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Polarized Partition Properties on the Second Level of the Projective Hierarchy. Yurii Khomskii University of Amstedam Joint work with J org Brendle (Kobe University, Japan) RIMS Set Theory Workshop 2009, Kyoto, Japan Polarized Partition

  1. Polarized Partitions Polarized partition properties have been studied by Henle, Llopis, DiPrisco, Todorˇ cevi´ c and Zapletal. Easy observations: 1. In order for ( � n → � m ) to hold even for very simple partitions, � m . n ≫ � 2. Γ ( � m ) . n → � m ) = ⇒ Γ ( � ω → � Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 9/3

  2. Polarized Partitions Polarized partition properties have been studied by Henle, Llopis, DiPrisco, Todorˇ cevi´ c and Zapletal. Easy observations: 1. In order for ( � n → � m ) to hold even for very simple partitions, � m . n ≫ � 2. Γ ( � m ) . n → � m ) = ⇒ Γ ( � ω → � m ′ ) , for all m, m ′ ≥ 2 . ω → � 3. Γ ( � ω → � m ) ⇐ ⇒ Γ ( � Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 9/3

  3. Polarized Partitions Polarized partition properties have been studied by Henle, Llopis, DiPrisco, Todorˇ cevi´ c and Zapletal. Easy observations: 1. In order for ( � n → � m ) to hold even for very simple partitions, � m . n ≫ � 2. Γ ( � m ) . n → � m ) = ⇒ Γ ( � ω → � m ′ ) , for all m, m ′ ≥ 2 . ω → � 3. Γ ( � ω → � m ) ⇐ ⇒ Γ ( � m ) , then for every other � m ′ there is � n ′ such that If Γ ( � n → � Γ ( � n ′ → � m ′ ) Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 9/3

  4. Polarized Partitions Polarized partition properties have been studied by Henle, Llopis, DiPrisco, Todorˇ cevi´ c and Zapletal. Easy observations: 1. In order for ( � n → � m ) to hold even for very simple partitions, � m . n ≫ � 2. Γ ( � m ) . n → � m ) = ⇒ Γ ( � ω → � m ′ ) , for all m, m ′ ≥ 2 . ω → � 3. Γ ( � ω → � m ) ⇐ ⇒ Γ ( � m ) , then for every other � m ′ there is � n ′ such that If Γ ( � n → � Γ ( � n ′ → � m ′ ) Use coding function ϕ ( x ) := �� x (0) , . . . , x ( i 1 ) � , � x ( i 1 + 1) , . . . , x ( i 1 + i 2 ) � , . . . � . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 9/3

  5. Polarized Partitions Polarized partition properties have been studied by Henle, Llopis, DiPrisco, Todorˇ cevi´ c and Zapletal. Easy observations: 1. In order for ( � n → � m ) to hold even for very simple partitions, � m . n ≫ � 2. Γ ( � m ) . n → � m ) = ⇒ Γ ( � ω → � m ′ ) , for all m, m ′ ≥ 2 . ω → � 3. Γ ( � ω → � m ) ⇐ ⇒ Γ ( � m ) , then for every other � m ′ there is � n ′ such that If Γ ( � n → � Γ ( � n ′ → � m ′ ) Use coding function ϕ ( x ) := �� x (0) , . . . , x ( i 1 ) � , � x ( i 1 + 1) , . . . , x ( i 1 + i 2 ) � , . . . � . From now on, use generic notations ( � ω → � m ) and ( � n → � m ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 9/3

  6. Polarized Partitions In [DiPrisco & Todorˇ cevi´ c, 2003]: ( � ω → � m ) and ( � n → � m ) hold for analytic sets. Explicit bounds � n computed from � m (using Ackermann-like function). Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 10/3

  7. Polarized Partitions In [DiPrisco & Todorˇ cevi´ c, 2003]: ( � ω → � m ) and ( � n → � m ) hold for analytic sets. Explicit bounds � n computed from � m (using Ackermann-like function). On the other hand, easy to find counterexample using AC (i.e. well-ordering of ω ω ). Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 10/3

  8. Polarized Partitions In [DiPrisco & Todorˇ cevi´ c, 2003]: ( � ω → � m ) and ( � n → � m ) hold for analytic sets. Explicit bounds � n computed from � m (using Ackermann-like function). On the other hand, easy to find counterexample using AC (i.e. well-ordering of ω ω ). So, what about ∆ 1 2 / Σ 1 m ) and ∆ 1 2 / Σ 1 2 ( � ω → � 2 ( � n → � m ) ? Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 10/3

  9. Upper bound Fact. Γ ( Ramsey ) = ⇒ Γ ( � ω → � m ) . Proof. Given A , let X ∈ ω ↑ ω be homogeneous for A ∩ ω ↑ ω . Then divide ran ( X ) into X 0 , X 1 , . . . such that | X i | = m i . Now H := � X 0 , X 1 , . . . � witnesses that A satisfies ( � ω → � m ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 11/3

  10. Eventually different reals Theorem. (Brendle) If ∆ 1 2 ( � ω → � m ) then ∀ a there is an eventually different real over L[ a ] . i.e. an x such that ∀ y ∈ ω ω ∩ L[ a ] ∀ ∞ n ( x ( n ) � = y ( n )) Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 12/3

  11. Eventually different reals Theorem. (Brendle) If ∆ 1 2 ( � ω → � m ) then ∀ a there is an eventually different real over L[ a ] . i.e. an x such that ∀ y ∈ ω ω ∩ L[ a ] ∀ ∞ n ( x ( n ) � = y ( n )) Proof. • Suppose not, fix a such that ∀ x ∃ y ∈ L[ a ] s.t. ∃ ∞ n ( x ( n ) = y ( n )) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 12/3

  12. Eventually different reals Theorem. (Brendle) If ∆ 1 2 ( � ω → � m ) then ∀ a there is an eventually different real over L[ a ] . i.e. an x such that ∀ y ∈ ω ω ∩ L[ a ] ∀ ∞ n ( x ( n ) � = y ( n )) Proof. • Suppose not, fix a such that ∀ x ∃ y ∈ L[ a ] s.t. ∃ ∞ n ( x ( n ) = y ( n )) . • W.l.o.g., assume that ∀ x ∃ y ∈ L[ a ] s.t. ∃ ∞ n [ x ( n ) = y ( n ) & x ( n + 1) = y ( n + 1)] . Let y x denote the < L[ a ] -least such real. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 12/3

  13. Eventually different reals Theorem. (Brendle) If ∆ 1 2 ( � ω → � m ) then ∀ a there is an eventually different real over L[ a ] . i.e. an x such that ∀ y ∈ ω ω ∩ L[ a ] ∀ ∞ n ( x ( n ) � = y ( n )) Proof. • Suppose not, fix a such that ∀ x ∃ y ∈ L[ a ] s.t. ∃ ∞ n ( x ( n ) = y ( n )) . • W.l.o.g., assume that ∀ x ∃ y ∈ L[ a ] s.t. ∃ ∞ n [ x ( n ) = y ( n ) & x ( n + 1) = y ( n + 1)] . Let y x denote the < L[ a ] -least such real. • Let A := { x | first n at which x ( n ) = y x ( n ) is even } . This is ∆ 1 2 ( a ) using the fact that < L[ a ] is ∆ 1 2 ( a ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 12/3

  14. Eventually different reals Theorem. (Brendle) If ∆ 1 2 ( � ω → � m ) then ∀ a there is an eventually different real over L[ a ] . i.e. an x such that ∀ y ∈ ω ω ∩ L[ a ] ∀ ∞ n ( x ( n ) � = y ( n )) Proof. • Suppose not, fix a such that ∀ x ∃ y ∈ L[ a ] s.t. ∃ ∞ n ( x ( n ) = y ( n )) . • W.l.o.g., assume that ∀ x ∃ y ∈ L[ a ] s.t. ∃ ∞ n [ x ( n ) = y ( n ) & x ( n + 1) = y ( n + 1)] . Let y x denote the < L[ a ] -least such real. • Let A := { x | first n at which x ( n ) = y x ( n ) is even } . This is ∆ 1 2 ( a ) using the fact that < L[ a ] is ∆ 1 2 ( a ) . • Let H be homogeneous for A , w.l.o.g. [ H ] ⊆ A . But if x ∈ [ H ] then let us change finitely many digits of x to produce a new real x ′ , such that the first n at which x ′ ( n ) = y x ( n ) is odd but still x ′ ∈ [ H ] . It is easy to see that y x = y x ′ , hence x ′ / ∈ A : contradiction. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 12/3

  15. Diagram of implications Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 13/3

  16. Diagram of implications Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 13/3

  17. Diagram of implications Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 13/3

  18. Diagram of implications Question: which implications cannot be reversed? Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 13/3

  19. Diagram of implications Question: which implications cannot be reversed? Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 14/3

  20. Mathias model Theorem. (Brendle-Kh) Let L R ω 1 be the Mathias model , i.e., the ω 1 -iteration with countable support of Mathias forcing = ∆ 1 2 ( Ramsey ) but ¬ ∆ 1 starting from L . Then L R ω 1 | 2 ( � n → � m ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 15/3

  21. Diagram of implications Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 16/3

  22. Mathias model Theorem. (Brendle-Kh) Let L R ω 1 be the Mathias model , i.e., the ω 1 -iteration with countable support of Mathias forcing = ∆ 1 2 ( Ramsey ) but ¬ ∆ 1 starting from L . Then L R ω 1 | 2 ( � n → � m ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 17/3

  23. Mathias model Theorem. (Brendle-Kh) Let L R ω 1 be the Mathias model , i.e., the ω 1 -iteration with countable support of Mathias forcing = ∆ 1 2 ( Ramsey ) but ¬ ∆ 1 starting from L . Then L R ω 1 | 2 ( � n → � m ) . Proof • Clearly ∆ 1 2 ( Ramsey ) holds in L R ω 1 . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 17/3

  24. Mathias model Theorem. (Brendle-Kh) Let L R ω 1 be the Mathias model , i.e., the ω 1 -iteration with countable support of Mathias forcing = ∆ 1 2 ( Ramsey ) but ¬ ∆ 1 starting from L . Then L R ω 1 | 2 ( � n → � m ) . Proof • Clearly ∆ 1 2 ( Ramsey ) holds in L R ω 1 . → [ ω ] <ω | ∀ i | S ( i ) | ≤ 2 i } . Mathias forcing satisfies the Laver • Let C := { S : ω − property : For every y ∈ M ∩ ω ω and ˙ x s.t. � ∀ i ˙ x ( i ) ≤ y ( i ) , there is an S ∈ C ∩ M s.t. � ∀ i ˙ x ( i ) ∈ S ( i ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 17/3

  25. Mathias model Theorem. (Brendle-Kh) Let L R ω 1 be the Mathias model , i.e., the ω 1 -iteration with countable support of Mathias forcing = ∆ 1 2 ( Ramsey ) but ¬ ∆ 1 starting from L . Then L R ω 1 | 2 ( � n → � m ) . Proof • Clearly ∆ 1 2 ( Ramsey ) holds in L R ω 1 . → [ ω ] <ω | ∀ i | S ( i ) | ≤ 2 i } . Mathias forcing satisfies the Laver • Let C := { S : ω − property : For every y ∈ M ∩ ω ω and ˙ x s.t. � ∀ i ˙ x ( i ) ≤ y ( i ) , there is an S ∈ C ∩ M s.t. � ∀ i ˙ x ( i ) ∈ S ( i ) . 2 -well-ordering of L ∩ ω ω to define a ∆ 1 • Use the ∆ 1 2 -well-ordering of L ∩ C . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 17/3

  26. Mathias model Theorem. (Brendle-Kh) Let L R ω 1 be the Mathias model , i.e., the ω 1 -iteration with countable support of Mathias forcing = ∆ 1 2 ( Ramsey ) but ¬ ∆ 1 starting from L . Then L R ω 1 | 2 ( � n → � m ) . Proof • Clearly ∆ 1 2 ( Ramsey ) holds in L R ω 1 . → [ ω ] <ω | ∀ i | S ( i ) | ≤ 2 i } . Mathias forcing satisfies the Laver • Let C := { S : ω − property : For every y ∈ M ∩ ω ω and ˙ x s.t. � ∀ i ˙ x ( i ) ≤ y ( i ) , there is an S ∈ C ∩ M s.t. � ∀ i ˙ x ( i ) ∈ S ( i ) . 2 -well-ordering of L ∩ ω ω to define a ∆ 1 • Use the ∆ 1 2 -well-ordering of L ∩ C . • Use that to define a ∆ 1 2 set A which explicitly violates ( � n → � m ) , where the m i grow faster then 2 i . This set is well-defined because of the Laver property. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 17/3

  27. Diagram of implications Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 18/3

  28. Diagram of implications Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 18/3

  29. A model for ∆ 1 2 ( � n → � m ) Goal. Force a model in which ∆ 1 2 ( � ω → � m ) is true but ∆ 1 2 ( Ramsey ) is false. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 19/3

  30. A model for ∆ 1 2 ( � n → � m ) Goal. Force a model in which ∆ 1 2 ( � ω → � m ) is true but ∆ 1 2 ( Ramsey ) is false. Stronger. Force a model in which ∆ 1 2 ( � n → � m ) is true but ∆ 1 2 ( Miller ) is false. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 19/3

  31. Diagram of implications Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 20/3

  32. A model for ∆ 1 2 ( � n → � m ) Goal. Force a model in which ∆ 1 2 ( � ω → � m ) is true but ∆ 1 2 ( Ramsey ) is false. Stronger. Force a model in which ∆ 1 2 ( � n → � m ) is true but ∆ 1 2 ( Miller ) is false. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 21/3

  33. A model for ∆ 1 2 ( � n → � m ) Goal. Force a model in which ∆ 1 2 ( � ω → � m ) is true but ∆ 1 2 ( Ramsey ) is false. Stronger. Force a model in which ∆ 1 2 ( � n → � m ) is true but ∆ 1 2 ( Miller ) is false. Which properties must such a forcing have? Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 21/3

  34. A model for ∆ 1 2 ( � n → � m ) Goal. Force a model in which ∆ 1 2 ( � ω → � m ) is true but ∆ 1 2 ( Ramsey ) is false. Stronger. Force a model in which ∆ 1 2 ( � n → � m ) is true but ∆ 1 2 ( Miller ) is false. Which properties must such a forcing have? 1. Proper and ω ω -bounding. for all ˙ x there is a y in the ground model and a p s.t. p � ∀ n ˙ x ( n ) ≤ y ( n ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 21/3

  35. A model for ∆ 1 2 ( � n → � m ) Goal. Force a model in which ∆ 1 2 ( � ω → � m ) is true but ∆ 1 2 ( Ramsey ) is false. Stronger. Force a model in which ∆ 1 2 ( � n → � m ) is true but ∆ 1 2 ( Miller ) is false. Which properties must such a forcing have? 1. Proper and ω ω -bounding. for all ˙ x there is a y in the ground model and a p s.t. p � ∀ n ˙ x ( n ) ≤ y ( n ) . 2. If ∀ a there is a generic over L[ a ] , then ∆ 1 m ) holds. 2 ( � n → � Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 21/3

  36. Creature forcing Such a forcing notion exists! Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 22/3

  37. Creature forcing Such a forcing notion exists! Creature forcing, due to [Kellner-Shelah, 2009] and [Shelah-Zapletal, unpublished]. We shall refer to it as P KSZ . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 22/3

  38. Creature forcing Such a forcing notion exists! Creature forcing, due to [Kellner-Shelah, 2009] and [Shelah-Zapletal, unpublished]. We shall refer to it as P KSZ . Construction of P KSZ : Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 22/3

  39. Creature forcing Such a forcing notion exists! Creature forcing, due to [Kellner-Shelah, 2009] and [Shelah-Zapletal, unpublished]. We shall refer to it as P KSZ . Construction of P KSZ : • At each n , a small ǫ n is given, and we construct a local partial order P n as follows: Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 22/3

  40. Creature forcing Such a forcing notion exists! Creature forcing, due to [Kellner-Shelah, 2009] and [Shelah-Zapletal, unpublished]. We shall refer to it as P KSZ . Construction of P KSZ : • At each n , a small ǫ n is given, and we construct a local partial order P n as follows: - Let F ( n ) ∈ ω be a ‘large’ upper bound. P n consists of ‘conditions’ or ‘creatures’ of the form ( c, k ) with c ⊆ F ( n ) and k ∈ ω such that log 2 ( | c | ) − k ≥ 1 Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 22/3

  41. Creature forcing Such a forcing notion exists! Creature forcing, due to [Kellner-Shelah, 2009] and [Shelah-Zapletal, unpublished]. We shall refer to it as P KSZ . Construction of P KSZ : • At each n , a small ǫ n is given, and we construct a local partial order P n as follows: - Let F ( n ) ∈ ω be a ‘large’ upper bound. P n consists of ‘conditions’ or ‘creatures’ of the form ( c, k ) with c ⊆ F ( n ) and k ∈ ω such that log 2 ( | c | ) − k ≥ 1 - ( c ′ , k ′ ) ≤ n ( c, k ) iff c ′ ⊆ c and k ′ ≥ k . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 22/3

  42. Creature forcing Such a forcing notion exists! Creature forcing, due to [Kellner-Shelah, 2009] and [Shelah-Zapletal, unpublished]. We shall refer to it as P KSZ . Construction of P KSZ : • At each n , a small ǫ n is given, and we construct a local partial order P n as follows: - Let F ( n ) ∈ ω be a ‘large’ upper bound. P n consists of ‘conditions’ or ‘creatures’ of the form ( c, k ) with c ⊆ F ( n ) and k ∈ ω such that log 2 ( | c | ) − k ≥ 1 - ( c ′ , k ′ ) ≤ n ( c, k ) iff c ′ ⊆ c and k ′ ≥ k . • Let a n := 2 1 /ǫ n . For each ( c, k ) ∈ P n , norm n ( c, k ) := log a n (log 2 ( | c | ) − k ) Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 22/3

  43. Creature forcing Such a forcing notion exists! Creature forcing, due to [Kellner-Shelah, 2009] and [Shelah-Zapletal, unpublished]. We shall refer to it as P KSZ . Construction of P KSZ : • At each n , a small ǫ n is given, and we construct a local partial order P n as follows: - Let F ( n ) ∈ ω be a ‘large’ upper bound. P n consists of ‘conditions’ or ‘creatures’ of the form ( c, k ) with c ⊆ F ( n ) and k ∈ ω such that log 2 ( | c | ) − k ≥ 1 - ( c ′ , k ′ ) ≤ n ( c, k ) iff c ′ ⊆ c and k ′ ≥ k . • Let a n := 2 1 /ǫ n . For each ( c, k ) ∈ P n , norm n ( c, k ) := log a n (log 2 ( | c | ) − k ) • If F ( n ) is large enough, then ∃ ( c, k ) ∈ P n s.t. norm n ( c, k ) ≥ n . [To be precise: F ( n ) ≥ 2 ((2 1 /ǫn ) n ) ] Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 22/3

  44. Creature forcing Now let P KSZ consist of conditions p such that: Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 23/3

  45. Creature forcing Now let P KSZ consist of conditions p such that: • There is stem( p ) ∈ ω <ω , ∀ n ≥ | stem( p ) | : p ( n ) ∈ P n and norm n ( p ( n )) → ∞ . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 23/3

  46. Creature forcing Now let P KSZ consist of conditions p such that: • There is stem( p ) ∈ ω <ω , ∀ n ≥ | stem( p ) | : p ( n ) ∈ P n and norm n ( p ( n )) → ∞ . • p ′ ≤ p iff - stem( p ′ ) ⊇ stem( p ) - For n with | stem( p ) | ≤ n < | stem( p ′ ) | we have p ′ ( n ) ∈ first coordinate of p ( n ) - For n ≥ | stem( p ′ ) | we have p ′ ( n ) ≤ n p ( n ) Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 23/3

  47. Creature forcing Now let P KSZ consist of conditions p such that: • There is stem( p ) ∈ ω <ω , ∀ n ≥ | stem( p ) | : p ( n ) ∈ P n and norm n ( p ( n )) → ∞ . • p ′ ≤ p iff - stem( p ′ ) ⊇ stem( p ) - For n with | stem( p ) | ≤ n < | stem( p ′ ) | we have p ′ ( n ) ∈ first coordinate of p ( n ) - For n ≥ | stem( p ′ ) | we have p ′ ( n ) ≤ n p ( n ) Remark: P KSZ adds a generic real, but the generic filter is not determined from the generic real in the usual way, and P KSZ is not in general representable as BOREL( ω ω ) /I for a σ -ideal I . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 23/3

  48. Proper and ω ω -bounding Theorem. (Kellner-Shelah, Shelah-Zapletal) If P KSZ is as above, and moreover 1 ∀ n : ǫ n ≤ n · � j<n F ( j ) then P KSZ is proper and ω ω -bounding. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 24/3

  49. Proper and ω ω -bounding Theorem. (Kellner-Shelah, Shelah-Zapletal) If P KSZ is as above, and moreover 1 ∀ n : ǫ n ≤ n · � j<n F ( j ) then P KSZ is proper and ω ω -bounding. The proof uses two properties from the general theory of creature forcings: for each n , P n satisfies “ ǫ n -bigness ” and “ ǫ n -halving ”. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 24/3

  50. Forcing ∆ 1 2 ( � n → � m ) Theorem. (Brendle-Kh) If for every a there is a P KSZ -generic over L[ a ] then ∆ 1 2 ( � m → � n ) holds. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 25/3

  51. Forcing ∆ 1 2 ( � n → � m ) Theorem. (Brendle-Kh) If for every a there is a P KSZ -generic over L[ a ] then ∆ 1 2 ( � m → � n ) holds. Proof • For p ∈ P KSZ let [ p ] := { x ∈ ω ω | stem( p ) ⊆ x and ∀ n ≥ | stem( p ) | : x ( n ) ∈ 1 st coordinate of p ( n ) } . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 25/3

  52. Forcing ∆ 1 2 ( � n → � m ) Theorem. (Brendle-Kh) If for every a there is a P KSZ -generic over L[ a ] then ∆ 1 2 ( � m → � n ) holds. Proof • For p ∈ P KSZ let [ p ] := { x ∈ ω ω | stem( p ) ⊆ x and ∀ n ≥ | stem( p ) | : x ( n ) ∈ 1 st coordinate of p ( n ) } . • P KSZ satisfies pure decision : for every φ and p ∈ P KSZ there is q ≤ p with the same stem as p s.t. q � φ or q � ¬ φ . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 25/3

  53. Forcing ∆ 1 2 ( � n → � m ) Theorem. (Brendle-Kh) If for every a there is a P KSZ -generic over L[ a ] then ∆ 1 2 ( � m → � n ) holds. Proof • For p ∈ P KSZ let [ p ] := { x ∈ ω ω | stem( p ) ⊆ x and ∀ n ≥ | stem( p ) | : x ( n ) ∈ 1 st coordinate of p ( n ) } . • P KSZ satisfies pure decision : for every φ and p ∈ P KSZ there is q ≤ p with the same stem as p s.t. q � φ or q � ¬ φ . • Let A ⊆ ω ω be a ∆ 1 2 ( a ) -set, defined by Σ 1 2 ( a ) formulas φ and ψ . By downward Π 1 3 -absoluteness, the sentence “ ∀ x ( φ ( x ) ↔ ¬ ψ ( x )) ” holds in all generic extensions of L[ a ] . Using this fact and pure decision, find a condition p in L[ a ] , with empty stem, s.t. p � φ ( ˙ x gen ) or p � ψ ( ˙ x gen ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 25/3

  54. Forcing ∆ 1 2 ( � n → � m ) Theorem. (Brendle-Kh) If for every a there is a P KSZ -generic over L[ a ] then ∆ 1 2 ( � m → � n ) holds. Proof • For p ∈ P KSZ let [ p ] := { x ∈ ω ω | stem( p ) ⊆ x and ∀ n ≥ | stem( p ) | : x ( n ) ∈ 1 st coordinate of p ( n ) } . • P KSZ satisfies pure decision : for every φ and p ∈ P KSZ there is q ≤ p with the same stem as p s.t. q � φ or q � ¬ φ . • Let A ⊆ ω ω be a ∆ 1 2 ( a ) -set, defined by Σ 1 2 ( a ) formulas φ and ψ . By downward Π 1 3 -absoluteness, the sentence “ ∀ x ( φ ( x ) ↔ ¬ ψ ( x )) ” holds in all generic extensions of L[ a ] . Using this fact and pure decision, find a condition p in L[ a ] , with empty stem, s.t. p � φ ( ˙ x gen ) or p � ψ ( ˙ x gen ) . • W.l.o.g. assume the former, and work in L[ a ] from now on. Let M ≺ H θ be countable and q ≤ p a ( M, P KSZ ) -Master condition. By pure decision, q has empty stem as well. Moreover, every x ∈ [ q ] is M -generic and by standard absoluteness arguments [ q ] ⊆ A follows. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 25/3

  55. Forcing ∆ 1 2 ( � n → � m ) Theorem. (Brendle-Kh) If for every a there is a P KSZ -generic over L[ a ] then ∆ 1 2 ( � m → � n ) holds. Proof • For p ∈ P KSZ let [ p ] := { x ∈ ω ω | stem( p ) ⊆ x and ∀ n ≥ | stem( p ) | : x ( n ) ∈ 1 st coordinate of p ( n ) } . • P KSZ satisfies pure decision : for every φ and p ∈ P KSZ there is q ≤ p with the same stem as p s.t. q � φ or q � ¬ φ . • Let A ⊆ ω ω be a ∆ 1 2 ( a ) -set, defined by Σ 1 2 ( a ) formulas φ and ψ . By downward Π 1 3 -absoluteness, the sentence “ ∀ x ( φ ( x ) ↔ ¬ ψ ( x )) ” holds in all generic extensions of L[ a ] . Using this fact and pure decision, find a condition p in L[ a ] , with empty stem, s.t. p � φ ( ˙ x gen ) or p � ψ ( ˙ x gen ) . • W.l.o.g. assume the former, and work in L[ a ] from now on. Let M ≺ H θ be countable and q ≤ p a ( M, P KSZ ) -Master condition. By pure decision, q has empty stem as well. Moreover, every x ∈ [ q ] is M -generic and by standard absoluteness arguments [ q ] ⊆ A follows. • Since q has empty stem, it witnesses that A satisfies ( � n → � m ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 25/3

  56. Forcing ∆ 1 2 ( � n → � m ) Corollary. An ω 1 -iteration of P KSZ , starting from L , gives a model in which ∆ 1 m ) holds but ∆ 1 2 ( Miller ) fails. 2 ( � n → � Notice that the bounds “ � n ” have been explicitly computed beforehand: they are the F ( n ) ’s from the definition of P KSZ . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 26/3

  57. Diagram of implications Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 27/3

  58. Other properties Definition. A real x ∈ [ ω ] ω is splitting over M if for all a ∈ [ ω ] ω ∩ M , both a ∩ x and a \ x are infinite. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 28/3

  59. Other properties Definition. A real x ∈ [ ω ] ω is splitting over M if for all a ∈ [ ω ] ω ∩ M , both a ∩ x and a \ x are infinite. Theorem. (Shelah-Zapletal) P KSZ does not add splitting reals. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 28/3

  60. Other properties Definition. A real x ∈ [ ω ] ω is splitting over M if for all a ∈ [ ω ] ω ∩ M , both a ∩ x and a \ x are infinite. Theorem. (Shelah-Zapletal) P KSZ does not add splitting reals. By another result of Zapletal, the conjunction “ ω ω -bounding and not adding splitting reals” is preserved in ω 1 -iterations, so: Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 28/3

  61. Diagram of implications Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 29/3

  62. Open questions for ∆ 1 2 Open questions 1. Is the implication ∆ 1 2 ( � ω → � m ) ⇒ ∃ ev. diff. reals strict? Conjecture: ∆ 1 2 ( � ω → � m ) fails in the Random model. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 30/3

  63. Open questions for ∆ 1 2 Open questions 1. Is the implication ∆ 1 2 ( � ω → � m ) ⇒ ∃ ev. diff. reals strict? Conjecture: ∆ 1 2 ( � ω → � m ) fails in the Random model. 2. Is there a characterization of ∆ 1 2 ( � ω → � m ) and ∆ 1 2 ( � n → � m ) in terms of transcendence over L ? Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 30/3

  64. The property on the Σ 1 2 level Recall that for Ramsey, Sacks, Miller and Laver measurability, ∆ 1 2 and Σ 1 2 are equivalent. Question: Are ∆ 1 2 and Σ 1 2 equivalent for the polarized partition properties? Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 31/3

  65. What we do know Theorem. If Σ 1 2 ( � ω → � m ) then ∀ a ∃ H s.t. ∀ x ∈ [ H ] : x is eventually different over L[ a ] . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 32/3

  66. What we do know Theorem. If Σ 1 2 ( � ω → � m ) then ∀ a ∃ H s.t. ∀ x ∈ [ H ] : x is eventually different over L[ a ] . Theorem. In the Mathias model, Σ 1 2 ( Ramsey ) holds while Σ 1 2 ( � n → � m ) fails. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 32/3

  67. Diagram of implications Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 33/3

  68. Forcing Σ 1 2 ( � n → � m ) Can we extend the result about P KSZ to Σ 1 2 ? Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 34/3

  69. Forcing Σ 1 2 ( � n → � m ) Can we extend the result about P KSZ to Σ 1 2 ? Not a priori, since P KSZ only adds one generic real. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 34/3

  70. Forcing Σ 1 2 ( � n → � m ) Can we extend the result about P KSZ to Σ 1 2 ? Not a priori, since P KSZ only adds one generic real. [DiPrisco & Todorˇ cevi´ c] use a forcing P DPT adding a whole generic product H G with the following property: For all Borel sets B in the ground model, ( ∗ ) B ∩ [ H G ] is relatively clopen in [ H G ] . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 34/3

  71. Forcing Σ 1 2 ( � n → � m ) Theorem. (Brendle-Kh) An ω 1 -iteration of P DPT starting from L give a model where Σ 1 m ) holds. 2 ( � n → � Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 35/3

  72. Forcing Σ 1 2 ( � n → � m ) Theorem. (Brendle-Kh) An ω 1 -iteration of P DPT starting from L give a model where Σ 1 m ) holds. 2 ( � n → � Proof. • Let A be Σ 1 2 ( a ) . Using Shoenfield trees, we find a partition A = � α<ω 1 A α into Borel sets with codes in L[ a ] . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 35/3

  73. Forcing Σ 1 2 ( � n → � m ) Theorem. (Brendle-Kh) An ω 1 -iteration of P DPT starting from L give a model where Σ 1 m ) holds. 2 ( � n → � Proof. • Let A be Σ 1 2 ( a ) . Using Shoenfield trees, we find a partition A = � α<ω 1 A α into Borel sets with codes in L[ a ] . • Since by the property ( ∗ ) of P DPT there is a product H in V s.t. every A α ∩ [ H ] is relatively clopen in [ H ] , by compactness A is a union of finitely many clopen sets (in [ H ] ) and so it is in fact Borel (in [ H ] ). Then it follows easily that A satisfies ( � n → � m ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 35/3

  74. Forcing Σ 1 2 ( � n → � m ) Theorem. (Brendle-Kh) An ω 1 -iteration of P DPT starting from L give a model where Σ 1 m ) holds. 2 ( � n → � Proof. • Let A be Σ 1 2 ( a ) . Using Shoenfield trees, we find a partition A = � α<ω 1 A α into Borel sets with codes in L[ a ] . • Since by the property ( ∗ ) of P DPT there is a product H in V s.t. every A α ∩ [ H ] is relatively clopen in [ H ] , by compactness A is a union of finitely many clopen sets (in [ H ] ) and so it is in fact Borel (in [ H ] ). Then it follows easily that A satisfies ( � n → � m ) . Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 35/3

  75. Forcing Σ 1 2 ( � n → � m ) Only problem: it is difficult to see whether P DPT is ω ω -bounding. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 36/3

  76. Forcing Σ 1 2 ( � n → � m ) Only problem: it is difficult to see whether P DPT is ω ω -bounding. So instead, we can combine elements of P DPT with P KSZ to produce a new forcing notion P which is still proper and ω ω -bounding (higher bounds but same idea) and moreover adds a product with the ( ∗ ) property. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 36/3

  77. Forcing Σ 1 2 ( � n → � m ) Only problem: it is difficult to see whether P DPT is ω ω -bounding. So instead, we can combine elements of P DPT with P KSZ to produce a new forcing notion P which is still proper and ω ω -bounding (higher bounds but same idea) and moreover adds a product with the ( ∗ ) property. Corollary. There is a model where Σ 1 2 ( � n → � m ) holds but Σ 1 2 ( Miller ) fails. Polarized Partition Properties on the Second Level of theProjective Hierarchy. – p. 36/3

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