Sliders SDF 60th Birthday Celebration Natasha Dobrinen University - - PowerPoint PPT Presentation

Sliders SDF 60th Birthday Celebration

Natasha Dobrinen University of Denver

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 1 / 27

Remeniscences

Thanks Sy for giving me a job at KGRC. (2004-2007)

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 2 / 27

Remeniscences

Thanks Sy for giving me a job at KGRC. (2004-2007) It was first time I worked around more than one other set theorist.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 2 / 27

Remeniscences

Thanks Sy for giving me a job at KGRC. (2004-2007) It was first time I worked around more than one other set theorist. Lovely to do research all day!

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 2 / 27

So, what are sliders?

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 3 / 27

Pulled Pork Sliders

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 4 / 27

Chicken Sliders

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 5 / 27

Hamburger Sliders

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 6 / 27

Sliders come in many forms

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 7 / 27

Sliders come in many forms

Yet, all sliders of the same form are indistinguishable from each other.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 7 / 27

Sliders come in many forms

Yet, all sliders of the same form are indistinguishable from each other. In mathematics, sliders are formally known as indiscernibles.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 7 / 27

History

During a working lunch at an AIMS Conference in December 2004, Sy introduced me to indiscernibles.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 8 / 27

History

During a working lunch at an AIMS Conference in December 2004, Sy introduced me to indiscernibles. (He also introduced me to the working lunch.)

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 8 / 27

History

During a working lunch at an AIMS Conference in December 2004, Sy introduced me to indiscernibles. (He also introduced me to the working lunch.) The indiscernibles were pleasant, though undistinguished lunch guests.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 8 / 27

History

During a working lunch at an AIMS Conference in December 2004, Sy introduced me to indiscernibles. (He also introduced me to the working lunch.) The indiscernibles were pleasant, though undistinguished lunch guests. Since that initial introduction, indiscernibles keep sliding into key positions in my work.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 8 / 27

Sy and I were interested in the following:

• Problem. Given models V ⊆ W of ZFC, when does having a new

subset of κ in W \ V make (Pκ+(λ))W \ (Pκ+(λ))V stationary in W ? i.e. When is the ground model co-stationary?

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 9 / 27

Sy and I were interested in the following:

• Problem. Given models V ⊆ W of ZFC, when does having a new

subset of κ in W \ V make (Pκ+(λ))W \ (Pκ+(λ))V stationary in W ? i.e. When is the ground model co-stationary? Pκ(λ) = {x ⊆ λ : |x| < κ}.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 9 / 27

Sy and I were interested in the following:

• Problem. Given models V ⊆ W of ZFC, when does having a new

subset of κ in W \ V make (Pκ+(λ))W \ (Pκ+(λ))V stationary in W ? i.e. When is the ground model co-stationary? Pκ(λ) = {x ⊆ λ : |x| < κ}. C ⊆ Pκ(λ) is club if it is closed under < κ-unions and ⊆-cofinal in Pκ(λ).

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 9 / 27

Sy and I were interested in the following:

• Problem. Given models V ⊆ W of ZFC, when does having a new

subset of κ in W \ V make (Pκ+(λ))W \ (Pκ+(λ))V stationary in W ? i.e. When is the ground model co-stationary? Pκ(λ) = {x ⊆ λ : |x| < κ}. C ⊆ Pκ(λ) is club if it is closed under < κ-unions and ⊆-cofinal in Pκ(λ). S ⊆ Pκ(λ) is stationary if S meets every club set.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 9 / 27

Background

[Abraham/Shelah 1983]: ccc forcings adding a new subset of ℵ0 [Gitik 1985]: models V ⊆ W where W has a new subset of ℵ0 make the ground model co-stationary for Pκ(λ), for all cardinals ℵ1 < κ < λ in the larger model.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 10 / 27

Background

[Abraham/Shelah 1983]: ccc forcings adding a new subset of ℵ0 [Gitik 1985]: models V ⊆ W where W has a new subset of ℵ0 make the ground model co-stationary for Pκ(λ), for all cardinals ℵ1 < κ < λ in the larger model. What if the larger model has a new subset of ℵ1 but no new subsets of ℵ0?

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 10 / 27

Background

[Abraham/Shelah 1983]: ccc forcings adding a new subset of ℵ0 [Gitik 1985]: models V ⊆ W where W has a new subset of ℵ0 make the ground model co-stationary for Pκ(λ), for all cardinals ℵ1 < κ < λ in the larger model. What if the larger model has a new subset of ℵ1 but no new subsets of ℵ0? Sy knew that Erd˝

• s cardinals would be necessary if we add no new

ω-sequences, because of a covering theorem of [Magidor 1990].

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 10 / 27

Background

[Abraham/Shelah 1983]: ccc forcings adding a new subset of ℵ0 [Gitik 1985]: models V ⊆ W where W has a new subset of ℵ0 make the ground model co-stationary for Pκ(λ), for all cardinals ℵ1 < κ < λ in the larger model. What if the larger model has a new subset of ℵ1 but no new subsets of ℵ0? Sy knew that Erd˝

• s cardinals would be necessary if we add no new

ω-sequences, because of a covering theorem of [Magidor 1990]. Erd˝

• s cardinals involve indiscernibles.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 10 / 27

Background

[Abraham/Shelah 1983]: ccc forcings adding a new subset of ℵ0 [Gitik 1985]: models V ⊆ W where W has a new subset of ℵ0 make the ground model co-stationary for Pκ(λ), for all cardinals ℵ1 < κ < λ in the larger model. What if the larger model has a new subset of ℵ1 but no new subsets of ℵ0? Sy knew that Erd˝

• s cardinals would be necessary if we add no new

ω-sequences, because of a covering theorem of [Magidor 1990]. Erd˝

• s cardinals involve indiscernibles.

This was the beginning of our work on finding the equiconsistency of co-stationarity of the ground model and broader work in which indiscernibles play an important role.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 10 / 27

Indiscernibles

• Def. M a structure, X ⊆ M linearly ordered by <.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 11 / 27

Indiscernibles

• Def. M a structure, X ⊆ M linearly ordered by <.

X, < is a set of indiscernibles for M iff for all ϕ(v1, . . . , vn) in the language of M, for all x1 < · · · < xn and y1 < · · · < yn in X, M | = ϕ[x1, . . . , xn] iff M | = ϕ[y1, . . . , yn].

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 11 / 27

α-Erd˝

• s cardinals

κ is α-Erd˝

• s if for each structure M in a countable language with universe

κ (endowed with Skolem functions), for each club C ⊆ κ there is a set I ⊆ C of order type α such that

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 12 / 27

α-Erd˝

• s cardinals

κ is α-Erd˝

• s if for each structure M in a countable language with universe

κ (endowed with Skolem functions), for each club C ⊆ κ there is a set I ⊆ C of order type α such that I is a set of indiscernibles for M and

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 12 / 27

α-Erd˝

• s cardinals

κ is α-Erd˝

• s if for each structure M in a countable language with universe

κ (endowed with Skolem functions), for each club C ⊆ κ there is a set I ⊆ C of order type α such that I is a set of indiscernibles for M and I is remarkable: whenever α0 < · · · < αn; β0 < · · · < βn are from I, αi−1 < βi, τ is a term, and τ M(α0, . . . , αn) < αi, then

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 12 / 27

α-Erd˝

• s cardinals

κ is α-Erd˝

• s if for each structure M in a countable language with universe

κ (endowed with Skolem functions), for each club C ⊆ κ there is a set I ⊆ C of order type α such that I is a set of indiscernibles for M and I is remarkable: whenever α0 < · · · < αn; β0 < · · · < βn are from I, αi−1 < βi, τ is a term, and τ M(α0, . . . , αn) < αi, then τ M(α0, . . . , αn) = τ M(α0, . . . , αi−1, βi, . . . , βn).

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 12 / 27

• Thm. [Dobrinen/Friedman 06] Suppose that in V , λ > κ, κ is regular,

and λ is κ-Erd˝

• s. Let Cκ be κ-Cohen forcing

(or any (λ, λ, κ)-distributive partial ordering adding a new subset of κ). Then (Pκ+(µ))V Cκ \ (Pκ+(µ))V is stationary in V Cκ for all µ ≥ λ.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 13 / 27

• Thm. [Dobrinen/Friedman 06] Suppose that in V , λ > κ, κ is regular,

and λ is κ-Erd˝

• s. Let Cκ be κ-Cohen forcing

(or any (λ, λ, κ)-distributive partial ordering adding a new subset of κ). Then (Pκ+(µ))V Cκ \ (Pκ+(µ))V is stationary in V Cκ for all µ ≥ λ. Pushing the κ down to smaller cardinals involved gleaning tree coding from some work of [Baumgartner 1991].

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 13 / 27

Let C denote ℵ1-Cohen forcing. Thm (Global Gitik). [Dobrinen/Friedman 06] The following are equiconsistent:

1 There is a proper class of ω1-Erd˝

• s cardinals.

2 (Pκ(λ))V C \ (Pκ(λ))V is stationary in V C, for all regular κ ≥ ℵ2

and λ > κ.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 14 / 27

Let C denote ℵ1-Cohen forcing. Thm (Global Gitik). [Dobrinen/Friedman 06] The following are equiconsistent:

1 There is a proper class of ω1-Erd˝

• s cardinals.

2 (Pκ(λ))V C \ (Pκ(λ))V is stationary in V C, for all regular κ ≥ ℵ2

and λ > κ. Note: There are still many open problems in this line of work.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 14 / 27

Let C denote ℵ1-Cohen forcing. Thm (Global Gitik). [Dobrinen/Friedman 06] The following are equiconsistent:

1 There is a proper class of ω1-Erd˝

• s cardinals.

2 (Pκ(λ))V C \ (Pκ(λ))V is stationary in V C, for all regular κ ≥ ℵ2

and λ > κ. Note: There are still many open problems in this line of work. Indiscernibles were also important in our work on the internal consistency strength of co-stationarity of the ground model [Dobrinen/Friedman 2008].

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 14 / 27

More work using indiscernibles

• Thm. [Dobrinen/Friedman 10] The following are equiconsistent:

1 κ is a measurable cardinal and the tree property holds at κ++. 2 κ is a weakly compact hypermeasurable cardinal. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 15 / 27

More work using indiscernibles

• Thm. [Dobrinen/Friedman 10] The following are equiconsistent:

1 κ is a measurable cardinal and the tree property holds at κ++. 2 κ is a weakly compact hypermeasurable cardinal.

• Thm. [Dobrinen/Friedman 10] Suppose 0# exists. Then there is an

inner model in which the tree property holds at the double successor of every strongly inaccessible cardinal.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 15 / 27

More work using indiscernibles

• Thm. [Dobrinen/Friedman 10] The following are equiconsistent:

1 κ is a measurable cardinal and the tree property holds at κ++. 2 κ is a weakly compact hypermeasurable cardinal.

• Thm. [Dobrinen/Friedman 10] Suppose 0# exists. Then there is an

inner model in which the tree property holds at the double successor of every strongly inaccessible cardinal. The proofs of such theorems heavily involve the Silver indiscernibles for building the generics.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 15 / 27

And now for something discernibly different

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 16 / 27

Ramsey Theory

Ramsey Theorem. For each k, n ≥ 1 and coloring c : [ω]k → n, there is an infinite M ⊆ ω such that c restricted to [M]k monochromatic. That is, M is homogeneous.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 17 / 27

Ramsey Theory

Ramsey Theorem. For each k, n ≥ 1 and coloring c : [ω]k → n, there is an infinite M ⊆ ω such that c restricted to [M]k monochromatic. That is, M is homogeneous. What about colorings into infinitely many colors?

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 17 / 27

Order Indiscernibility

Erd˝

• s-Rado Canonization Theorem.

For each k ≥ 1 and each equivalence relation E on [ω]k, there is an infinite M ⊆ ω such that E ↾ [M]k is canonical; i.e. E ↾ [M]k is given by Ek

I for some I ⊆ k.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 18 / 27

Order Indiscernibility

Erd˝

• s-Rado Canonization Theorem.

For each k ≥ 1 and each equivalence relation E on [ω]k, there is an infinite M ⊆ ω such that E ↾ [M]k is canonical; i.e. E ↾ [M]k is given by Ek

I for some I ⊆ k.

For a, b ∈ [ω]k, a Ek

I b iff ∀i ∈ I, ai = bi.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 18 / 27

Order Indiscernibility

Erd˝

• s-Rado Canonization Theorem.

For each k ≥ 1 and each equivalence relation E on [ω]k, there is an infinite M ⊆ ω such that E ↾ [M]k is canonical; i.e. E ↾ [M]k is given by Ek

I for some I ⊆ k.

For a, b ∈ [ω]k, a Ek

I b iff ∀i ∈ I, ai = bi.

• Thm. [Dobrinen/Mijares/Trujillo 1] For any product of n + 1 many

Fra¨ ıss´ e classes of ordered structures with the Ramsey Property, the canonical equivalence relations are given by EI0,...,In.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 18 / 27

Order Indiscernibility

Erd˝

• s-Rado Canonization Theorem.

For each k ≥ 1 and each equivalence relation E on [ω]k, there is an infinite M ⊆ ω such that E ↾ [M]k is canonical; i.e. E ↾ [M]k is given by Ek

I for some I ⊆ k.

For a, b ∈ [ω]k, a Ek

I b iff ∀i ∈ I, ai = bi.

• Thm. [Dobrinen/Mijares/Trujillo 1] For any product of n + 1 many

Fra¨ ıss´ e classes of ordered structures with the Ramsey Property, the canonical equivalence relations are given by EI0,...,In. The proofs of these theorems involve sliding of points between fixed points; in essence, indiscernibility.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 18 / 27

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ⊏ Y ⊆ X}.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 19 / 27

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ⊏ Y ⊆ X}. X ⊆ [ω]ω is Ramsey iff for each [s, X], there is a s ⊏ Y ⊆ X such that either [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 19 / 27

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ⊏ Y ⊆ X}. X ⊆ [ω]ω is Ramsey iff for each [s, X], there is a s ⊏ Y ⊆ X such that either [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

• Thm. [Ellentuck 1974] Every X ⊆ [ω]ω with the property of Baire (in

the Ellentuck topology) is Ramsey.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 19 / 27

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ⊏ Y ⊆ X}. X ⊆ [ω]ω is Ramsey iff for each [s, X], there is a s ⊏ Y ⊆ X such that either [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

• Thm. [Ellentuck 1974] Every X ⊆ [ω]ω with the property of Baire (in

the Ellentuck topology) is Ramsey. Galvin-Prikry Theorem: All (metrically) Borel sets are Ramsey.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 19 / 27

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ⊏ Y ⊆ X}. X ⊆ [ω]ω is Ramsey iff for each [s, X], there is a s ⊏ Y ⊆ X such that either [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

• Thm. [Ellentuck 1974] Every X ⊆ [ω]ω with the property of Baire (in

the Ellentuck topology) is Ramsey. Galvin-Prikry Theorem: All (metrically) Borel sets are Ramsey. Silver Theorem: All (metrically) Suslin sets are Ramsey.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 19 / 27

The Next Topological Ramsey Space: R1 [D/T 1]

T1

1 2 3 4 5

. . .

• ,
• 1

, 1

• 1

, 2

• 2

, 3

• 2

, 4

• 2

, 5

• 3

, 6

• 3

, 7

• 3

, 8

• 3

, 9

• 4

, 1

• 4

, 1 1

• 4

, 1 2

• 4

, 1 3

• 4

, 1 4

• 5

, 1 5

• 5

, 1 6

• 5

, 1 7

• 5

, 1 8

• 5

, 1 9

• 5

, 2

• X ∈ R1 iff X is a subtree of T1 and X ∼

= T1. For X, Y ∈ R1, Y ≤1 X iff Y ⊆ X.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 20 / 27

T1

1 2 3 4 5

. . .

• ,
• 1

, 1

• 1

, 2

• 2

, 3

• 2

, 4

• 2

, 5

• 3

, 6

• 3

, 7

• 3

, 8

• 3

, 9

• 4

, 1

• 4

, 1 1

• 4

, 1 2

• 4

, 1 3

• 4

, 1 4

• 5

, 1 5

• 5

, 1 6

• 5

, 1 7

• 5

, 1 8

• 5

, 1 9

• 5

, 2

• The Erd˝
• s-Rado type theorems are building blocks for canonical

equivalence relations on barriers on a certain collection of (new) topological Ramsey spaces.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 21 / 27

T1

1 2 3 4 5

. . .

• ,
• 1

, 1

• 1

, 2

• 2

, 3

• 2

, 4

• 2

, 5

• 3

, 6

• 3

, 7

• 3

, 8

• 3

, 9

• 4

, 1

• 4

, 1 1

• 4

, 1 2

• 4

, 1 3

• 4

, 1 4

• 5

, 1 5

• 5

, 1 6

• 5

, 1 7

• 5

, 1 8

• 5

, 1 9

• 5

, 2

• The Erd˝
• s-Rado type theorems are building blocks for canonical

equivalence relations on barriers on a certain collection of (new) topological Ramsey spaces. Numbers of Canonical Equivalence Relations on Finite Rank Barriers [D/T 1]

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 21 / 27

T1

1 2 3 4 5

. . .

• ,
• 1

, 1

• 1

, 2

• 2

, 3

• 2

, 4

• 2

, 5

• 3

, 6

• 3

, 7

• 3

, 8

• 3

, 9

• 4

, 1

• 4

, 1 1

• 4

, 1 2

• 4

, 1 3

• 4

, 1 4

• 5

, 1 5

• 5

, 1 6

• 5

, 1 7

• 5

, 1 8

• 5

, 1 9

• 5

, 2

• The Erd˝
• s-Rado type theorems are building blocks for canonical

equivalence relations on barriers on a certain collection of (new) topological Ramsey spaces. Numbers of Canonical Equivalence Relations on Finite Rank Barriers [D/T 1] 1-approximations: 3 = 21 + 1

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 21 / 27

T1

1 2 3 4 5

. . .

• ,
• 1

, 1

• 1

, 2

• 2

, 3

• 2

, 4

• 2

, 5

• 3

, 6

• 3

, 7

• 3

, 8

• 3

, 9

• 4

, 1

• 4

, 1 1

• 4

, 1 2

• 4

, 1 3

• 4

, 1 4

• 5

, 1 5

• 5

, 1 6

• 5

, 1 7

• 5

, 1 8

• 5

, 1 9

• 5

, 2

• The Erd˝
• s-Rado type theorems are building blocks for canonical

equivalence relations on barriers on a certain collection of (new) topological Ramsey spaces. Numbers of Canonical Equivalence Relations on Finite Rank Barriers [D/T 1] 1-approximations: 3 = 21 + 1 2-approximations: 15 = (21 + 1)(22 + 1)

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 21 / 27

T1

1 2 3 4 5

. . .

• ,
• 1

, 1

• 1

, 2

• 2

, 3

• 2

, 4

• 2

, 5

• 3

, 6

• 3

, 7

• 3

, 8

• 3

, 9

• 4

, 1

• 4

, 1 1

• 4

, 1 2

• 4

, 1 3

• 4

, 1 4

• 5

, 1 5

• 5

, 1 6

• 5

, 1 7

• 5

, 1 8

• 5

, 1 9

• 5

, 2

• The Erd˝
• s-Rado type theorems are building blocks for canonical

equivalence relations on barriers on a certain collection of (new) topological Ramsey spaces. Numbers of Canonical Equivalence Relations on Finite Rank Barriers [D/T 1] 1-approximations: 3 = 21 + 1 2-approximations: 15 = (21 + 1)(22 + 1) k-approximations: (21 + 1)(22 + 1) · · · (2k + 1).

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 21 / 27

The space R2

T2

0, 0 1, 1 2, 2 2, 3 3, 4 3, 5 3, 6 1 2 3

. . .

0, 0, 0 1, 1, 1 1, 1, 2 2, 2, 3 2, 2, 4 2, 2, 5 2, 3, 6 2, 3, 7 2, 3, 8 2, 3, 9 3, 4, 10 3, 4, 11 3, 4, 12 3, 4, 13 3, 4, 14 3, 5, 15 3, 5, 16 3, 5, 17 3, 5, 18 3, 5, 19 3, 5, 20 3, 6, 21 3, 6, 22 3, 6, 23 3, 6, 24 3, 6, 25 3, 6, 26 3, 6, 27

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 22 / 27

Theorems in [Dobrinen/Todorcevic 2]

Ramsey-Classification Thms. For each α < ω1, every equivalence relation on a barrier on the topological Ramsey space Rα is canonical

• n some open set.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 23 / 27

Theorems in [Dobrinen/Todorcevic 2]

Ramsey-Classification Thms. For each α < ω1, every equivalence relation on a barrier on the topological Ramsey space Rα is canonical

• n some open set.

‘Canonical’ essentially means built in a recursive manner from Erd˝

• s-Rado equivalence relations.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 23 / 27

Theorems in [Dobrinen/Todorcevic 2]

Ramsey-Classification Thms. For each α < ω1, every equivalence relation on a barrier on the topological Ramsey space Rα is canonical

• n some open set.

‘Canonical’ essentially means built in a recursive manner from Erd˝

• s-Rado equivalence relations.

Ramsey-classification theorems for equivalence relations on barriers were used to classify all Rudin-Keisler isomorphism types of ultrafilters within the Tukey type of ultrafilters with weak partition properties.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 23 / 27

Classification of Tukey vs Rudin-Keisler for ultrafilters

• Def. U ≥RK V iff ∃f : ω → ω such that {X ⊆ ω : f −1(X) ∈ U} = V.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 24 / 27

Classification of Tukey vs Rudin-Keisler for ultrafilters

• Def. U ≥RK V iff ∃f : ω → ω such that {X ⊆ ω : f −1(X) ∈ U} = V.

U ≥T V iff ∃g : U → V such that for each filter base B ⊆ U, g(B) is a filter base for V.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 24 / 27

Classification of Tukey vs Rudin-Keisler for ultrafilters

• Def. U ≥RK V iff ∃f : ω → ω such that {X ⊆ ω : f −1(X) ∈ U} = V.

U ≥T V iff ∃g : U → V such that for each filter base B ⊆ U, g(B) is a filter base for V. U ≥RK V ⇒ U ≥T V.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 24 / 27

Classification of Tukey vs Rudin-Keisler for ultrafilters

• Def. U ≥RK V iff ∃f : ω → ω such that {X ⊆ ω : f −1(X) ∈ U} = V.

U ≥T V iff ∃g : U → V such that for each filter base B ⊆ U, g(B) is a filter base for V. U ≥RK V ⇒ U ≥T V.

• Thm. [Dobrinen/Todorcevic 1,2] For each α < ω1, there is an

ultrafilter Uα which is a rapid p-point, has partition properties, and the cofinal types below it form a chain of order-type (α + 1)∗.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 24 / 27

Classification of Tukey vs Rudin-Keisler for ultrafilters

• Def. U ≥RK V iff ∃f : ω → ω such that {X ⊆ ω : f −1(X) ∈ U} = V.

U ≥T V iff ∃g : U → V such that for each filter base B ⊆ U, g(B) is a filter base for V. U ≥RK V ⇒ U ≥T V.

• Thm. [Dobrinen/Todorcevic 1,2] For each α < ω1, there is an

ultrafilter Uα which is a rapid p-point, has partition properties, and the cofinal types below it form a chain of order-type (α + 1)∗. Moreover, the isomorphism types within these cofinal types are completely classified as tree ultrafilters, where branching occurs according to p-points from a precise countable collection determined by the canonization theorem.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 24 / 27

General Framework in [Dobrinen/Mijares/Trujillo 1,2]

A general framework is being developed in [Dobrinen/Mijares/Trujillo 1,2] for

1 Constructing general topological Ramsey spaces; Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 25 / 27

General Framework in [Dobrinen/Mijares/Trujillo 1,2]

A general framework is being developed in [Dobrinen/Mijares/Trujillo 1,2] for

1 Constructing general topological Ramsey spaces; 2 Canonization theorems for equivalence relations on barriers; Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 25 / 27

General Framework in [Dobrinen/Mijares/Trujillo 1,2]

A general framework is being developed in [Dobrinen/Mijares/Trujillo 1,2] for

1 Constructing general topological Ramsey spaces; 2 Canonization theorems for equivalence relations on barriers; 3 Classification of all isomorphism types within the Tukey types of the

associated ultrafilters;

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 25 / 27

General Framework in [Dobrinen/Mijares/Trujillo 1,2]

A general framework is being developed in [Dobrinen/Mijares/Trujillo 1,2] for

1 Constructing general topological Ramsey spaces; 2 Canonization theorems for equivalence relations on barriers; 3 Classification of all isomorphism types within the Tukey types of the

associated ultrafilters;

4 Finding initial structures in Tukey types besides chains. Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 25 / 27

General Framework in [Dobrinen/Mijares/Trujillo 1,2]

A general framework is being developed in [Dobrinen/Mijares/Trujillo 1,2] for

1 Constructing general topological Ramsey spaces; 2 Canonization theorems for equivalence relations on barriers; 3 Classification of all isomorphism types within the Tukey types of the

associated ultrafilters;

4 Finding initial structures in Tukey types besides chains.

More on this in Barcelona.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 25 / 27

References

[Dobrinen/Friedman 2006] Co-stationarity of the ground model. JSL. [Dobrinen/Friedman 2008] Internal consistency and global co-stationarity

• f the ground model. JSL.

[Dobrinen/Friedman 2010] The consistency strength of the tree property at the double successor of a measurable cardinal. Fundamenta. [Dobrinen/Todorcevic 1,2] New Ramsey-classification theorems and their applications to the Tukey theory of ultrafilters, Parts 1 and 2, To appear. Transactions AMS. [Dobrinen/Mijares/Trujillo 1,2] General framework for topological Ramsey spaces, Ramsey-classification theorems, and applications to Tukey theory

• f ultrafilters. In preparation.

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 26 / 27

Happy 60th Birthday Sy!

Happy wishes as you slide into the next decade!

Natasha Dobrinen Sliders SDF 60th Birthday Celebration University of Denver 27 / 27