk (n<ω)weset

[(Sn)n<ω]= Sn × Xk . n<ω n

Mn is meager in Xk (n<ω),M⊆ Mn × Xk . k≤n n<ω n

We will need that Gω(A) is determined for co-analytic A, as well. To this end, we introduce an unfolding of Gω, as follows. ⊆ × ω Definition 3.5. With the notation of Definition 3.2, for every F ( k<ω Xk) ω , G ω(F ) denotes the game with payoff set F in which players I and II play

I : U0(0),y0 (U0(2),U1(0)),y1 ... (Ui(2(n − i)))i≤n,yn ...

II: U0(1) (U0(3),U1(1)) ... (Ui(2(n − i)+1))i≤n ... − − ∈G ∈ where for every n<ω,((Ui(2(k i)))i≤k, (Ui(2(k i)+1))i≤k)k≤n ω and yn ω, and player I wins the game if and only if for every k<ω, i<ω Uk(i)isasingleton

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and Uk(i) , (yk)k<ω ∈ F. k<ω i<ω ⊆G For every pruned tree σ ω,set ω W (σ)= ((xk)k<ω, (yk)k<ω) ∈ Xk × ω : k<ω

∃((Ui(2(k−i)))i≤k,yk, (Ui(2(k−i)+1))i≤k)k

It is obvious that Vn ⊆ Un is an open set which is dense in Un (n<ω). We have (xi)i<ω ∈ [V] if and only if (xi)i≤n ∈ Vn (n<ω). So for every n<ω, by (xi)i≤n ∈ Vn we have (xi)i≤n ∈ Un,andby(xi)i≤n+k ∈ Vn+k (k<ω)we have (xi)i≤n ∈/ Rn(k)(k<ω). Thus (xi)i≤n ∈ Bn (n<ω), which completes the proof.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 32 TAMAS´ MATRAI´ ⊆ Corollary 3.8. With the notation of Deﬁnition 3.3,letA k<ω Xk be arbitrary and let U be a non-empty open tower. Then (1) [U] \ A ∈ M if and only if there is a non-empty open tower V such that V is dense in U and [V] ⊆ A. \ ∈ M U (2) ( k<ω Xk) A if and only if there is a dense open tower such that [U] ⊆ A. U \ ⊆ × ⊆ Proof. For 1, ﬁrst let [ ] A n<ω(Mn n

Proof. Let U =(Un)n<ω. For 1, it is enough to construct a strategy σ ∈ Σ(I)such that W (σ) ⊆ [U]. We define σ ⊆Gω by induction, as follows. Let (U0(0)) ∈ σ if and only if (U0(0)) ∈Gω and U0(0) ⊆ U0. Let n<ωbe arbitrary and suppose that σ ∩ lev2n+1(Gω) is already defined. For every U ∈ lev2n+1(σ), Ui(2(n − i)+1)(i ≤ n)andUi(2(n +1− i)) (i ≤ n + 1), let U ((Ui(2(n − i)+1))i≤n) ∈ σ and U ((Ui(2(n − i)+1))i≤n, (Ui(2(n +1− i)))i≤n+1) ∈ σ if and only if U ((Ui(2(n − i)+1))i≤n, (Ui(2(n +1− i)))i≤n+1) ∈Gω − ⊆ and i≤n+1 Ui(2(n +1 i)) Un+1. This completes the inductive step of the definition of σ. It is obvious that σ ⊆Gω and that for every n<ω, U ∈ lev2n+1(σ)andu ∈Uω, U u ∈Gω implies U u ∈ σ.Toseethatσ is pruned, let n<ω, U ∈ lev2n+1(σ) − − ∈ and (Ui(2(n i)+1))i≤n be arbitrary such that U ((Ui(2(n i)+1))i≤n) σ. Since U (2(n − i)) ⊆ U and U ΔPr (U )ismeager, i≤n i n n k≤n Xk n+1

Ui(2(n − i)+1) × Xn+1 ∩ Un+1 = ∅. i≤n

In particular, there are Ui(2(n +1− i)) (i ≤ n + 1) such that Ui(2(n +1− i)) ⊆ − ≤ − −(n+1−i) ≤ Ui(2(n i)+1)(i n), diamXi (Ui(2(n +1 i))) < 2 (i n +1)and − ⊆ i≤n+1 Ui(2(n +1 i)) Un+1.Thus U ((Ui(2(n − i)+1))i≤n, (Ui(2(n +1− i)))i≤n+1) ∈ σ, which concludes σ ∈ Σ(I). ⊆ U ∈ It remains to prove W (σ) [ ]. Let (xk)k<ω W (σ) be arbitrary, say ∈ ∈ xk = i<ω Uk(i)(k<ω)forsomeU [σ]. Then for every n<ω,(xk)k≤n − ⊆ ∈ × i≤n Ui(2(n i)) Un. This shows (xk)k<ω Un ( n

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Since the spaces Xk (k<ω)areChoquet,inGω both players can refine their − quasi-strategies in such a way that the resulting sequence ((Ui(2(k i)))i≤k, − (Ui(2(k i)+1))i≤k)k<ω satisfies i<ω Uk(i) is a singleton for every k<ω.In particular, for every P ∈{I,II}, A ∈W(P ) implies A = ∅. So by Proposition 3.9.1, if U is a non-empty open tower, then [U] = ∅. The proof of the following proposition closely follows the proofs of [11, Theorem 8.33, p. 51] and [11, Theorem 21.8, p. 153]. Proposition 3.10. With the notation of Definition 3.3 and Definition 3.5, (1) if A ∈W(I), then there is a non-empty open tower U such that [U] ⊆ A; (2) if A ∈W(II), then there is a dense open tower U such that [U] ⊆ A; (3) if F ∈W(II),thenPr (F ) ∈ M. k<ω Xk Proof. To see 1, let σ ∈ Σ(I) be such that W (σ) ⊆ A. We define a tree τ ⊆ σ by induction, as follows. Let U0 ⊆ lev1(σ) be a maximal family of pairwise disjoint open sets, and set lev1(τ)=U0.Letn<ωand suppose that lev2n+1(τ)isalready defined. Let Un+1 ⊆{u ∈Uω : ∃U ∈ lev2n+1(τ) ∃v ∈Uω (U (v, u) ∈ σ)}

be a maximal family of pairwise disjoint open sets. For every u ∈ Un+1 fix one v(u) ∈Uω such that U (v(u),u) ∈ σ and set lev2n+3(τ)={U (v(u),u) ∈ σ : U ∈ lev2n+1(τ),u∈ Un+1}. This completes the inductive step of the definition of τ. Observe that by requiring the members of Un to be pairwise disjoint, for every n<ωand u ∈ Un there is a ∈ unique U lev 2n+1(τ) such that U(2n)=u. Let Un = Un (n<ω); we show that U =(Un)n<ω is a non-empty open tower and [U] ⊆ A. ItisobviousthatU0 = ∅.Letn<ωbe arbitrary and let u ⊆ Un be an arbitrary non-empty open set; we show (Pr (U )) ∩ u = ∅.By k≤n Xk n+1 passing to a subset we can assume u ⊆ u for some u ∈ Un. Then there is a unique U ∈ lev2n+1(τ) such that U(2n)=u . By the definition of Gω, there is a v ∈Uω such that U v ∈Gω and v ⊆ u.Sinceσ is a strategy of player I, there is a w ∈Uω such that U (v, w) ∈ σ. In particular Pr (w) ⊆ v, hence Pr (w) ⊆ u. k≤n Xk k≤n Xk By the maximality of Un+1, there is a w ∈ Un+1 such that w ∩ w = ∅.This shows (Pr (Un+1)) ∩ u = ∅.Sinceu ⊆ Un was arbitrary, we obtained k≤n Xk U \ Pr (U ) is nowhere dense in X .SincePr (U ) ⊆ U n k≤n Xk n+1 k≤n k k≤n Xk n+1 n follows immediately from the definition, we proved UnΔPr (Un+1)ismeager k≤n Xk U in k≤n Xk (n<ω). Thus is a non-empty open tower. We also obtained that τ is a pruned tree. Since τ ⊆ σ,wehaveW (τ) ⊆ W (σ) ⊆ A,soitremainstosee [U] ⊆ W (τ). Let (xk)k<ω ∈ [U] be arbitrary. For every n<ω,byUn being pairwise disjoint, there is a unique v2n ∈ Un such that (xk)k≤n ∈ v2n. By the definition of τ, v2n ∈ Un meansthatthereisaVn ∈ lev2n+1(τ) with Vn(2n)=v2n.Butthen (xk)k≤i ∈ Vn(2i), which implies Vn(2i)=v2i (i

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 34 TAMAS´ MATRAI´ ∈ U lev2n+2(τ) satisfying U(2n +1)= u.SetUn = Un (n<ω). Since σ is a strategy of player II, the maximality of Un implies Un is dense in k≤n Xk. Hence U =(Un)n<ω is a dense open tower, and as in the proof of 1, we have [U] ⊆ W (τ) ⊆ W (σ) ⊆ A. To see 3, let σ ∈ Σ(II) be such that W (σ) ∩ F = ∅. For every y ∈ ω<ω we say U ∈Gω is compatible with σ, y if |U| =2|y| and (U(2i),y(i),U(2i +1))i<|y| ∈ σ. <ω For every y ∈ ω \ {∅} we construct a tree τy ⊆Gω of height 2|y| such that

(i) every U ∈ lev2|y|(τy)iscompatiblewithσ, y; (ii) {U(2|y|−1): U ∈ lev2|y|(τy)} is a family of pairwise disjoint open sets, and for every U, V ∈ lev2|y|(τy), U(2|y|−2) = V (2|y|−2) implies U(2|y|−1) ∩ V (2|y|−1) = ∅. (iii) y y implies τy is the restriction of τy to sequences of length ≤ 2|y|; (iv) τy is maximal with these properties. <ω − Set τ∅ = ∅.Lety ∈ ω \ {∅} be arbitrary, set y = y||y|−1 and suppose that τy− is already deﬁned. Let Uy ⊆{u ∈Uω : ∃U ∈ τy− ∃v ∈Uω (U (v, u)iscompatiblewithσ, y)}

be a maximal family of pairwise disjoint open sets. For every u ∈ Uy fix one v(u) ∈Uω such that U (v(u),u) is compatible with σ, y and set lev2|y|(τy)={U (v(u),u) ∈Gω : U ∈ lev2|y|−2(τy− ),u∈ Uy}. <ω This completes the inductive step of the definition of τy (y ∈ ω \ {∅}). It is obvious from the definition that (i-iv) hold. As in the proof of statement ∈ <ω \ {∅} 2, for every y ω the maximality of Uy implies Uy = Uy is dense in k<|y| Xk. Thus with n+1 Mn = {( Xk) \ Uy : y ∈ ω }, k<|y| M= (M × X ) satisfies M∈M. It remains to show that Pr (F ) n<ω n n

Proof of Theorem 3.6. For 1, if F ∈W(I), then by omitting the yis, player I gets a strategy in G showing Pr (F ) ∈W(I). Statement 2 is Proposition ω k<ω Xk 3.10.3. The following is an immediate corollary of Theorem 3.4 and Theorem 3.6.

Corollary 3.11. With the notation of Deﬁnition 3.2 and Deﬁnition 3.5,letA ⊆ X and F ⊆ ( X ) × ωω be arbitrary. k<ω k k<ω k G \ ∈ M (1) If the game ω(A) is determined, then either ( k<ω Xk) A or there is a non-empty open tower U such that A ∩ [U]=∅.

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(2) If the game G (F ) is determined, then either Pr (F ) ∈ M or there ω k<ω Xk is a non-empty open tower U such that [U] ⊆ Pr (F ). k<ω Xk The following is to be compared to [21, Theorem 6.10], [22, Proposition 3.29, p. 31] and [30, Section 5.1.3, p. 231]. In Proposition 4.8 we will show that the complexity assumptions in this result are consistently optimal. Corollary 3.12. With the notation of Deﬁnition 3.3, ⊆ ∈ M (1) if A k<ω Xk is an analytic set, then either A or there is a non- empty open tower U such that [U] ⊆ A. ⊆ \ ∈ M (2) if A k<ω Xk is a co-analytic set, then either ( k<ω Xk) A or there is a non-empty open tower U such that A ∩ [U]=∅;inparticular, Gω(A) is determined. Proof. For 1, let F ⊆ ( X )×ωω be a closed set such that Pr (F )=A. k<ω k k<ω Xk G Then the game ω(F ) is closed, hence determined, i.e., the statement follows from Corollary 3.11.2. Statement 2 follows from 1 by taking complements. We note that Corollary 3.12.1 has an alternative proof based on Borel determinacy, as follows. Since the σ-ideal M is generated by closed sets, by [24, Theorem ∈ M ⊆ ∈ M 1, p. 1023] either A or there is a Gδ set G A such that G/ .The G \ game ω(( k<ω Xk) G) is determined, so we conclude that there is a non-empty open tower U such that [U] ⊆ G ⊆ A. However, we believe that by avoiding Borel G determinacy and by presenting the unfolded game ω we give a better insight to the fusion game Gω. 3.2. Perfect set property games. For simplicity, in this section we assume X is a non-empty perfect Polish space and we ﬁx a countable base U in X which consists of non-empty open sets. We study the following games.

N Deﬁnition 3.13. Let 0

I :((Uk(0, 0),Uk(0, 1)))k

II:(ik(0))k

where for every k

I :(U0(0, 0),U0(0, 1)) ((U0(1, 0),U0(1, 1)), (U1(0, 0),U1(0, 1))) ...

II: i0(0) (i0(1),i1(0)) ...

... (Uk(n − k, 0),Uk(n − k, 1)))k≤n ...

... (ik(n − k))k≤n ...

where for every k, l < ω and j<2wehaveik(l)∈{0, 1}, Uk(l, j)∈U, diamX (Uk(l, j)) −(k+l) < 2 , Uk(l, 0) ∩ Uk(l, 1) = ∅ and Uk(l +1,j) ⊆ Uk(l, ik(l)). Player I wins

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Notice that P1 is the usual Perfect Set Property game (see e.g. [11, Section 21.A, p. 149]). The quasi-strategies of the players in the games PN (0

Deﬁnition 3.14. Let 0

diamX (C)=max{diamX (Uk(j)): k0 such that for every N N-cube C with diamX (C) <δthere is a t ∈ 2 with F ∩C(t)=∅.Weset

N N FN = F ⊆ X : ∃Fn ⊆ X (n<ω) Fn is N-cube free (n<ω), F ⊆ Fn , n<ω ω n+1 Fω = F ⊆ X : ∃Fn ⊆ X (n<ω)

ω\(n+1) Fn is (n + 1)-cube free (n<ω), F ⊆ Fn × X . n<ω

ω N N n A function f :(2 ) → X is cube preserving if for every n<ωand sk ∈ 2 −n (k

≤ω Notice that F1 =[X] . It is easy to see that every cube preserving function is continuous and injective. The statement corresponding to Lemma 3.7 is the following. Lemma 3.15. With the notation of Deﬁnition 3.14, N (1) F ⊆ X is N-cube free if and only if clXN (F ) is N-cube free. F F (2) N (0

ω ω\(n+1) (3) Fω = F ⊆ X : ∃Fn ∈ Fn+1 (n<ω) F ⊆ Fn × X . n<ω Proof. Since C(t) is open for every N-cube C and t ∈ 2N , the ﬁrst statement follows. Then by deﬁnition, 2 holds for FN (0

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Proposition 3.16. With the notation of Definition 3.13 and Definition 3.14, N (1) for every A ⊆ X ,playerI has a winning quasi-strategy in PN (A) if and only if there is a cube preserving function f :(2ω)N → XN such that f (2ω)N ⊆ A; ω (2) for every A ⊆ X ,playerI has a winning quasi-strategy in Pω(A) if and only if there is a cube preserving function f :(2ω)ω → Xω such that f (2ω)ω ⊆ A. Proof. For 1, suppose first that player I has a winning quasi-strategy τ. By passing to a non-empty pruned subtree we can assume that τ is a strategy, i.e., for every n<ωand s ∈ lev2n(τ) there is a unique N-cube C with s C∈τ. For every ∈ n C ∈ ω n<ωand sk 2 (k

C f((σk)k

Proposition 3.17. With the notation of Definition 3.13 and Definition 3.14, N (1) for every A ⊆ X ,playerII has a winning quasi-strategy in PN (A) if and only if A ∈ FN ; ω (2) for every A ⊆ X ,playerII has a winning quasi-strategy in Pω(A) if and only if A ∈ Fω. Proof. For 1, first suppose that player II has a winning quasi-strategy τ.Let N U∅ = X . For every 0

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We show Ft (t ∈ τ)areN-cube free. Fix t ∈ τ and let C =((Uk(0),Uk(1)))k

[C] ∩ Ut =[((Uk(0) ∩ Vk,Uk(1) ∩ Vk))k

N So since Ft ⊆ Ut, we can assume [C] ⊆ Ut.Thent C∈τ, hence there is an s ∈ 2 C ∈ ∩C ∅ with t s τ. By deﬁnition, this implies Ft (s)= , as required. ⊆ ∈ Since τ is countable, it remains to show A t∈τ Ft. Suppose (xk)k

Corollary 3.18. With the notation of Deﬁnition 3.13 and Deﬁnition 3.14, N (1) for every analytic set A ⊆ X ,eitherA ∈ FN or there is a cube preserving function f :(2ω)N → XN such that f (2ω)N ⊆ A; ω (2) for every analytic set A ⊆ X ,either A ∈ Fω or there is a cube preserving function f :(2ω)ω → Xω such that f (2ω)ω ⊆ A.

Proof. To see 1, by Lemma 3.15.2, the σ-ideal FN is generated by closed sets. So by [24, Theorem 1, p. 1023], either A ∈ FN or there is a Gδ set G ⊆ A such that G/∈ FN . The game PN (G) is determined, so the statement follows from Proposition 3.16.1 and Proposition 3.17.1. Statement 2 follows by an analogous argument.

4. Infinite dimensional perfect set theorems Our inﬁnite dimensional perfect set theorems are based on the following easy observation. Recall that for every set X and α ≤ ω, ISα(X)={(xn)n<α ∈ α X : xn = xm (n

Theorem 4.1. Let X be a non-empty Choquet space such that X has no isolated points and there is a metric d on X whose balls are open in X.LetA ⊆ Xω satisfy ω X \ A ∈ M. Then there is a non-empty perfect set P ⊆ X such that ISω(P ) ⊆ A.

The proof of Theorem 4.1 uses the following version of Mycielski’s Theorem (see e.g. [20, Theorem 1, p. 141] and [11, Exercise 19.5, p. 130]).

Theorem 4.2. Let X be a non-empty Choquet space such that X has no isolated n+1 points and there is a metric d on X whose balls are open in X.LetMn ⊆ X (n<ω) be meager sets. Then there is a non-empty perfect set P ⊆ X such that for every n<ωwe have ISn+1(P ) ∩ Mn = ∅.

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⊆ n+1 ω \ ⊆ Proof of Theorem 4.1. Let Mn X (n<ω) be meager sets such that X A × ω\(n+1) ⊆ n<ω(Mn X ). By Theorem 4.2, there is a non-empty perfect set P X such that for every n<ωwe have ISn+1(P ) ∩ Mn = ∅. We show that P fulfills the requirements. ω If (xi)i<ω ∈ X \ A, then there is an n<ωsuch that (xi)i<ω ∈ Mn.Then (xi)i≤n ∈/ ISn+1(P ), hence (xi)i<ω ∈/ ISω(P ), as required. Corollary 3.12.2 say that for co-analytic A, Xω \ A ∈ M holds if there is no non- empty open tower U such that [U] ∩ A = ∅. In the sequel we give various sufficient conditions for this. ⊆ n+1 4.1. Largeness in category. Recall that for every Sn X (n<ω)wehave × ω\(n+1) ∈ n+1 ⊆ n+2 [(Sn)n<ω]= n<ω(Sn X ). If n<ω,(xi)i≤n X and S X ,then we set { ∈ ≤ ∈ } [S](xi)i≤n = xn+1 X :(xi)i n+1 S . The most important additional property our topological spaces have to satisfy is the following. Definition 4.3. A topological space X has the Kuratowski-Ulam property if for every n<ωand for every meager set M ⊆ Xn+2, n+1 { ≤ ∈ } (xi)i n X :[M](xi)i≤n is non-meager in X is meager in Xn+1. By [11, Theorem 8.41, p. 53], every second countable topological space has the Kuratowski-Ulam property. In particular, all of our results hold for Polish spaces and for the canonical refinement of Polish topologies turning a countable family of analytic sets into clopen sets (see e.g. [11, Theorem 25.18, p. 203]). Our main technical notion is the following.

Definition 4.4. Let X be a topological space. We call W =(Wn)n<ω a flag if for n+1 every n<ω, Wn ⊆ X and Wn =PrXn+1 (Wn+1). A flag W is of second category everywhere if W0 ⊆ X is of second category everywhere, and for every n<ωand ≤ ∈ (xi)i n Wn we have [Wn+1](xi)i≤n is of second category everywhere in X. If W is a flag and U is an open tower, then we say W is co-meager in U if U0 \W0 ≤ ∈ ∩ \ is meager, and for every n<ωand (xi)i n Wn Un we have [Un+1 Wn+1](xi)i≤n is meager. Lemma 4.5. Let X be a topological space and let U be a non-empty open tower. If X has the Kuratowski-Ulam property, then there is a flag W =(Wn)n<ω which is co-meager in U and Wn ⊆ Un (n<ω). Moreover, if X is Polish, then Wn (n<ω) can be taken Gδ. n+1 Proof. We define Tn(i) ⊆ X (n, i < ω) by induction, as follows. Set Tn(0) = UnΔPrXn+1 (Un+1)(n<ω). Let i<ωand suppose that Tn(i)(n<ω) are defined. Then let { ∈ n+1 } Tn(i +1)= (xi)i≤n X :[Tn+1(i)](x ) ≤ is non-meager (n<ω). i i n \ × n−k We show that Wn = Un i<ω k≤n(Tk(i) X )(n<ω) fulfill the requirements. It is obvious that Wn ⊆ Un (n<ω). Next we show that U0 \ W0 is meager and ≤ ∈ \ that for every n<ωand (xi)i n Wn we have [Un+1 Wn+1](xi)i≤n is meager. Since U is an open tower, Tn(0) (n<ω) are meager. Using the Kuratowski-Ulam

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n+1 property of X,itiseasytoseethatforeveryn, i < ω, Tn(i) ⊆ X is meager. Hence U0 \ W0 is meager, as required. Now let n<ωand let (xi)i≤n ∈ Wn.By(xi)i≤n ∈/ Tn(j +1)(j<ω)wehave \ [Tn+1(j)](xi)i≤n is meager (j<ω). So [Un+1 Wn+1](xi)i≤n is meager, as required. To see that W is a flag, we have to show Wn =PrXn+1 (Wn+1)(n<ω). Let first n<ωand (xi)i≤n ∈ Wn be arbitrary. By (xi)i≤n ∈/ Tn(0) we have ∅ \ [Un+1](xi)i≤n = . As we have seen above, [Un+1 Wn+1](xi)i≤n is meager, in particular [W ] = ∅, as required. If n<ωand (x ) ≤ ∈ W is arbitrary, n+1 (xi)i≤n i i n+1 n+1 ∈ ∈ × n+1−k then by (xi)i≤n+1 Un+1 and (xi)i≤n+1 / i<ω k≤n+1(Tk(i) X )we ∈ ∈ × n−k ∈ get (xi)i≤n Un and (xi)i≤n / i<ω k≤n(Tk(i) X ), i.e. (xi)i≤n Wn,as required. Finally, if X is Polish, then Tn(0) (n<ω)areFσ. By Montgomery’s Theorem (see e.g. [11, Exercise 22.22, p. 174]), Tn(i)(n, i < ω)areFσ, so by definition, Wn (n<ω)areGδ. Corollary 4.6. Let X be a topological space with the Kuratowski-Ulam property, and let U be a non-empty open tower. Let V be flag which is of second category everywhere. Then [U] ∩ [V] = ∅.

Proof. By Lemma 4.5, there is a flag W which is co-meager in U and Wn ⊆ Un (n<ω). By induction on n<ω, we define xn ∈ X (n<ω) such that (xi)i≤n ∈ Wn+1 ∩ Vn+1 (n<ω). Then by (xn)n<ω ∈ [W] ∩ [V]and[W] ⊆ [U] the statement follows. Since U0 = ∅, U0 \ W0 is meager and V0 is of second category everywhere, we have W0 ∩ V0 = ∅.Letx0 ∈ W0 ∩ V0 be arbitrary. Let n<ωand suppose that xi (i ≤ n) are defined such that (xi)i≤n ∈ Wn+1 ∩ W ∅ ⊆ Vn+1.Since is a flag, we get [Wn+2](xi)i≤n = .ByWn+2 Un+2 this implies ∅ \ [Un+2](xi)i≤n = .Moreover,[Un+2 Wn+2](xi)i≤n is meager, so since [Vn+2](xi)i≤n ∩ ∅ is of second category everywhere in X,wehave[Vn+2](xi)i≤n [Wn+2](xi)i≤n = . ∈ ∩ Let xn+1 [Vn+2](xi)i≤n [Wn+2](xi)i≤n be arbitrary. This completes the inductive step of the construction and finishes the proof. Corollary 4.7. Let X be a non-empty Choquet space such that X has no isolated points, X has the Kuratowski-Ulam property and there is a metric d on X whose balls are open in X.LetA ⊆ Xω.

(1) Suppose Gω(A) is determined and there is an H ⊆ X which is of second category everywhere and ISω(H) ⊆ A. Then there exists a non-empty perfect set P ⊆ X such that ISω(P ) ⊆ A. (2) Suppose A is co-analytic and there is an H ⊆ X which is non-meager and ISω(H) ⊆ A. Then there exists a non-empty perfect set P ⊆ X such that ISω(P ) ⊆ A. n+1 Proof. To see 1, set Vn = {(xi)i≤n ∈ H : xi = xj (i

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clopen sets in the usual way. Even if largeness in the Baire category is not preserved G during such reﬁnement of topologies, this observation shows that the game ω(A) ∩ ⊆ ⊆ is informative for A = C k<ω Ak,whereC X is co-analytic and Ak X (k<ω) are analytic. Finally we show that the complexity assumptions in Corollary 3.12 are consistently optimal. The assumption of the following proposition holds e.g. in L (see e.g. [9, Corollary 25.28, p. 495]). 1 ⊆ ω Proposition 4.8. Assume there exists a Σ2 set D 2 which does not have the Baire property. Then there exists a co-analytic set A ⊆ (2ω)ω such that A/∈ M but [U] ⊆ A for every non-empty open tower U. Proof. By passing to a relative open subset we can assume D is non-meager and 2ω \ D is of second category everywhere. Let C ⊆ 2ω × 2ω be a co-analytic set such that projection of C to the ﬁrst coordinate is D.Let ω ω A = {(xi)i<ω ∈ (2 ) :(x0, (xi(0))0

ω Un = {x ∈ 2 : x(0) = x1(i)} (n<ω) i≤n ω n+1 and Vn = Wn+1 ∩ ({x0}×(2 ) )(n<ω). Then U =(Un)n<ω is a non-empty open tower and {x0}×[U] ⊆ A, while V =(Vn)n<ω is a co-meager ﬂag with {x0}×[V] ⊆ [W]. By Corollary 4.6 we have [U] ∩ [V] = ∅, i.e. A ∩ [W] = ∅,andso A ⊆ M.ThisprovesA/∈ M and completes the proof. 4.2. Largeness in cardinality. We show that in the iterated perfect set model and in Cohen extensions the existence of a homogeneous set of suﬃciently large cardinality implies the existence of a non-empty perfect homogeneous set. Theorem 4.9. Let V be a model obtained from a model of the Continuum Hy- ω pothesis by adding ω2 Sacks reals. Let A ⊆ R be a co-analytic set such that there is an A-homogeneous set of cardinality ω2. Then there exists a non-empty perfect A-homogeneous set.

Proof. Let H ∈ [R]ω2 be A-homogeneous. By [18, Theorem, p. 581], there is a non- empty perfect set X ⊆ R such that H ∩ X is of second category everywhere in X. Then the statement follows from Corollary 4.7.2 applied to A ∩ Xω and H ∩ X.

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For every cardinal μ, C[μ]={f ⊆ μ × ω → 2: |f| <ω} denotes the forcing for adding μ many Cohen reals. We will use the elementary properties of the Cohen forcing stated in [9, Lemma 26.4, p. 514], [14, Lemma 2.2, p. 250] and [14, Theorem 2.1, p. 252] without further reference. ℵ Theorem 4.10. In V ,letκ =2 0 and let κ<λ≤ μ be arbitrary cardinals. In V C[μ],letA ⊆ (2ω)ω be a co-analytic set such that there is an A-homogeneous set of cardinality λ. Then there exists a non-empty perfect A-homogeneous set. C[μ] Proof. In V ,letR = {rα : α<μ} be the μ many Cohen reals added to V .First ∈ λ ∈ λ { ∈ } suppose H [R] , i.e., that there is an I [μ] such that H = rα : α I . In V C[μ],letU = {U ⊆ 2ω : U open, |U ∩ H| <λ}.SetU = U. Then there is ≤ω an IU ∈ [μ] such that U ∈ V [{rα : α ∈ IU }]. Since {rα : α/∈ IU } are Cohen reals ω over V [{rα : α ∈ IU }], U cannot be dense in 2 . So we can find a non-empty open set O ⊆ 2ω \ U. C[μ] We show that in V , every non-meager Gδ set G ⊆ O satisfies |G ∩ H| = λ. Given such a G,letW ⊆ O be a non-empty open set such that G is dense in W . ≤ω Let IG,W ∈ [μ] be such that G, W ∈ V [{rα : α ∈ IG,W }]. Since W ⊆ O,bythe definition of O we have |H ∩W | = λ.ThesetHG,W = {rα : α ∈ IG,W } is countable. So H ∩ W \ HG,W has cardinality λ, and each member of this set is a Cohen real over V [{rα : α ∈ IG,W }]. Thus H ∩ W \ HG,W ⊆ G, so the statement follows. By Theorem 4.1, in order to conclude the existence of a non-empty perfect A- homogeneous set, it is enough to show that Oω \ A ∈ M. By Corollary 3.12.2, this follows if we show that A ∩ [U] = ∅ for every non-empty open tower U =(Un)n<ω n+1 with Un ⊆ O (n<ω). So let U be such an open tower. By Lemma 4.5, there is a flag W =(Wn)n<ω which is co-meager in U andinwhichWn (n<ω)areGδ. The proof will be complete if we show A ∩ [W] = ∅. ≤ω Let IW ∈ [μ] be such that W ∈ V [{rα : α ∈ IW}]. We define a sequence (hn)n<ω ∈ ISω(H) ∩ [W] by induction, as follows. Since W0 ⊆ O is a non-meager Gδ set, we have |H ∩ W0| = λ.Leth0 ∈ H ∩ W0 be arbitrary. Let n<ωbe arbitrary and suppose that hi (i ≤ n) are defined such that ≤ ∈ ∩ ⊆ (hi)i n ISn+1(H) Wn. By definition, [Wn+1](hi)i≤n O is a non-meager Gδ set. | ∩ | ∈ ∩ \{ ≤ } So H [Wn+1](hi)i≤n = λ,thuswecanpickhn+1 H [Wn+1](hi)i≤n hi : i n . This completes the inductive step of the definition of (hn)n<ω and completes the proofofthespecialH ∈ [R]λ case. In the general case, let H ∈ [2ω]λ be A-homogeneous. For every h ∈ H fix an ≤ω Ih ∈ [μ] such that h ∈ V [{rα : α ∈ Ih}]. By the standard Δ-system argument and by extending V , we can assume Ih ∩Ih = ∅ (h, h ∈ H, h = h ). By passing to a subset of H, we can assume in addition that tp(Ih)=tp(Ih )=η<ω1 (h, h ∈ H). ω η ω For every h ∈ H,letfh :(2 ) → 2 be a Borel function in V such that fh((rα)α∈Ih )=h. By passing again to a subset of H, we can assume fh = fh = f ω η (h, h ∈ H). Let X ⊆ (2 ) be a co-meager Gδ set such that f|X is continuous. ∈ ω η Observe that for every h H,(rα)α∈Ih is a Cohen real in (2 ) , so in particular ∈ ∈ (rα)α∈Ih X (h H). ω ω Set A = {(xk)k<ω ∈ X :(f(xk))k<ω ∈ A ∩ ISω(2 )}. As we observed above, { ∈ }⊆ H = (rα)α∈Ih : h H X is an A-homogeneous set of Cohen reals of cardinality λ.SinceA is co-analytic, by the special case of Theorem 4.10 proved above, there exists a non-empty perfect A-homogeneous set P ⊆ X. ω Set P = f[P]. By definition, if (pk)k<ω ∈ ISω(P), then (f(pk))k<ω ∈ ISω(2 ). Hence f is injective on P, i.e. by the continuity of f, P is a non-empty perfect

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set. Similarly, for every (pk)k<ω ∈ ISω(P)wehave(f(pk))k<ω ∈ A,soP is A- homogeneous. This completes the proof.

Proof of Theorem 1.3. Let 1 <α<ω1 be an ordinal. By [23, Claim 3.9, p. 39], ω 2 it is consistent with ZFC that there exists an Fσ set C ⊆ [2 ] such that there exists a C-homogeneous set of cardinality ℵα but there is no non-empty perfect C-homogeneous set. Then by Theorem 1.1, there exists a symmetric open set ω ω U ⊆ (2 ) such that there exists a U-homogeneous set of cardinality ℵα but there is no non-empty perfect U-homogeneous set. ℵ0 On the other hand, by starting from a model with 2 = ℵ1 and by adding ℵα+1 ℵ0 many Cohen reals, we get a model in which 2 = ℵα+1, and Theorem 4.10 implies the statement.

5. An application: Definability of Tukey maps In this section a set is called bounded if it is bounded from above. Similarly, directed means upward directed. Let (P, ≤)and(Q, ≤) be directed partial orders. We say that (P, ≤)isTukey reducible to (Q, ≤), (P, ≤) ≤T (Q, ≤) in notation, if there is a function f : P → Q such that for every unbounded set A ⊆ P , f[A] ⊆ Q is unbounded. Such an f is called a Tukey map.If(P, ≤) ≤T (Q, ≤)and(Q, ≤) ≤T (P, ≤), then (P, ≤)and(Q, ≤) are called Tukey equivalent,(P, ≤) ≡T (Q, ≤) in notation. In the sequel we do not write out the partial order when it is obvious from the context. An equivalent definition of Tukey reducibility is that P ≤T Q if and only if there is a function g : Q → P such that for every cofinal set A ⊆ Q, g[A] ⊆ P is cofinal. This characterization indicates that Tukey reductions provide information about the cofinal types of directed partial orders and explains why Tukey equivalence classes are also called cofinal types. Tukey reducibility turns out to be the right tool for the comparison of cofinal types. Not only does the existence of a Tukey reduction between two directed partial orders relate many of their structural properties (e.g. it is easy to see, using the equivalent definitions given above, that P ≤T Q implies add(Q) ≤ add(P ) and cof(P ) ≤ cof(Q)), but in addition, Tukey reductions account for many known inequalities between cardinal invariants (e.g. all inequalities in the Cichoń diagram can be witnessed by Tukey maps; see e.g. [5] and [1]). So a natural question arises: how many cofinal types of directed partial orders are there? The following result indicates that such a general endeavor has to face independence. As usual, for every cardinal κ,[κ]<ω denotes the set of finite subsets of κ partially ordered by inclusion. We remark that [κ]<ω is the maximal cofinal type of directed partial orders of cardinality ≤ κ (see e.g. [29, Theorem 5.1, p. 13]). Theorem 5.1 ([27, Theorem 9, p. 718]). (1) If the Continuum Hypothesis holds, then there are 2ω1 many different cofinal types of directed partial orders of cardinality ω1. <ω (2) It is consistent with ZFC that {1,ω,ω1,ω× ω1, [ω1] } are the only cofinal types of directed orders of cardinality ≤ ω1. Therefore it is reasonable to restrict the cofinal diversity problem to classes of directed orders which carry additional structures (see e.g. [5], [6]). One possible restriction is to assume definability properties. Accordingly, in the present section we study analytic ideals on ω, i.e. such families I⊆P(ω) which form an ideal

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under the partial order ⊆ and which are analytic subsets of P(ω), endowed with the Cantor space topology. In Section 5.4 we will examine how restrictive this assumption is (see Proposition 5.25 and Proposition 5.26). Recall that in the deﬁnition of Tukey reducibility the reducing functions are not required to possess any regularity properties. However, Tukey maps are not unique; e.g. if f is a Tukey map and f ≤ f pointwise, then f is also a Tukey map. So it is reasonable to ask whether a Tukey reduction between analytic ideals can be witnessed by “nice” Tukey maps (see [5, Problem 3N (c), p. 212], [16, Question 2, p. 193] and [28, Question 6.69] for analogous problems).

Problem 5.2. Let I, J⊆P(ω) be analytic ideals satisfying I≤T J .Isthere then a “deﬁnable” Tukey map f : I→J?

Depending on I and J , “definable” may mean continuous, Borel measurable, Souslin measurable (i.e. measurable with respect to the σ-algebra generated by analytic sets), Baire measurable, Lebesgue measurable, etc. An affirmative answer to this problem could allow the use of descriptive set theoretic methods for the study of an originally non-definable object. Surprisingly, Problem 5.2 has an affirmative answer for many analytic ideals. In [25] the notion of basic directed partial orders was introduced and the following result was proved.

Theorem 5.3 ([25, Theorem 5.3, p. 1890]). Let P, Q be basic directed partial orders satisfying P ≤T Q. Then there is a Souslin measurable Tukey map f : P → Q. We will recall the deﬁnition of basic directed partial orders in Section 5.4 (see Deﬁnition 5.27). Here we only mention that every analytic P -ideal on ω is basic. However, there are many analytic ideals on ω which are not basic in any topology; we will call such ideals non-basic. In [16, Section 7, p. 190] a sequence of non-basic Borel ideals was constructed which is strictly decreasing in the Tukey hierarchy. In Section 5.4 we will prove the following.

Proposition 5.4. The structure (P(ω), ⊆) embeds into the family of non-basic Fσ ideals on ω partially ordered by ≤T . ℵ As we pointed out above, [2 0 ]<ω is the maximal Tukey type among directed ℵ partial orders of cardinality ≤ 2 0 , so in particular among analytic ideals. As we will recall in Section 5.4, this maximal coﬁnal type admits a representation as an Fσ ideal on ω.

Proposition 5.5 ([16, Proposition 3, p. 185]). There exists an Fσ ideal Imax ⊆ ℵ0 <ω P(ω) such that Imax ≡T [2 ] .

As we will see in Section 5.4, Imax is not basic; in particular, Theorem 5.3 does not apply to its Tukey reductions. Therefore the following special case of Problem 5.2 is of particular interest (see e.g. [28, Question 6.69]).

Problem 5.6. Let I⊆P(ω) be an analytic ideal satisfying Imax ≤T I.Isthere then a “deﬁnable” Tukey map f : Imax →I? The purpose of this section is to show that even Problem 5.6 is independent of ZFC.

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Theorem 5.7. Let I⊆P(ω) be an arbitrary analytic ideal. (1) Let V be a model obtained from a model of the Continuum Hypothesis by adding ω2 Sacks reals. Then in V ,ifImax ≤T I, then there is a continuous Tukey map f : Imax →I. (2) If the Continuum Hypothesis holds, then there is an analytic ideal J ⊆P(ω) such that Imax ≤T J,butiff : Imax → J is a Tukey map, then f[Imax] has no non-empty perfect subsets. In particular, a Tukey map f : Imax → J cannot be Lebesgue measurable or have the Baire property.

As a corollary, in Section 5.2 we obtain that it is consistent with ZFC that Imax has the primality property (see Definition 5.13 and Theorem 5.15). Presently, we do not know about any other special cases of Problem 5.2 where the same independence phenomenon appears. What makes Tukey reductions of Imax particularly easy to describe is the following simple characterization. Definition 5.8. Let (P, ≤) be a directed partial order. A set H ⊆ P is called strongly unbounded if every A ∈ [H]ω is unbounded in (P, ≤). Proposition 5.9 ([16, Section 1, p. 174]). Let (P, ≤) be a directed partial order <ω and let κ be an arbitrary cardinal. Then [κ] ≤T P if and only if there exists a strongly unbounded set H ∈ [P ]κ. 5.1. A consistent positive answer to Problem 5.6. We will need the following simple observation. Lemma 5.10. Let P and Q be directed partial orders and let f : P → Q be a Tukey map. If H ⊆ P is strongly unbounded, then f|H is finite-to-one and f[H] ⊆ Q is also strongly unbounded. Proof. By definition, every A ∈ [H]ω is unbounded. Since f is a Tukey map, f[A] cannot be a singleton, i.e., f is finite-to-one. The second statement immediately follows from the definition. The following implies Theorem 5.7.1 and also shows that the conclusion of Theo- rem 5.7.1 holds for every projective ideal under suitable large cardinal assumptions. Theorem 5.11. Let Γ be a projective pointclass such that every I ∈ Γ(P(ω)) has ω the Baire property, and for every A ∈ Γ(P(ω) ) the game Gω(A) is determined. ℵ Let κ ≤ 2 0 be a cardinal such that for every H ∈ [P(ω)]κ there exists a non-empty perfect set Q ⊆P(ω) such that H ∩ Q is of second category everywhere in Q.Let <ω I ⊆P(ω) be an ideal with P(ω) \ I ∈ Γ(P(ω)) such that [κ] ≤T I. Then there is a continuous Tukey reduction from [2ω]<ω to I. Proof. Let f :[κ]<ω → I be a Tukey map and let H = f[κ]. By Lemma 5.10, f is finite-to-one, hence H has cardinality κ. So there exists a non-empty perfect set Q ⊆P(ω) such that H ∩ Q is of second category everywhere in Q.Since I ∩ Q has the Baire property in Q,byH ∩ Q ⊆ I ∩ Q there is a co-meager Gδ set X ⊆ I ∩ Q ⊆P (ω) satisfyingH ∩ X is of second category everywhere in X. ∈ ω ∈ P ω Set A = (xi)i<ω X : i<ω xi / I . Since the union function from (ω) to P(ω)isBorelandP(ω) \ I ∈ Γ(P(ω)), we have A ∈ Γ(Xω). Then the game Gω(A) is determined and ISω(H ∩ X) ⊆ A, so by Corollary 4.7.1 there is a non- empty perfect set P ⊆ X such that ISω(P ) ⊆ A. Then any continuous bijection f˜:2ω → P is a continuous Tukey map from [2ω]<ω to I.

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ω <ω Proof of Theorem 5.7.1. Recall that Imax ≡T [2 ] . By [18, Theorem, p. 581], ℵ in the iterated perfect set model for every H ∈ [2ω]2 0 there exists a non-empty perfect set Q ⊆P(ω) such that H ∩ Q is of second category everywhere in Q.So 1 by Corollary 3.12.2, the statement follows from Theorem 5.11 with Γ = Π1 and ℵ0 κ =2 = ω2.

5.2. Consistent primality of Imax. It is easy to see that the least upper bound in the Tukey order of two ideals I, J⊆P(ω) is their direct sum, or disjoint union, defined as follows. Definition 5.12. Let I, J⊆P(ω) be arbitrary ideals. Let E = {2n: n<ω}. Then I⊕J ⊆P(ω) is the ideal defined by A ∈I⊕J ⇔{n<ω:2n ∈ A ∩ E}∈I and {n<ω:2n +1∈ A \ E}∈J. For a complete description of the cofinal types of analytic ideals, ideals having the following primality property are of particular importance. Definition 5.13. We say that an ideal I⊆P(ω)hastheprimality property if for every ideal J , K⊆P(ω), I≤T J⊕Kimplies I≤T J or I≤T K. It is reasonable to ask which ideals have the primality property, and especially whether Imax has the primality property (see e.g. [28, Question 6.68]). Note that by [27, Theorem 6, p. 715] the primality of Imax fails among non-definable ideals. On the other hand, by [28, Theorem 6.71], the primality of Imax holds for Souslin measurable Tukey reductions, as follows. Theorem 5.14 ([28, Theorem 6.71]). Let I, J⊆P(ω) be analytic ideals such that there exists a Souslin-measurable Tukey reduction of Imax to I⊕J. Then either Imax ≤T I or Imax ≤T J . So by Theorem 5.7.1, we get the following. Theorem 5.15. Let V be a model obtained from a model of the Continuum Hy- pothesis by adding ω2 Sacks reals. In V ,letI, J⊆P(ω) be analytic ideals such that Imax ≤T I⊕J. Then either Imax ≤T I or Imax ≤T J .

Proof. By Theorem 5.7.1, there exists a continuous Tukey reduction of Imax to I⊕J. So the statement follows from Theorem 5.14. 5.3. A consistent negative answer to Problem 5.6. In this section we construct the ideal J of Theorem 5.7.2. To this end, the main technical step is to observe that it is enough to construct an ideal of compact sets with the same properties. Deﬁnition 5.16. Let K(2ω) denote the space of compact subsets of the Cantor set endowed with the Vietoris topology. A family I⊆K(2ω) is called an ideal of compact sets if I is closed under taking ﬁnite unions and I is closed downward, i.e. for every K, L ∈K(2ω), K ⊆ L ∈Iimplies K ∈I. Throughout this section we use the following notation.

<ω ω <ω Deﬁnition 5.17. For every s ∈ 2 ,setNs = {x ∈ 2 : s x}.LetΩ=2 .We deﬁne Φ: K(2ω) →P(Ω),

Φ(K)={s ∈ Ω: Ns ∩ K = ∅}.

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For every A⊆P(Ω), set A↓ = {B ∈P(Ω): ∃A ∈A(B ⊆ A)}. For every I⊆K(2ω) we deﬁne I =(Φ[I])↓. Lemma 5.18. With the notation of Deﬁnition 5.17, we have the following: (1) Φ is continuous, and for every K, L ∈K(2ω), K ⊆ L ⇔ Φ(K) ⊆ Φ(L).In particular, Φ: K(2ω) → Φ[K(2ω)] is a homeomorphism. (2) If I⊆K(2ω) is an ideal, then I ⊆P(Ω) is also an ideal. (3) If A⊆P(Ω) is analytic, then A↓ ⊆P(Ω) is also analytic. (4) If I⊆K(2ω) is an analytic ideal, then I ⊆P(Ω) is also an analytic ideal. (5) 3 and 4 also hold if we replace “analytic” with “Fσ”. Proof. It is obvious that for every K, L ∈K(2ω), K ⊆ L ⇔ Φ(K) ⊆ Φ(L). In particular, Φ is injective. For every s ∈ Ω, Φ−1({A ∈P(Ω): s ∈ A})={K ∈ ω K(2 ): K ∩ Ns = ∅} is clopen in the Vietoris topology, thus Φ is continuous. Since K(2ω) is compact, Φ is a homeomorphism. For 2, it is obvious that I is closed downward. To see that it is closed under taking unions, pick Ai ∈ I (i<2). Then there are Ki ∈I(i<2) such that Ai ⊆ Φ(Ki)(i<2). By 1, A0 ∪ A1 ⊆ Φ(K0) ∪ Φ(K1)=Φ(K0 ∪ K1), so the statement follows. To see 3, observe that {(A, B) ∈P(Ω) ×P(Ω): B ⊆ A} is a closed relation and A↓ is the projection of {(A, B) ∈P(Ω) ×P(Ω): B ⊆ A ∈A}to the second coordinate. Hence A↓ is analytic if A is analytic, and by P(Ω) being compact, A↓ is Fσ if A is Fσ. Finally, 4 and 5 follow from 1, 2 and 3.

The ideals I and I are coﬁnally similar, as well.

ω Lemma 5.19. Let I⊆K(2 ) be an analytic ideal. Then I≡T I. Moreover: (1) Φ: I→I and Φ−1 :Φ[I] →I are continuous Tukey maps; (2) there is a Tukey reduction Φ−1 ◦ f : I →Iwhere f : I → I is a Souslin measurable Tukey map such that H ⊆ f(H) ∈ Φ[I](H ∈ I). Proof. To see 1, let H⊆Ibe an arbitrary set. If Φ[H] ⊆ I is bounded, then there is an L ∈Isuch that Φ(K) ⊆ Φ(L)(K ∈H). By Lemma 5.18.1, this implies K ⊆ L (K ∈H), i.e. H is also bounded. Hence Φ is a Tukey map, which is continuous by Lemma 5.18.1. The argument for Φ−1 is identical. For 2, consider the set P = {(A, B) ∈ I×Φ[I]: A ⊆ B}.ThenP ⊆P(Ω)×P(Ω) is analytic, so by the Jankov-von Neumann Uniformization Theorem (see e.g. [11, Theorem 18.1, p. 120]) P has a Souslin measurable uniformizing function f : I → Φ[I]. Also, H ⊆ f(H) ∈ Φ[I](H ∈ I)holds. Since f pointwise dominates the identity function, it is a Tukey map. Then by 1, Φ−1 ◦ f : I →I is also a Tukey map, which completes the proof.

Corollary 5.20. Let I⊆K(2ω) be an analytic ideal. <ω <ω (1) For every cardinal κ we have [κ] ≤T I⇔[κ] ≤T I. (2) There exists a non-empty perfect strongly unbounded subset of I if and only if there exists a non-empty perfect strongly unbounded subset of I. Proof. Statement 1 follows from Lemma 5.19. To see 2, suppose ﬁrst that P ⊆I is a non-empty perfect strongly unbounded set. By Lemma 5.18.1, Φ is continuous

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and injective. So Φ[P ] is also non-empty and perfect. By Lemma 5.19.1, Φ is a Tukey map, i.e. by Lemma 5.10, Φ[P ] is strongly unbounded. Finally let P ⊆ I be a non-empty perfect strongly unbounded set. Let f be the function of Lemma 5.19.2. Then f|P is Souslin measurable. Hence it has the Baire property, so by passing to a non-empty perfect subset of P we can assume that f|P is continuous (see e.g. [11, Theorem 8.38, p. 52]). By Lemma 5.10 and Lemma 5.19.2, f|P is ﬁnite-to one. Thus f[P ] ⊆ Φ[I]is an uncountable compact strongly unbounded set; in particular, it has a non-empty perfect subset Q, which is also strongly unbounded. So by Lemma 5.19.1 and Lemma 5.10, Φ−1(Q) ⊆Iis a non-empty perfect strongly unbounded set.

Our main theorem is the following.

Theorem 5.21. ThereisananalyticidealJ⊆K(2ω) with the following properties: <ω (1) [ω1] ≤T J . <ω (2) [ω2] ≤T J . (3) There is no non-empty perfect strongly unbounded subset of J .

The following corollary answers [16, Conjecture 2, p. 194] in the negative.

Corollary 5.22. There is an analytic ideal J ⊆P(ω) with the following properties: <ω (1) [ω1] ≤T J. <ω (2) [ω2] ≤T J. (3) There is no non-empty perfect strongly unbounded subset of J.

Proof. Let J be the ideal of Theorem 5.21 and set J =(Φ[J ])↓. By Lemma 5.18.4, J is an analytic ideal. By Corollary 5.20, 1, 2 and 3 follows from Theorem 5.21.

Also, Theorem 5.7.2 easily follows.

Proof of Theorem 5.7.2. Let J be the ideal of Corollary 5.22. Assume the Contin- <ω uum Hypothesis holds. Then Imax ≡T [ω1] ≤T J by Corollary 5.22.1. Let f : Imax → J be an arbitrary Tukey map. By Lemma 5.10, f[Imax] ⊆ J is strongly unbounded, so every subset of f[Imax] is strongly unbounded, as well. Hence by Corollary 5.22.3, f[Imax] has no non-empty perfect subsets. If f is either Lebesgue measurable or has the Baire property, then there is an uncountable Borel set B ⊆Imax such that f|B is continuous. By Lemma 5.10, f[B] ⊆ J is an uncountable strongly unbounded analytic set; in particular, it has a non-empty perfect strongly unbounded subset. This contradicts Corollary 5.22.3.

It remains to prove Theorem 5.21. Recall the definition of the Cantor-Bendixson derivative (see e.g. [11, Definition 6.10, p. 33]): for every K ∈K(2ω), set K = {x ∈ K : x ∈ cl2ω (K \{x})}. We define, by induction on n<ω, the iterated derivatives: K(0) = K, K(n+1) =(K(n)) (n<ω). Also recall that there exists an analytic ω ω equivalence relation E ⊆ 2 × 2 such that E has exactly ω1 many equivalence classes (see e.g. [9, Exercise 32.3, p. 625]).

Deﬁnition 5.23. Set R = {K ∈K(2ω): ∃n<ω(K(n) = ∅)}.LetE ⊆ 2ω × 2ω be an analytic equivalence relation with exactly ω1 many equivalence classes. We

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deﬁne

(5.1) L = {K ∈K(2ω): K ∈R, ∀x, y ∈ K ((x, y) ∈ E)}.

It is known that R is a Borel subset of K(2ω) (see [3] and [4] for its exact Borel class). However, due to the universal quantiﬁcation in the deﬁnition of L,the following is not straightforward.

Proposition 5.24. L⊆K(2ω) is analytic.

K ω n →K ω Proof. For every n<ω,letϑn : (2 ) (2 ) denote the union function, i.e. ∈K ω n for every (Ki)i

and (5.3) L L ∪{ ∈K ω ∃ ∈ ∈ \ ∈L+ } n+1 = 0 L (2 ): u, v L (u = v, (u, v) E, L Nu|m n (m<ω)) .

L L+ ⊆K ω It follows from the definition that n, n (2 )(n<ω) are analytic sets and L ⊆L+ L L+ n n (n<ω). So the proof will be complete if we show = n<ω n . L+ ⊆L ∈L+ To see n<ω n , we show by induction on n that for every L n we have L(n+1) = ∅ and (x, y) ∈ E (x, y ∈ L). The statement is obvious for n =0;solet L+ ∈L+ n<ωbe arbitrary and suppose that the statement holds for n .LetL n+1 be arbitrary. Suppose first L ∈Ln+1, and let u, v ∈ L be as in (5.3). By the inductive hypothesis, L(n) ⊆{u} and (x, y) ∈ E (x, y ∈ L \{u}). Thus L(n+1) = ∅,and (u, v) ∈ E, u = v imply (x, y) ∈ E (x, y ∈ L). ∈L In the general case, for some m<ω,letL = i 1, then it is a finite set, say L = {ui : i

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ω n ω Proof of Theorem 5.21. Recall that for every n<ω, ϑn : K(2 ) →K(2 ) denotes the union function. With the set L defined in (5.1), set n J = ϑn[L ]. n<ω We show that J fulfills the requirements. Since L is closed downward and J is closed under taking finite unions, J is an ideal of compact sets. The functions ϑn (n<ω) are continuous. Hence by Proposition 5.24, J is analytic. It is immediate from the definition that for every K ∈K(2ω), K ∈J if and only if (i) K intersects only finitely many equivalence classes of E; (ii) for every equivalence class C ⊆ 2ω of E, K ∩ C ∈K(2ω)and(K ∩ C)(n) = ∅ for some n<ω. Let H ∈ [2ω]ω1 be an arbitrary set such that (x, y) ∈/ E (x, y ∈ H), and set H =[H]1. By (i), H⊆J is strongly unbounded, i.e., by Proposition 5.9, 1 holds. To see 2, by Proposition 5.9 we have to show that if H∈[J ]ω2 ,thenH is ω2 not strongly unbounded. So let H∈[J ] be arbitrary. Let {Eα : α<ω1} be an enumeration of the equivalence classes of E. By (i) and (ii), there are n<ω, <ω ω2 A ∈ [ω1] and H ∈ [H] such that for every K ∈H,

(n) K ∩ Eα = ∅⇔α ∈ A and (K ∩ Eα) = ∅ (α ∈ A). X J∩K Consider the space = α∈A( (Eα)) and the set X = (Kα)α∈A : Kα ∈J∩K(Eα), Kα ∈H . α∈A Then X is a separable metric space and X ⊆X is an uncountable set. Since X ∈X ∈ is Lindelöf, there is a (Kα)α∈A and an injective sequence (Kα(m))α∈A X ∈ (m<ω) such that limm<ω Kα(m)=Kα (α A). Set H(m)= α∈A Kα(m) ∪ ∈J (m<ω), H = α∈A Kα and L = H n<ω H(m). Observe that H ; in particular, there is a d<ωsuch that H(d) = ∅.Also,H(m) ∈H implies H(m)(n) = ∅ (m<ω). ω We have L ∩ Eα = ∅⇔α ∈ A (α<ω1)andL ∩ Eα ∈K(2 )(α ∈ A). Moreover, L(n) ⊆ H, hence L(n+d) = ∅. By (i) and (ii) this implies L ∈J. Since ((Kα(m))α∈A)m<ω is injective, {H(m): m<ω}⊆His an infinite set. By H(m) ⊆ L (m<ω), H is not strongly unbounded, as stated. Finally, observe that by Shoenfield’s Absoluteness (see e.g. [9, Theorem 25.20, p. 490]), a non-empty perfect strongly unbounded subset of J would remain strongly unbounded in any forcing extension. Hence 3 follows from 2.

5.4. Miscellanea. In the previous sections we restricted our investigations to analytic ideals I⊆(P(ω), ⊆). Unfortunately, there are simple directed partial orders which are not coﬁnally similar to any such ideal. The following may be considered as a negative answer to [16, Question 1, p. 193].

Proposition 5.25. Let 1 denote the one element set with the trivial order, let ω denote the ﬁrst inﬁnite ordinal with its usual well-order, and let (ωω, ≤) be the set

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of all functions from ω to ω, partially ordered by eventual dominance.

(1) If I⊆(P(ω), ⊆) is an arbitrary directed set, then I≤T 1 or ω ≤T I. ω ω ω (2) The directed partial order (ω , ≤ ) satisfies ≤ ⊆ ω × ω is Fσ, but it is ω incomparable with ω in the Tukey order. In particular, (ω , ≤ ) ≡T I for every directed set I⊆(P(ω), ⊆). ∈ ≤ ∈ ⊆ Proof. To see 1, let S = I.IfS I,then I T 1. If S/I,let(Sn)n<ω I be an increasing sequence satisfying S = n<ω Sn; such a sequence exists since I is directed. Then f : ω → I, f(n)=Sn (n<ω)isaTukeymap. ω For 2, ω ≤T (ω , ≤ ) follows from the fact that every countable subset of ω ω ω (ω , ≤ ) is bounded. To see (ω , ≤ ) ≤T ω let f :(ω , ≤ ) → ω be an arbitrary function. Then there is an n<ωsuch that f −1({n}) ⊆ ωω is non-meager. Since for every ϕ ∈ ωω, {ψ ∈ ωω : ψ ≤ ϕ} is meager, such a set cannot be bounded. Hence f is not a Tukey map. The second statement follows from 1. On the other hand, closed directed partial orders on analytic spaces admit rep- resentations as analytic ideals on ω. Proposition 5.26. Let (I,≤) be a topological space endowed with a directed partial order such that I is a continuous image of a Polish space and ≤⊆ I × I is closed. Then there is an analytic ideal I ⊆ (P(ω), ⊆) such that (I,≤) ≡T I. Proof. Let G ⊆ 2ω be an arbitrary set homeomorphic to ωω, and let f : G → I be a continuous surjection (see e.g. [11, Theorem 7.9, p. 38]). For every a ∈ I,set La = {b ∈ I : b ≤ a}.NotethatLa ⊆ I (a ∈ I) are closed. We show that ω −1 I = {K ∈K(2 ): ∃a ∈ I (K ⊆ cl2ω (f (La))} ω is an analytic ideal of compact subsets of 2 such that (I,≤) ≡T I. Then, with the notation of Definition 5.17, by Lemma 5.19 I =(Φ[I])↓ fulfills the requirements. It is obvious that I is closed downward. Since I is directed, I is closed under taking finite unions. Next we show that I is analytic. Let F(G)denotetheset of closed subsets of G endowed with the Effros Borel structure (see e.g. [11, Sec- tion 12.C, p. 75]). Recall that a sub-basis of the corresponding topology consists of a set of the form {F ∈F(G): F ∩ U = ∅},whereU varies over the open subsets of G. First we show that −1 R = {(F, a) ∈F(G) × I : F ⊆ f (La)}

is a closed set. Let (F, a) ∈F(G) × I,(Fn,an) ∈ R (n<ω) be such that ∈ ⊆ −1 limn<ω(Fn,an)=(F, a). Let x F be arbitrary; then by Fn f (Lan )(n<ω) ∈ −1 there are xn f (Lan )(n<ω) such that limn<ω xn = x.Sincef is continuous, f(x) = limn<ω f(xn). By ≤⊆ I × I being closed, f(xn) ≤ an (n<ω) implies −1 −1 f(x) ≤ a.Thusx ∈ f (La). Since x ∈ F was arbitrary, F ⊆ f (La); thus (F, a) ∈ R follows. ω Observe that {(K, F) ∈K(2 ) ×F(G): K ⊆ cl2ω (F )} is a Borel set (see e.g. the proof of [11, Theorem 12.6, p. 75]). Hence ω −1 S = {(K, F, a) ∈K(2 ) ×F(G) × I : K ⊆ cl2ω (F ),F⊆ f (La)} is also Borel. It is obvious that ω −1 T = {(K, a) ∈K(2 ) × I : K ⊆ cl2ω (f (La))} =PrK(2ω )×I (S),

so T is an analytic set. Since I is analytic, I =PrK(2ω )(T ) is analytic, as well.

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−1 We show that g : I →I, g(a)=cl2ω (f (La)) (a ∈ I)isaTukeymap.Let A ⊆ I be arbitrary. If g[A] ⊆Iis bounded, then there is a b ∈ I satisfying −1 −1 −1 −1 cl2ω (f (La)) ⊆ cl2ω (f (Lb)) (a ∈ A). Then f (a) ⊆ cl2ω (f (Lb)) ∩ G = −1 f (Lb)(a ∈ A), so a ≤ b (a ∈ A). Thus g maps unbounded sets into unbounded sets, as stated. −1 Finally, let h: I→I be any function satisfying K ⊆ cl2ω (f (Lh(K))) (K ∈I); we show that h is a Tukey map. Let A⊆Ibe arbitrary. If h[A] ⊆ I is bounded, then there is a b ∈ I satisfying h(K) ≤ b; hence Lh(K) ⊆ Lb (K ∈A). Then −1 K ⊆ cl2ω (f (Lb)) (K ∈A); i.e. h maps unbounded sets into unbounded sets, as required. Next we present a simple proof of Proposition 5.5. Proof of Proposition 5.5. Observe that for every n<ω,[2ω]≤n ⊆K(2ω)isclosed. ω <ω ω Hence [2 ] ⊆K(2 )isanFσ ideal of compact sets. Then, with the notation of ω <ω ↓ Deﬁnition 5.17, by Lemma 5.18.5 and Lemma 5.19, Imax = (Φ[[2 ] ]) fulﬁlls the requirements.

As we mentioned above, Imax is not basic. To see this, let us recall the definition of basic directed partial orders. Recall that a set is called bounded if it is bounded from above. Definition 5.27 ([25, Definition, p. 1881]). Let (D, ≤) be a separable metric space endowed with a partial order. Then (D, ≤)isbasic if (1) each pair of elements of D has the least upper bound with respect to ≤,and the binary operation of least upper bound from D × D to D is continuous; (2) each bounded sequence has a converging subsequence; (3) each converging sequence has a bounded subsequence.

Since every infinite subset of Imax is unbounded, in a topology making Imax satisfy Definition 5.27.3 there are no injective convergent sequences. The only such metric topology is the discrete topology, which is not separable in this case. We close this section with the proof of Proposition 5.4. The construction originates from [16, Theorem 6, p. 183], and an analogous construction was used in [17] to show that the structure (P(ω), ⊆ ) embeds into the family of Fσ ideals on ω partially ordered by Borel reduction. Our main improvement compared to [16] and [17], which makes it possible to omit definability assumptions, is that our proof for non-reducibility is purely combinatorial. We define sequences (bj )j<ω and (mj)j<ω by induction on j, as follows. Set j·mj m0 =0;ifj<ωand mj is already defined, set bj =2 and mj+1 = mj + bj . Let Ij =[mj ,mj+1)(j<ω). Let log stand for a logarithm of base 2. For every ∈ ω S [ω] and N<ω,letIS(N)= j∈S∩N Ij and IS = j∈S Ij . For every j<ω and x ⊆ ω, log(|x ∩ Ij | +1) xj = , x =supxj . mj +1 j<ω F { ⊆ ∞} F We define S = x IS :supj<ω x j < . For every N<ω,let S (N)= {x ∈FS : x≤N}. We will use the property that for arbitrary n, j < ω and (x ) ⊆P(ω), i i

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Proposition 5.4 is an immediate corollary of the following two statements.

ω Proposition 5.28. For every S ∈ [ω] , FS ⊆P(ω) is an Fσ ideal which is not basic in any topology.

ω Proposition 5.29. For every S, T ∈ [ω] , FS ≤T FT if and only if S ⊆ T .

Proof of Proposition 5.28. It is obvious that FS (N) ⊆P(ω)(N<ω) are closed sets, so FS is an Fσ ideal. Suppose there is a topology on FS which makes it basic. <ω For every N<ω,let(xi(N))i<ω ⊆ [ω] be a sequence such that

(1) for every i<ωwe have xi(N) ⊆ IS, N − 1 ≤xi(N)≤N; (2) for every j<ωwe have |{i<ω: x (N) ∩ I = ∅}| ≤ 1. i j ⊆ Set Xi(N)= i≤k<ω xk(N)(i, N < ω).Thenby1and2wehaveXi(N) IS and Xi(N)≤N, i.e. Xi(N) ∈FS (i, N < ω). Fix N<ω. The sequence (xk(N))k<ω is bounded by X0(N), so by Definition ∈ ω 5.27.2, it has a convergent subsequence (xk(N))k∈IN for some IN [ω] . + { ∈F ⊆ } By [25, Lemma 3.1, p. 1882], for every i<ω, Xi (N)= y S : y Xi(N) is ∈ compact. Since the convergent sequence (xk(N))k IN is eventually a subsequence + ∈ + {∅} ∅ of Xi (N)(i<ω), limk∈IN xk(N) i<ω Xi (N)= . Thus limi∈IN xi(N)= (N<ω). Since a basic topology is metric, by an easy diagonalization argument we can find a function ϕ: ω → ω such that ϕ(N) ∈ IN (N<ω) and limN<ω xϕ(N)(N)= ∅.Thus(xϕ(N)(N))N<ω is a convergent sequence, which by (1) has no bounded subsequence. This contradicts Definition 5.27.3.

The non-reduction part of Proposition 5.29 is based on the following. For every ≤ω t ∈ 2 we set [t =1]={n<ω: t(n)=1}. Recall Ns = {x ∈P(ω): x ∩|s| =[s = 1]} (s ∈ 2<ω).

ω Lemma 5.30. Let S, T ∈ [ω] satisfy S ∩ T = ∅.LetA ⊆FS (1), n ∈ S and mn s ∈ 2 be such that Ns ∩FS (1) = ∅ and let A be of second category everywhere in Ns ∩FS (1) in the relative topology of FS(1).LetN<ω,andletf : A →FT (N) m be an arbitrary function. Then there is a t ∈ 2 n such that [t =1] ⊆ T , t≤N, { ∈ | }⊆F ≥ − B = x A: f(x) mn = t S(1) is of second category and B n n 2.

mn Proof. Let T = {t ∈ 2 :[t =1]⊆ T,t≤N}. For every t ∈T,setBt = {x ∈ | } ∈T F A: f(x) mn = t . If there is a t such that Bt is of second category in S (1) and Btn ≥ n − 2, then we are done. So suppose no such t ∈T exists. Set C = {Bt : t ∈T,Bt is of second category in FS (1)}.

Then by (5.4), using |T | ≤ 2mn ,

mn mn ≤ log(2 ) − log(2 ) − C n supt∈T Bt +

ω Proof of Proposition 5.29. First let S, T ∈ [ω] satisfy S ⊆ T .Setf : FS →FT , f(x)=x ∩ IT (x ∈FS). It is easy to verify that f is a Tukey map.

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ω Now let S, T ∈ [ω] satisfy S ⊆ T . As we have seen, FS\T ≤T FS.Soitis enough to prove FS\T ≤T FT ; that is, we can assume S ∩ T = ∅. Let f : FS →FT be an arbitrary function. We ﬁnd N<ωand ni <ω, <ω si,ti ∈ 2 , Ai ⊆FS(1) (i<ω) such that

mn (1) ni ∈ S, ni ni, ∈ ni+1 ∩F ∅ si+1 2 such that Nsi+1 S (1) = and Ai+1 is of second category everywhere ∩F in Nsi+1 S (1). Then 1, 3 and 4 hold, and for 2 it remains to show ti−1 ti if i>0. However, this follows from Ai+1 ⊆ Ai using the inductive hypothesis f(x)| = t − (x ∈ A ). So the proof is complete. mni−1 i 1 i

6. Problems In our present work we did not apply some well-understood methods for studying inﬁnite dimensional perfect set theorems. We close this paper with a survey of possible further research directions and state some related open problems.

6.1. General symmetric topologies, analytic relations. Apart from the usual product topology, there are several topologies on Rω which are symmetric (i.e. open sets remain open under arbitrary permutation of coordinates) and which are important in applications, e.g. the box topology or the topologies induced by the p (1 ≤ p ≤∞) norms. There is no reason to believe that the product topology is the most appropriate for the formulation of optimal inﬁnite dimensional perfect

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set theorems. It would be interesting to find weaker largeness assumptions in these finer topologies than our Xω \A ∈ M in Theorem 4.1, which still imply the existence of a non-empty perfect A-homogeneous set. E.g. it is easy to construct a symmetric dense open set A ⊆ Rω such that there is no infinite A-homogeneous set. But in Rω endowed with the box topology, for every symmetric dense open set there is an infinite homogeneous set. Compare this with the remarks following Corollary 2.3. Note also that ironically, all of our perfect set theorems hold for co-analytic relations, while the rank approach of [13] and [23] is able to handle analytic relations. Observe that to every construction using finite approximations, one can associate the tree of finite approximations ordered by end-extension such that the ill-foundedness of this tree is equivalent with the existence of the limit object of the construction. Therefore the infinite dimensional counterpart of [13, Proposition 4.1, p. 151] is natural to formulate and easy to prove. The more involved task would be to study the existence of universal relations as in [13, Section 5] and to characterize the resulting rank as in [23]. It seems that these investigations can be carried out for any of the above mentioned refinements of the product topology, as well.

6.2. More fusion games. It seems informative to study fusions of other games of descriptive set theory, especially those of Separation games and Wadge games. We remark that the way we increased the dimension in our fusion scheme was completely arbitrary; different schemes characterize other notions of smallness. We propose an explicit modification of Gω of Definition 3.2 which seems partic- Gm ⊆G ularly interesting to study. Consider the game ω ω, where, in addition, in th − the (n +1) move player I is required to play (Ui(2(n +1 i)))i≤n+1 such that ⊆ − Un+1(0) i≤n Ui(2(n i) + 1); otherwise the game is unchanged. For player II this game is easier to win, still one can show that the existence of a winning strategy Gm for player II in ω (A) implies the existence of a non-empty perfect A-homogeneous set. Unfortunately we could not characterize the existence of a winning strategy for Gm player I in ω (A). Nevertheless, we expect that fusion games modified in such ways can provide sharper results.

6.3. Other Choquet topologies and forcings. The Ellentuck topology and the density topologies are Choquet, in particular Theorem 4.1 can be applied to them. However, they fail the Kuratowski-Ulam property, which is crucial for Lemma 4.4 and so for all the results of Section 4.1 and Section 4.2. E.g. our methods do not allow us to prove the counterparts of Theorem 1.2 and Theorem 4.10, in which “meager” is replaced by “Lebesgue null” and “Cohen” is replaced by “random”.1 Problem 6.1.1 Let A ⊆ Rω be a co-analytic set. Does the existence of an A- homogeneous set of positive outer Lebesgue measure imply the existence of a non- empty perfect A-homogeneous set? Problem 6.2.1 Let V be a model obtained from a model of the Continuum Hy- ω pothesis by adding ω2 random reals. In V ,letA ⊆ R be a co-analytic set. Does the existence of an A-homogeneous set of cardinality ω2 imply the existence of a non-empty perfect A-homogeneous set?

1Recently J. Zapletal [31] showed that these analogous results hold, and he gave aﬃrmative answers to Problem 6.1 and Problem 6.2.

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Note that the finite dimensional analogue of Problem 6.1 holds for every Lebesgue measurable set A by the measure version of Mycielski’s Theorem (see e.g. [28, Theorem 6.40]), while the finite dimensional analogue of Problem 6.2 holds for every analytic set A by [23, Fact 1.16, p. 23]. These problems may be related to the following. Problem 6.3. Let A ⊆ 2ω × 2ω be a co-meager set. Is it true in ZFC that there is an A-homogeneous set which is of second category everywhere? If e.g. cof(M)=cov(M), then by an easy transfinite argument, for every co- meager set A ⊆ 2ω ×2ω one can construct an A-homogeneous set which is of second category everywhere. Also note the counterpart of Problem 6.3 involving Lebesgue measure fails. Theorem 6.4 ([2], [7]). Let V be a model obtained from a model of the Continuum ω ω Hypothesis by adding ω2 Cohen reals. Then in V , there is an Fσ set A ⊆ 2 ×2 of Lebesgue measure 1 such that there exists no A-homogeneous set of positive outer Lebesgue measure. The proof of Theorem 6.4 is based on the observation that in Cohen extensions, if ω ω for an Fσ set A ⊆ 2 ×2 there is an A-homogeneous set of positive outer Lebesgue measure, then there is an A-homogeneous set of positive Lebesgue measure. Thus it is likely that Problem 6.1 has an affirmative answer in Cohen extensions. However, the proof of Theorem 6.4 does not have a straightforward modification valid for random extensions, so we could not obtain a negative answer to Problem 6.3. Problem 6.3 was motivated by the question of whether the largeness assumption in Theorem 1.2 is the natural analogue of the largeness assumption of Mycielski’s Theorem. As we pointed out above, this is consistently true, since for sets A ⊆ 2ω × 2ω having the Baire property, being co-meager and having an A-homogeneous set which is of second category everywhere are consistently equivalent. ℵ In Cohen extensions there are perfectly meager sets of cardinality 2 0 ,sothe largeness assumption in Theorem 1.2 is consistently not optimal. It would be interesting to know whether the converse is also consistent; the following is also related to Problem 6.3. Problem 6.5. Is it consistent with ZFC that H ⊆ R is perfectly meager if and only ω if there is an open set A ⊆ R such that ISω(H) ⊆ A but there is no non-empty perfect A-homogeneous set? 6.4. Local results. In [15] the following “local” infinite dimensional perfect set theorem was obtained (see also [28, Corollary 6.48]). We call a sequence (xn)n<ω ∈ ω R rapidly increasing if 0

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6.5. Coﬁnal types of analytic ideals. A deeper analysis of the construction in the proof of Theorem 5.21 reveals it uses that E of Deﬁnition 5.23 is an equivalence relation in an essential way. Thus this method cannot yield a Borel ideal satisfying the conditions of Theorem 5.21. On the other hand, the construction in the proof of Theorem 2.1 is the base step of the construction of an Fσ ideal satisfying the conditions of Theorem 5.21. Therefore we expect a positive answer to the following problems. <ω Problem 6.7. Is there an Fσ ideal I⊆P(ω) such that [ω1] ≤T I but I has no non-empty perfect strongly unbounded subsets?

Problem 6.8 ([16, Conjecture 1, p. 194]). Is there an Fσ ideal I⊆P(ω) such that <ω ω [ω1] ≤T I but ω ≤T I?

We also expect the consistency of the failure of the primality property for Imax.

Problem 6.9. Is it consistent with ZFC that Imax fails the primality property, i.e. there exist analytic ideals I, J⊆P(ω) such that Imax ≤T I⊕J but Imax ≤T I and Imax ≤T J ? At the present stage of research, one could wonder whether for every analytic <ω ideal I⊆P(ω), [ω2] ≤T I implies that I has a non-empty perfect strongly unbounded subset. However, we expect that the aﬃrmative answer to Problem 6.7 will be based on a construction which is ﬂexible enough to rule out such speculations. References

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Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada E-mail address: [email protected] Current address:Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 13-15 Reáltanoda Street, Budapest H-1053 Hungary

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