Perfect Set Games and Colorings on Generalized Baire Spaces - - PowerPoint PPT Presentation

Perfect Set Games and Colorings on Generalized Baire Spaces

Dorottya Sziráki 5th Workshop on Generalized Baire Spaces Bristol, 4 February 2020

Dorottya Sziráki Perfect set games and colorings on κκ

Perfectness for the κ-Baire space

Assume κ<κ = κ.

Dorottya Sziráki Perfect set games and colorings on κκ

Perfectness for the κ-Baire space

Assume κ<κ = κ. A subset of κκ is closed iff it is the set of branches [T] = {x ∈ κκ : x↾α ∈ T for all α < κ}

• f a subtree T of <κκ.

Dorottya Sziráki Perfect set games and colorings on κκ

Perfectness for the κ-Baire space

Assume κ<κ = κ. A subset of κκ is closed iff it is the set of branches [T] = {x ∈ κκ : x↾α ∈ T for all α < κ}

• f a subtree T of <κκ.

Example A subset of a topological space is perfect in the usual sense iff it is closed and contains no isolated points. Xω = {x ∈ κ2 : |{α < κ : x(α) = 0}| < ω} is perfect in this usual sense, but |Xω| = κ.

Dorottya Sziráki Perfect set games and colorings on κκ

Perfectness for the κ-Baire space

Assume κ<κ = κ. A subset of κκ is closed iff it is the set of branches [T] = {x ∈ κκ : x↾α ∈ T for all α < κ}

• f a subtree T of <κκ.

Example A subset of a topological space is perfect in the usual sense iff it is closed and contains no isolated points. Xω = {x ∈ κ2 : |{α < κ : x(α) = 0}| < ω} is perfect in this usual sense, but |Xω| = κ. Definition A subtree T of <κκ is a strongly κ-perfect tree if T is <κ-closed and every node of T extends to a splitting node. A set X ⊆ κκ is a strongly κ-perfect set if X = [T] for a strongly κ-perfect tree T.

Dorottya Sziráki Perfect set games and colorings on κκ

Väänänen’s perfect set game

Let X ⊆ κκ, let x0 ∈ κκ and let ω ≤ γ ≤ κ. Definition (Väänänen, 1991) The game Vγ(X, x0) has length γ and is played as follows: I U1 . . . Uα . . . II x0 x1 . . . xα . . . II first plays x0. In each round 0 < α < γ, I plays a basic open subset Uα of X, and then II chooses xα ∈ Uα with xα = xβ for all β < α. I has to play so that Uβ+1 ∋ xβ in each successor round β + 1 < γ and Uα =

β<α Uβ in each limit round α < γ.

II wins a given run of the game if II can play legally in all rounds α < γ.

Dorottya Sziráki Perfect set games and colorings on κκ

Väänänen’s perfect set game

Let X ⊆ κκ, let x0 ∈ κκ and let ω ≤ γ ≤ κ. Definition (Väänänen, 1991) The game Vγ(X, x0) has length γ and is played as follows: I U1 . . . Uα . . . II x0 x1 . . . xα . . . II first plays x0. In each round 0 < α < γ, I plays a basic open subset Uα of X, and then II chooses xα ∈ Uα with xα = xβ for all β < α. I has to play so that Uβ+1 ∋ xβ in each successor round β + 1 < γ and Uα =

β<α Uβ in each limit round α < γ.

II wins a given run of the game if II can play legally in all rounds α < γ.

Dorottya Sziráki Perfect set games and colorings on κκ

Perfect and scattered subsets of the κ-Baire space

Let X ⊆ κκ, and suppose ω ≤ γ ≤ κ. Definition (Väänänen, 1991) X is a γ-scattered set if I wins Vγ(X, x0) for all x0 ∈ X. X is a γ-perfect set if X is closed and II wins Vγ(X, x0) for all x0 ∈ X.

Dorottya Sziráki Perfect set games and colorings on κκ

Perfect and scattered subsets of the κ-Baire space

Let X ⊆ κκ, and suppose ω ≤ γ ≤ κ. Definition (Väänänen, 1991) X is a γ-scattered set if I wins Vγ(X, x0) for all x0 ∈ X. X is a γ-perfect set if X is closed and II wins Vγ(X, x0) for all x0 ∈ X. X is ω-perfect iff X is perfect in the usual sense (i.e., iff X closed and has no isolated points).

Dorottya Sziráki Perfect set games and colorings on κκ

Perfect and scattered subsets of the κ-Baire space

Let X ⊆ κκ, and suppose ω ≤ γ ≤ κ. Definition (Väänänen, 1991) X is a γ-scattered set if I wins Vγ(X, x0) for all x0 ∈ X. X is a γ-perfect set if X is closed and II wins Vγ(X, x0) for all x0 ∈ X. X is ω-perfect iff X is perfect in the usual sense (i.e., iff X closed and has no isolated points). X is ω-scattered iff X is scattered in the usual sense (i.e., each nonempty subspace contains an isolated point).

Dorottya Sziráki Perfect set games and colorings on κκ

Perfect and scattered subsets of the κ-Baire space

Let X ⊆ κκ, and suppose ω ≤ γ ≤ κ. Definition (Väänänen, 1991) X is a γ-scattered set if I wins Vγ(X, x0) for all x0 ∈ X. X is a γ-perfect set if X is closed and II wins Vγ(X, x0) for all x0 ∈ X. X is ω-perfect iff X is perfect in the usual sense (i.e., iff X closed and has no isolated points). X is ω-scattered iff X is scattered in the usual sense (i.e., each nonempty subspace contains an isolated point). Vγ(X, x0) may not be determined when γ > ω.

Dorottya Sziráki Perfect set games and colorings on κκ

κ-perfect sets vs. strongly κ-perfect sets

Dorottya Sziráki Perfect set games and colorings on κκ

κ-perfect sets vs. strongly κ-perfect sets

Example (Huuskonen) The following set is κ-perfect but is not strongly κ-perfect: Yω = {x ∈ κ3 : |{α < κ : x(α) = 2}| < ω}.

Dorottya Sziráki Perfect set games and colorings on κκ

κ-perfect sets vs. strongly κ-perfect sets

Example (Huuskonen) The following set is κ-perfect but is not strongly κ-perfect: Yω = {x ∈ κ3 : |{α < κ : x(α) = 2}| < ω}. Proposition Let X be a closed subset of κκ. X is κ-perfect ⇐ ⇒ X =

• i∈I

Xi for strongly κ-perfect sets Xi.

Dorottya Sziráki Perfect set games and colorings on κκ

Väänänen’s generalized Cantor-Bendixson theorem

Theorem (Väänänen, 1991) The following Cantor-Bendixson theorem for κκ is consistent relative to the existence of a measurable cardinal λ > κ: Every closed subset of κκ is the (disjoint) union of a κ-perfect set and a κ-scattered set, which is of size ≤ κ.

Dorottya Sziráki Perfect set games and colorings on κκ

Väänänen’s generalized Cantor-Bendixson theorem

Theorem (Väänänen, 1991) The following Cantor-Bendixson theorem for κκ is consistent relative to the existence of a measurable cardinal λ > κ: Every closed subset of κκ is the (disjoint) union of a κ-perfect set and a κ-scattered set, which is of size ≤ κ. Theorem (Galgon, 2016) Väänänen’s generalized Cantor-Bendixson theorem is consistent relative to the existence of an inaccessible cardinal λ > κ.

Dorottya Sziráki Perfect set games and colorings on κκ

Väänänen’s generalized Cantor-Bendixson theorem

Proposition (Sz) Väänänen’s generalized Cantor-Bendixson theorem is equivalent to the κ-perfect set property for closed subsets of κκ (i.e, the statement that every closed subset of κκ of size > κ has a κ-perfect subset). Remark: The κ-PSP for closed subsets of κκ is equiconsistent with the existence of an inaccessible cardinal λ > κ.

Dorottya Sziráki Perfect set games and colorings on κκ

Väänänen’s generalized Cantor-Bendixson theorem

Proposition (Sz) Väänänen’s generalized Cantor-Bendixson theorem is equivalent to the κ-perfect set property for closed subsets of κκ (i.e, the statement that every closed subset of κκ of size > κ has a κ-perfect subset). Remark: The κ-PSP for closed subsets of κκ is equiconsistent with the existence of an inaccessible cardinal λ > κ.

Sketch of the proof. Let X be a closed subset of κκ. Its set of κ-condensation points is defined to be CPκ(X) = {x ∈ X : |X ∩ Nx↾α| > κ for all α < κ}. If the κ-PSP holds for closed subsets of κκ, then CPκ(X) is a κ-perfect set and X − CPκ(X) is a κ-scattered set of size ≤ κ.

Dorottya Sziráki Perfect set games and colorings on κκ

Perfect and scattered trees

Let T be a subtree of <κ2, let t ∈ T, and let ω ≤ γ ≤ κ. Definition (Galgon, 2016) The game Gγ(T, t) has length γ and is played as follows: I δ0 i0 . . . δα iα . . . II t0 . . . tα . . . In each round α < γ, player I first plays δα < κ. Then II plays a node tα ∈ T of height ≥ δα, and I chooses iα < 2. II has to play so that t ⊆ t0, and tβ⌢iβ ⊆ tα for all β < α < γ. II wins a given run of the game if II can play legally in all rounds α < γ.

Dorottya Sziráki Perfect set games and colorings on κκ

Perfect and scattered trees

Let T be a subtree of <κ2, let t ∈ T, and let ω ≤ γ ≤ κ. Definition (Galgon, 2016) The game Gγ(T, t) has length γ and is played as follows: I δ0 i0 . . . δα iα . . . II t0 . . . tα . . . In each round α < γ, player I first plays δα < κ. Then II plays a node tα ∈ T of height ≥ δα, and I chooses iα < 2. II has to play so that t ⊆ t0, and tβ⌢iβ ⊆ tα for all β < α < γ. II wins a given run of the game if II can play legally in all rounds α < γ.

Dorottya Sziráki Perfect set games and colorings on κκ

Perfect and scattered trees

Definition (Galgon, 2016) T is a γ-scattered tree if player I wins Gγ(T, t) for all t ∈ T. T is a γ-perfect tree if player II wins Gγ(T, t) for all t ∈ T. Proposition Let T be a subtree of <κκ.

1

T is a κ-perfect tree ⇐ ⇒ [T] is a κ-perfect set.

Dorottya Sziráki Perfect set games and colorings on κκ

Perfect and scattered trees

Definition (Galgon, 2016) T is a γ-scattered tree if player I wins Gγ(T, t) for all t ∈ T. T is a γ-perfect tree if player II wins Gγ(T, t) for all t ∈ T. Proposition Let T be a subtree of <κκ.

1

T is a κ-perfect tree ⇐ ⇒ [T] is a κ-perfect set.

2 If the κ-PSP holds for closed subsets of κκ, then

T is a κ-scattered tree ⇐ ⇒ [T] is a κ-scattered set.

Dorottya Sziráki Perfect set games and colorings on κκ

γ-perfect sets and trees when γ ≤ κ

Theorem (Sz) Let T be a subtree of <κκ and let ω ≤ γ ≤ κ.

1 If [T] is a γ-perfect set, then T is a γ-perfect tree. Dorottya Sziráki Perfect set games and colorings on κκ

γ-perfect sets and trees when γ ≤ κ

Theorem (Sz) Let T be a subtree of <κκ and let ω ≤ γ ≤ κ.

1 If [T] is a γ-perfect set, then T is a γ-perfect tree. 2 If T is a γ-scattered tree, then [T] is a γ-scattered set. Dorottya Sziráki Perfect set games and colorings on κκ

γ-perfect sets and trees when γ ≤ κ

Theorem (Sz) Let T be a subtree of <κκ and let ω ≤ γ ≤ κ.

1 If [T] is a γ-perfect set, then T is a γ-perfect tree. 2 If T is a γ-scattered tree, then [T] is a γ-scattered set. 3 If κ is weakly compact and T ⊆ <κ2, then

T is a γ-perfect tree ⇐ ⇒ [T] is a γ-perfect set.

Dorottya Sziráki Perfect set games and colorings on κκ

γ-perfect sets and trees when γ ≤ κ

Theorem (Sz) Let T be a subtree of <κκ and let ω ≤ γ ≤ κ.

1 If [T] is a γ-perfect set, then T is a γ-perfect tree. 2 If T is a γ-scattered tree, then [T] is a γ-scattered set. 3 If κ is weakly compact and T ⊆ <κ2, then

T is a γ-perfect tree ⇐ ⇒ [T] is a γ-perfect set.

More generally: this holds if κ has the tree property and T is a κ-tree.

Dorottya Sziráki Perfect set games and colorings on κκ

γ-perfect sets and trees when γ ≤ κ

Theorem (Sz) Let T be a subtree of <κκ and let ω ≤ γ ≤ κ.

1 If [T] is a γ-perfect set, then T is a γ-perfect tree. 2 If T is a γ-scattered tree, then [T] is a γ-scattered set. 3 If κ is weakly compact and T ⊆ <κ2, then

T is a γ-perfect tree ⇐ ⇒ [T] is a γ-perfect set.

More generally: this holds if κ has the tree property and T is a κ-tree.

Question Is it consistent that 3 holds for “scattered” instead of “perfect”?

Dorottya Sziráki Perfect set games and colorings on κκ

γ-perfect sets and trees when γ ≤ κ

Theorem (Sz) Let T be a subtree of <κκ and let ω ≤ γ ≤ κ.

1 If [T] is a γ-perfect set, then T is a γ-perfect tree. 2 If T is a γ-scattered tree, then [T] is a γ-scattered set. 3 If κ is weakly compact and T ⊆ <κ2, then

T is a γ-perfect tree ⇐ ⇒ [T] is a γ-perfect set.

More generally: this holds if κ has the tree property and T is a κ-tree.

Question Is it consistent that 3 holds for “scattered” instead of “perfect”? Analogues of 1–3 hold for levels of “generalized Cantor-Bendixson hierar- chies” associated to subsets of κκ and to subtrees of <κκ (see next 4 slides).

Dorottya Sziráki Perfect set games and colorings on κκ

Generalizing the Cantor-Bendixson hierarchy

Let X ⊆ κκ, let x0 ∈ κκ, and let S be a tree without branches of length ≥ κ. Definition (Hyttinen; Väänänen) The S-approximation VS(X, x0) of Vκ(X, x0) is the following game. I s1, U1 . . . sα, Uα . . . II x0 x1 . . . xα . . . In each round α > 0, I first plays sα ∈ S such that sα >S sβ for all 0 < β < α. Then I plays Uα and II plays xα according to the same rules as in Vκ(X, x0). The first player who can not move loses, and the other player wins.

Dorottya Sziráki Perfect set games and colorings on κκ

Generalizing the Cantor-Bendixson hierarchy

Let X ⊆ κκ, let x0 ∈ κκ, and let S be a tree without branches of length ≥ κ. Definition (Hyttinen; Väänänen) The S-approximation VS(X, x0) of Vκ(X, x0) is the following game. I s1, U1 . . . sα, Uα . . . II x0 x1 . . . xα . . . In each round α > 0, I first plays sα ∈ S such that sα >S sβ for all 0 < β < α. Then I plays Uα and II plays xα according to the same rules as in Vκ(X, x0). The first player who can not move loses, and the other player wins. Let ScS(X) = {x ∈ X : I wins VS(X, x)}; KerS(X) = {x ∈ κκ : II wins VS(X, x)}.

Dorottya Sziráki Perfect set games and colorings on κκ

Generalizing the Cantor-Bendixson hierarchy

ScS(X) = {x ∈ X : I wins VS(X, x)}; KerS(X) = {x ∈ κκ : II wins VS(X, x)}. Given an ordinal α, let Bα = the tree of descending sequences of elements of α. X(α) denotes the αth Cantor-Bendixson derivative of X. Observation 1 (Väänänen) X(α) = X ∩ KerBα(X) = X − ScBα(X).

Dorottya Sziráki Perfect set games and colorings on κκ

Generalizing the Cantor-Bendixson hierarchy

ScS(X) = {x ∈ X : I wins VS(X, x)}; KerS(X) = {x ∈ κκ : II wins VS(X, x)}. Given an ordinal α, let Bα = the tree of descending sequences of elements of α. X(α) denotes the αth Cantor-Bendixson derivative of X. Observation 1 (Väänänen) X(α) = X ∩ KerBα(X) = X − ScBα(X). Corollary Kerω(X) =

• {KerS(X) : S is a tree without infinite branches};

Scω(X) =

• {ScS(X) : S is a tree without infinite branches}.

Dorottya Sziráki Perfect set games and colorings on κκ

Generalizing the Cantor-Bendixson hierarchy

ScS(X) = {x ∈ X : I wins VS(X, x)}; KerS(X) = {x ∈ κκ : II wins VS(X, x)}. Given an ordinal α, let Bα = the tree of descending sequences of elements of α. X(α) denotes the αth Cantor-Bendixson derivative of X. Observation 1 (Väänänen) X(α) = X ∩ KerBα(X) = X − ScBα(X). Theorem 2 (Hyttinen (1990); Väänänen (1991)) Kerκ(X) =

• {KerS(X) : S is a tree without branches of length ≥ κ};

Scκ(X) =

• {ScS(X) : S is a tree without branches of length ≥ κ}.

Dorottya Sziráki Perfect set games and colorings on κκ

Generalizing the Cantor-Bendixson hierarchy

ScS(X) = {x ∈ X : I wins VS(X, x)}; KerS(X) = {x ∈ κκ : II wins VS(X, x)}. Given an ordinal α, let Bα = the tree of descending sequences of elements of α. X(α) denotes the αth Cantor-Bendixson derivative of X. Observation 1 (Väänänen) X(α) = X ∩ KerBα(X) = X − ScBα(X). Theorem 2 (Hyttinen (1990); Väänänen (1991)) Kerκ(X) =

• {KerS(X) : S is a tree without branches of length ≥ κ};

Scκ(X) =

• {ScS(X) : S is a tree without branches of length ≥ κ}.

The sets X ∩ KerS(X) (resp. X − ScS(X)) can be seen as the “levels of a generalized Cantor-Bendixson hierarchy” for the set X associated to II (resp. I).

Dorottya Sziráki Perfect set games and colorings on κκ

Generalizing the Cantor-Bendixson hierarchy for trees

Theorem (Sz, part 1) There exists a family {G′

γ(T, t) : T is a subtree of <κκ, t ∈ T and ω ≤ γ ≤ κ}

• f games such that the following hold for all such T, t and γ.

The games G′

γ(T, t) and Gγ(T, t) are equivalent whenever T ⊆ <κ2.

Given a tree S without branches of length ≥ κ, let G′

S(T, t) denote the

S-approximation of G′

κ(T, t),1 and let

ScS(T) = {t ∈ T : I wins G′

S(T, t)};

KerS(T) = {t ∈ T : II wins G′

S(T, t)}.

Then the analogues of Observation 1 and Theorem 2 hold.2 The analogue of Theorem 2 is a special case of a general theorem due to Hyttinen (1990).

1This is defined analogously to the S-approximation VS(T, x). 2We consider the Cantor-Bendixson derivative of subtrees T of <κκ which was defined in:

• G. Galgon. Trees, refining, and combinatorial characteristics. PhD thesis, University of

California, Irvine, 2016.

Dorottya Sziráki Perfect set games and colorings on κκ

Comparing the Cantor-Bendixson hierarchies

ScS(T) = {t ∈ T : I wins G′

S(T, t)};

KerS(T) = {t ∈ T : II wins G′

S(T, t)}.

ScS(X) = {x ∈ X : I wins VS(X, x)}; KerS(X) = {x ∈ κκ : II wins VS(X, x)}.

Theorem (Sz, part 2) Let S be a tree without branches of length ≥ κ. Then

1

KerS([T]) ⊆ [KerS(T)] (i.e., if II wins VS([T], x) then II wins G′

S(T, t) when t x ∈ κκ.).

Dorottya Sziráki Perfect set games and colorings on κκ

Comparing the Cantor-Bendixson hierarchies

ScS(T) = {t ∈ T : I wins G′

S(T, t)};

KerS(T) = {t ∈ T : II wins G′

S(T, t)}.

ScS(X) = {x ∈ X : I wins VS(X, x)}; KerS(X) = {x ∈ κκ : II wins VS(X, x)}.

Theorem (Sz, part 2) Let S be a tree without branches of length ≥ κ. Then

1

KerS([T]) ⊆ [KerS(T)] (i.e., if II wins VS([T], x) then II wins G′

S(T, t) when t x ∈ κκ.).

2

[T] − ScS([T]) ⊆ [T − ScS(T)] (i.e., if I wins G′

S(T, t) then I wins VS([T], x) when t x ∈ κκ).

Dorottya Sziráki Perfect set games and colorings on κκ

Comparing the Cantor-Bendixson hierarchies

ScS(T) = {t ∈ T : I wins G′

S(T, t)};

KerS(T) = {t ∈ T : II wins G′

S(T, t)}.

ScS(X) = {x ∈ X : I wins VS(X, x)}; KerS(X) = {x ∈ κκ : II wins VS(X, x)}.

Theorem (Sz, part 2) Let S be a tree without branches of length ≥ κ. Then

1

KerS([T]) ⊆ [KerS(T)] (i.e., if II wins VS([T], x) then II wins G′

S(T, t) when t x ∈ κκ.).

2

[T] − ScS([T]) ⊆ [T − ScS(T)] (i.e., if I wins G′

S(T, t) then I wins VS([T], x) when t x ∈ κκ).

3

If κ has the tree property and T is a κ-tree, then KerS([T]) = [KerS(T)] (i.e., VS([T], x) and G′

S(T, t) are equivalent for II when t x ∈ κκ).

Dorottya Sziráki Perfect set games and colorings on κκ

Density in itself for the κ-Baire space

Definition A subset X ⊆ κκ is κ-dense in itself if X is a κ-perfect set. A subset X ⊆ κκ is strongly κ-dense in itself if X is a strongly κ-perfect set.

Dorottya Sziráki Perfect set games and colorings on κκ

Density in itself for the κ-Baire space

Definition A subset X ⊆ κκ is κ-dense in itself if X is a κ-perfect set. A subset X ⊆ κκ is strongly κ-dense in itself if X is a strongly κ-perfect set. Proposition (Sz) The following are equivalent for any X ⊆ κκ. X is κ-dense in itself. X =

i∈I Xi where each Xi is strongly κ-dense in itself.

X ⊆ Kerκ(X) (i.e., player II wins Vκ(X, x) for all x ∈ X.)

Dorottya Sziráki Perfect set games and colorings on κκ

Density in itself for the κ-Baire space

Theorem (Väänänen, 1991) If λ > κ is measurable and G is Col(κ, <λ)-generic, then in V [G], every subset of κκ of size κ+ contains a κ-dense in itself subset. (1)

Dorottya Sziráki Perfect set games and colorings on κκ

Density in itself for the κ-Baire space

Theorem (Väänänen, 1991) If λ > κ is measurable and G is Col(κ, <λ)-generic, then in V [G], every subset of κκ of size κ+ contains a κ-dense in itself subset. (1) Theorem (Schlicht, Sz) It is enough to assume that λ is weakly compact in the above theorem.

Dorottya Sziráki Perfect set games and colorings on κκ

Density in itself for the κ-Baire space

Theorem (Väänänen, 1991) If λ > κ is measurable and G is Col(κ, <λ)-generic, then in V [G], every subset of κκ of size κ+ contains a κ-dense in itself subset. (1) Theorem (Schlicht, Sz) It is enough to assume that λ is weakly compact in the above theorem. Question What is the consistency strength of (1)?

Dorottya Sziráki Perfect set games and colorings on κκ

A Cantor-Bendixson theorem for Gδ relations

R is a collection of finitary relations on a set X. Y ⊆ X is R-homogeneous if for all 1 ≤ k < ω and k-ary R ∈ R we have: (x1, . . . , xk) ∈ R for all pairwise distinct x1, . . . , xk ∈ Y .

Dorottya Sziráki Perfect set games and colorings on κκ

A Cantor-Bendixson theorem for Gδ relations

R is a collection of finitary relations on a set X. Y ⊆ X is R-homogeneous if for all 1 ≤ k < ω and k-ary R ∈ R we have: (x1, . . . , xk) ∈ R for all pairwise distinct x1, . . . , xk ∈ Y . Theorem (Kubiś, 2003; Doležal, Kubiś 2015) Let R be a countable set of Gδ relations on a Polish space X (i.e., every R ∈ R is an Gδ subset of kX for some 1 ≤ k < ω).

1

Either there exists a perfect R-homogeneous set, or there exists α < ω1 such that every R-homogeneous set Y has Cantor-Bendixson rank < α (i.e., Y (α) = ∅).

Dorottya Sziráki Perfect set games and colorings on κκ

A Cantor-Bendixson theorem for Gδ relations

R is a collection of finitary relations on a set X. Y ⊆ X is R-homogeneous if for all 1 ≤ k < ω and k-ary R ∈ R we have: (x1, . . . , xk) ∈ R for all pairwise distinct x1, . . . , xk ∈ Y . Theorem (Kubiś, 2003; Doležal, Kubiś 2015) Let R be a countable set of Gδ relations on a Polish space X (i.e., every R ∈ R is an Gδ subset of kX for some 1 ≤ k < ω).

1

Either there exists a perfect R-homogeneous set, or there exists α < ω1 such that every R-homogeneous set Y has Cantor-Bendixson rank < α (i.e., Y (α) = ∅).

2

If there exists an uncountable R-homogeneous set, then there exists a perfect R-homogeneous set.

Dorottya Sziráki Perfect set games and colorings on κκ

A Cantor-Bendixson theorem for Gδ relations

R is a collection of finitary relations on a set X. Y ⊆ X is R-homogeneous if for all 1 ≤ k < ω and k-ary R ∈ R we have: (x1, . . . , xk) ∈ R for all pairwise distinct x1, . . . , xk ∈ Y . Theorem (Kubiś, 2003; Doležal, Kubiś 2015) Let R be a countable set of Gδ relations on a Polish space X (i.e., every R ∈ R is an Gδ subset of kX for some 1 ≤ k < ω).

1

Either there exists a perfect R-homogeneous set, or there exists α < ω1 such that every R-homogeneous set Y has Cantor-Bendixson rank < α (i.e., Y (α) = ∅).

2

If there exists an uncountable R-homogeneous set, then there exists a perfect R-homogeneous set. This also holds for analytic spaces X.

Dorottya Sziráki Perfect set games and colorings on κκ

A Cantor-Bendixson theorem for Gδ relations

R is a collection of finitary relations on a set X. Y ⊆ X is R-homogeneous if for all 1 ≤ k < ω and k-ary R ∈ R we have: (x1, . . . , xk) ∈ R for all pairwise distinct x1, . . . , xk ∈ Y . Theorem (Kubiś, 2003; Doležal, Kubiś 2015) Let R be a countable set of Gδ relations on a Polish space X (i.e., every R ∈ R is an Gδ subset of kX for some 1 ≤ k < ω).

1

Either there exists a perfect R-homogeneous set, or there exists α < ω1 such that every R-homogeneous set Y has Cantor-Bendixson rank < α (i.e., Y (α) = ∅).

2

If there exists an uncountable R-homogeneous set, then there exists a perfect R-homogeneous set. This also holds for analytic spaces X. Recall that Y (α) = Y ∩ KerBα(Y ) = Y − ScBα(Y ).

Dorottya Sziráki Perfect set games and colorings on κκ

A dichotomy for infinitely many Π0

2(κ) relations R is a Π0

2(κ) relation on a topological space X iff

R is an intersection of ≤ κ many open subsets of kX for some 1 ≤ k < ω. Theorem (Sz) Assume ♦κ or κ is inaccessible. Let R be a collection of ≤ κ many Π0

2(κ) relations on a closed subset X of κκ.

Then either X has a κ-perfect R-homogeneous subset, or there exists a tree T without κ-branches, |T| ≤ 2κ, such that for all R-homogeneous Y ⊆ X, we have Y ∩ KerT (Y ) = ∅ (that is, player II does not win VT (Y, y) for any y ∈ Y ).

Dorottya Sziráki Perfect set games and colorings on κκ

A dichotomy for infinitely many Π0

2(κ) relations R is a Π0

2(κ) relation on a topological space X iff

R is an intersection of ≤ κ many open subsets of kX for some 1 ≤ k < ω. Theorem (Sz) Assume ♦κ or κ is inaccessible. Let R be a collection of ≤ κ many Π0

2(κ) relations on a closed subset X of κκ.

Then either X has a κ-perfect R-homogeneous subset, or there exists a tree T without κ-branches, |T| ≤ 2κ, such that for all R-homogeneous Y ⊆ X, we have Y ∩ KerT (Y ) = ∅ (that is, player II does not win VT (Y, y) for any y ∈ Y ). If κ is inaccessible, then there exists a tree T of size κ witnessing this.

Dorottya Sziráki Perfect set games and colorings on κκ

A dichotomy for infinitely many Π0

2(κ) relations R is a Π0

2(κ) relation on a topological space X iff

R is an intersection of ≤ κ many open subsets of kX for some 1 ≤ k < ω. Theorem (Sz) Assume ♦κ or κ is inaccessible. Let R be a collection of ≤ κ many Π0

2(κ) relations on a closed subset X of κκ.

Then either X has a κ-perfect R-homogeneous subset, or there exists a tree T without κ-branches, |T| ≤ 2κ, such that for all R-homogeneous Y ⊆ X, we have Y ∩ KerT (Y ) = ∅ (that is, player II does not win VT (Y, y) for any y ∈ Y ). If κ is inaccessible, then there exists a tree T of size κ witnessing this. Sketch of the proof First, show that if X has a κ-dense in itself R-homogeneous subset, then X has a κ-perfect R-homogeneous subset.

Dorottya Sziráki Perfect set games and colorings on κκ

Step 2

Let R be an arbitrary set of finitary relations on κκ. Lemma If X does not have a κ-dense in itself R-homogeneous subset, then there exists a tree T without κ-branches, |T| ≤ 2κ, such that Y ∩ KerT (Y ) = ∅ holds for all R-homogeneous Y ⊆ X.

Dorottya Sziráki Perfect set games and colorings on κκ

Step 2

Let R be an arbitrary set of finitary relations on κκ. Lemma If X does not have a κ-dense in itself R-homogeneous subset, then there exists a tree T without κ-branches, |T| ≤ 2κ, such that Y ∩ KerT (Y ) = ∅ holds for all R-homogeneous Y ⊆ X. Proof. The assumption holds iff II does not win Vκ(Y, x) for any R-homogeneous Y ⊆ κκ and x ∈ Y .

Dorottya Sziráki Perfect set games and colorings on κκ

Step 2

Let R be an arbitrary set of finitary relations on κκ. Lemma If X does not have a κ-dense in itself R-homogeneous subset, then there exists a tree T without κ-branches, |T| ≤ 2κ, such that Y ∩ KerT (Y ) = ∅ holds for all R-homogeneous Y ⊆ X. Proof. The assumption holds iff II does not win Vκ(Y, x) for any R-homogeneous Y ⊆ κκ and x ∈ Y . T0 = the tree of winning strategies τ of II in short games Vδ(X, x) (where δ < κ and x ∈ X) such that the set of all possible τ-moves of II R-homogeneous.

Dorottya Sziráki Perfect set games and colorings on κκ

Step 2

Let R be an arbitrary set of finitary relations on κκ. Lemma If X does not have a κ-dense in itself R-homogeneous subset, then there exists a tree T without κ-branches, |T| ≤ 2κ, such that Y ∩ KerT (Y ) = ∅ holds for all R-homogeneous Y ⊆ X. Proof. The assumption holds iff II does not win Vκ(Y, x) for any R-homogeneous Y ⊆ κκ and x ∈ Y . T0 = the tree of winning strategies τ of II in short games Vδ(X, x) (where δ < κ and x ∈ X) such that the set of all possible τ-moves of II R-homogeneous. T = σT0, the tree of ascending chains in T0.

Dorottya Sziráki Perfect set games and colorings on κκ

Step 2

Let R be an arbitrary set of finitary relations on κκ. Lemma If X does not have a κ-dense in itself R-homogeneous subset, then there exists a tree T without κ-branches, |T| ≤ 2κ, such that Y ∩ KerT (Y ) = ∅ holds for all R-homogeneous Y ⊆ X. Remark If all R-homogeneous sets Y ⊆ X are κ-scattered (i.e., I wins Vκ(Y, y) for all y ∈ Y ), then there exists a tree S without κ-branches, |S| ≤ 2κ such that ScS(Y ) = Y (i.e., I wins VS(Y, y) for all y ∈ Y ) for all R-homogeneous Y ⊆ κκ.

Dorottya Sziráki Perfect set games and colorings on κκ

A corollary

Corollary If λ > κ is weakly compact, and G is Col(κ, <λ)-generic, then in V [G]: Let X be a Σ1

1(κ) subset of κκ, let R a set of ≤ κ many Π0 2(κ) relations on X.

If X has an R-homogeneous subset of size > κ, then X has a κ-perfect R-homogeneous subset. (2) This was known for measurable λ > κ (Sz, Väänänen). Question What is the consistency strength of (2)?

Dorottya Sziráki Perfect set games and colorings on κκ

Thank you!

Dorottya Sziráki Perfect set games and colorings on κκ