Tutorial: Ramsey theory in Forcing
Natasha Dobrinen
University of Denver

RIMS, 2016

DobrinenRamsey theory in forcing

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Outline of Tutorial
Day 1

1
2

Introduction to topological Ramsey spaces
Classes of new topological Ramsey spaces which are dense in σ-closed
forcings yielding ultrafilters with complete combinatorics

DobrinenRamsey theory in forcing

University of Denver

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Outline of Tutorial
Day 1

1
2

Day 2

1
2

Introduction to topological Ramsey spaces
Classes of new topological Ramsey spaces which are dense in σ-closed
forcings yielding ultrafilters with complete combinatorics
Canonical Ramsey theory for equivalence relations on fronts
Applications to exact Tukey and Rudin-Keisler structures

DobrinenRamsey theory in forcing

University of Denver

2 / 44

Outline of Tutorial
Day 1

1
2

Day 2

1
2

Day 3

1
2

Introduction to topological Ramsey spaces
Classes of new topological Ramsey spaces which are dense in σ-closed
forcings yielding ultrafilters with complete combinatorics
Canonical Ramsey theory for equivalence relations on fronts
Applications to exact Tukey and Rudin-Keisler structures
Topological Ramsey spaces of strong trees
Applications to finding finite Ramsey degrees for universal relational
structures, including the universal triangle-free graphNational Science Foundation Grants DMS-1301665 and DMS-1600781 andSimons Foundation Grant 245286 sponsored much of Dobrinen’s research in thistutuorial.
Graphics by Timothy Trujillo.
DobrinenRamsey theory in forcing

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Day 1 Overview

1

Introduction to infinite dimensional Ramsey theory

DobrinenRamsey theory in forcing

University of Denver3 / 44

Day 1 Overview

1

Introduction to infinite dimensional Ramsey theory

2

The Ellentuck space

DobrinenRamsey theory in forcing

University of Denver3 / 44

Day 1 Overview

1

Introduction to infinite dimensional Ramsey theory

2

The Ellentuck space

3

Abstract Topological Ramsey Spaces

DobrinenRamsey theory in forcing

University of Denver3 / 44

Day 1 Overview

1

Introduction to infinite dimensional Ramsey theory

2

The Ellentuck space

3

Abstract Topological Ramsey Spaces

4

Connections with forcing and ultrafilters

DobrinenRamsey theory in forcing

University of Denver3 / 44

Day 1 Overview

1

Introduction to infinite dimensional Ramsey theory

2

The Ellentuck space

3

Abstract Topological Ramsey Spaces

4

Connections with forcing and ultrafilters

5

New topological Ramsey spaces dense inside σ-closed forcings which
add ultrafilters satisfying weak partition properties

DobrinenRamsey theory in forcing

University of Denver3 / 44

Day 1 Overview

1

Introduction to infinite dimensional Ramsey theory

2

The Ellentuck space

3

Abstract Topological Ramsey Spaces

4

Connections with forcing and ultrafilters

5

New topological Ramsey spaces dense inside σ-closed forcings which
add ultrafilters satisfying weak partition properties

6

A new Ramsey theorem motivated by this study

DobrinenRamsey theory in forcing

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Ramsey’s Theorem and Higher Dimensional Versions
Thm. (Ramsey) Given n, l ≥ 1 and a coloring c : [ω]n → l, there is an
infinite set M ⊆ ω such that c is monochromatic on [M]n .

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Ramsey’s Theorem and Higher Dimensional Versions
Thm. (Ramsey) Given n, l ≥ 1 and a coloring c : [ω]n → l, there is an
infinite set M ⊆ ω such that c is monochromatic on [M]n .
Ramsey’s Theorem can be extended to clopen sets on the Baire space.

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Ramsey’s Theorem and Higher Dimensional Versions
Thm. (Ramsey) Given n, l ≥ 1 and a coloring c : [ω]n → l, there is an
infinite set M ⊆ ω such that c is monochromatic on [M]n .
Ramsey’s Theorem can be extended to clopen sets on the Baire space.
Def. A set F ⊆ [ω]<ω is Nash-Williams if or a, b ∈ F, a 6< b.

DobrinenRamsey theory in forcing

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Ramsey’s Theorem and Higher Dimensional Versions
Thm. (Ramsey) Given n, l ≥ 1 and a coloring c : [ω]n → l, there is an
infinite set M ⊆ ω such that c is monochromatic on [M]n .
Ramsey’s Theorem can be extended to clopen sets on the Baire space.
Def. A set F ⊆ [ω]<ω is Nash-Williams if or a, b ∈ F, a 6< b.
Thm. (Nash-Williams) Every Nash-Williams set F ⊆ [ω]<ω is Ramsey:
Given a coloring c on a front F into 2 colors, there is an M ∈ [ω]ω suchthat c is monochromatic on F|M := {a ∈ F : a ⊆ M}.

DobrinenRamsey theory in forcing

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4 / 44Extensions of Ramsey’s Theorem to higher dimensions

The Nash-Williams Theorem was later extended by Galvin and Prikry to
all metrically Borel subsets of the Baire space.

DobrinenRamsey theory in forcing

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5 / 44Extensions of Ramsey’s Theorem to higher dimensions

The Nash-Williams Theorem was later extended by Galvin and Prikry to
all metrically Borel subsets of the Baire space.

Silver extended it to all metrically analytic subsets of the Baire space.

DobrinenRamsey theory in forcing

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5 / 44Extensions of Ramsey’s Theorem to higher dimensions

The Nash-Williams Theorem was later extended by Galvin and Prikry to
all metrically Borel subsets of the Baire space.

Silver extended it to all metrically analytic subsets of the Baire space.

The optimal extension of Ramsey’s Theorem to infinite dimensions isEllentuck’s Theorem.
This theorem uses a topology on the Baire space which refines the metric
topology.

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Ellentuck Space ([ω]ω , ⊆, r )Basis for topology: [a, X ] = {Y ∈ [ω]ω : a < Y ⊆ X }.
This is a refinement of the metric topology on the Baire space.

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Ellentuck Space ([ω]ω , ⊆, r )Basis for topology: [a, X ] = {Y ∈ [ω]ω : a < Y ⊆ X }.
This is a refinement of the metric topology on the Baire space.
Def. X ⊆ [ω]ω is Ramsey iff for each [a, X ], there is a < Y ⊆ X suchthat either [a, Y ] ⊆ X or [a, Y ] ∩ X = ∅.

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Ellentuck Space ([ω]ω , ⊆, r )Basis for topology: [a, X ] = {Y ∈ [ω]ω : a < Y ⊆ X }.
This is a refinement of the metric topology on the Baire space.
Def. X ⊆ [ω]ω is Ramsey iff for each [a, X ], there is a < Y ⊆ X suchthat either [a, Y ] ⊆ X or [a, Y ] ∩ X = ∅.
Thm. (Ellentuck) Every X ⊆ [ω]ω with the property of Baire isRamsey, and every meager set is Ramsey null.

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Connection with Mathias Forcing

Mathias forcing M has conditions ha, X i, where a ∈ [ω]<ω , X ∈ [ω]ω ,and max(a) < min(X ).
hb, Y i ≤ ha, X i iff b w a, Y ⊆ X , and b \ a ⊆ X .
Mathias forcing is equivalent to forcing using the basic open sets in the
Ellentuck space, ordered by ⊆.

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Connections with Forcing and Ultrafilters
Def. An ultrafilter U on ω is Ramsey if given any coloring c : [ω]n → l,there is a U ∈ U which is homogenous for c.

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Connections with Forcing and Ultrafilters
Def. An ultrafilter U on ω is Ramsey if given any coloring c : [ω]n → l,there is a U ∈ U which is homogenous for c.
P(ω)/fin, or equivalently, ([ω]ω , ⊆∗ ), forces a Ramsey ultrafilter.

DobrinenRamsey theory in forcing

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Connections with Forcing and Ultrafilters
Def. An ultrafilter U on ω is Ramsey if given any coloring c : [ω]n → l,there is a U ∈ U which is homogenous for c.
P(ω)/fin, or equivalently, ([ω]ω , ⊆∗ ), forces a Ramsey ultrafilter.
Ramsey ultrafilters have complete combinatorics. One way to statethis is that if there is a supercompact cardinal in V , then any Ramsey
ultrafilter in V is generic for the forcing ([ω]ω , ⊆∗ ) over the Solovay
model L(R).

DobrinenRamsey theory in forcing

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Connections with Forcing and Ultrafilters
Def. An ultrafilter U on ω is Ramsey if given any coloring c : [ω]n → l,there is a U ∈ U which is homogenous for c.
P(ω)/fin, or equivalently, ([ω]ω , ⊆∗ ), forces a Ramsey ultrafilter.
Ramsey ultrafilters have complete combinatorics. One way to statethis is that if there is a supercompact cardinal in V , then any Ramsey
ultrafilter in V is generic for the forcing ([ω]ω , ⊆∗ ) over the Solovay
model L(R).
For an ultrafilter U, let MU denote Mathias forcing where the tails are
members of U.

DobrinenRamsey theory in forcing

University of Denver

8 / 44

Connections with Forcing and Ultrafilters
Def. An ultrafilter U on ω is Ramsey if given any coloring c : [ω]n → l,there is a U ∈ U which is homogenous for c.
P(ω)/fin, or equivalently, ([ω]ω , ⊆∗ ), forces a Ramsey ultrafilter.
Ramsey ultrafilters have complete combinatorics. One way to statethis is that if there is a supercompact cardinal in V , then any Ramsey
ultrafilter in V is generic for the forcing ([ω]ω , ⊆∗ ) over the Solovay
model L(R).
For an ultrafilter U, let MU denote Mathias forcing where the tails are
members of U.
M is forcing equivalent to P(ω)/Fin ∗ MU̇ , where U is the Ramsey
ultrafilter forced by P(ω)/Fin.
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Key properties from the Ellentuck space can be abstracted to give ageneral notion of a topological Ramsey space.

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Abstract Topological Ramsey Spaces (R, ≤, r )R is a set. ≤ is quasi-order on R.
For each n, rn (·) := r (n, ·) is a restriction map on domain R giving then-th approximation to X .

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Abstract Topological Ramsey Spaces (R, ≤, r )R is a set. ≤ is quasi-order on R.
For each n, rn (·) := r (n, ·) is a restriction map on domain R giving then-th approximation to X .
ARn = {rn (X ) : X ∈ R}

DobrinenRamsey theory in forcing

AR =

S

n<ω ARn .

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Abstract Topological Ramsey Spaces (R, ≤, r )R is a set. ≤ is quasi-order on R.
For each n, rn (·) := r (n, ·) is a restriction map on domain R giving then-th approximation to X .
ARn = {rn (X ) : X ∈ R}

AR =

S

n<ω ARn .

For a ∈ AR, Y ∈ R, a < Y iff rn (Y ) = a for some n.

DobrinenRamsey theory in forcing

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Abstract Topological Ramsey Spaces (R, ≤, r )R is a set. ≤ is quasi-order on R.
For each n, rn (·) := r (n, ·) is a restriction map on domain R giving then-th approximation to X .
ARn = {rn (X ) : X ∈ R}

AR =

S

n<ω ARn .

For a ∈ AR, Y ∈ R, a < Y iff rn (Y ) = a for some n.
Basic open sets: [a, X ] = {Y ∈ R : a < Y ≤ X }.

DobrinenRamsey theory in forcing

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Abstract Topological Ramsey Spaces (R, ≤, r )R is a set. ≤ is quasi-order on R.
For each n, rn (·) := r (n, ·) is a restriction map on domain R giving then-th approximation to X .
ARn = {rn (X ) : X ∈ R}

AR =

S

n<ω ARn .

For a ∈ AR, Y ∈ R, a < Y iff rn (Y ) = a for some n.
Basic open sets: [a, X ] = {Y ∈ R : a < Y ≤ X }.
The topology on R generated
Q by the basic open sets is a refinement ofthe ‘metric topology’ on n<ω ARn .
DobrinenRamsey theory in forcing

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10 / 44The Axioms A.1 - A.4
A.1 (Sequencing)1r0 (A) = ∅ for all A ∈ R.

2

A 6= B implies rn (A) 6= rn (B) for some n.

3

rn (A) = rm (B) implies n = m and rk (A) = rk (B) for all k < n.

DobrinenRamsey theory in forcing

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11 / 44The Axioms A.1 - A.4
A.1 (Sequencing)1r0 (A) = ∅ for all A ∈ R.

2

A 6= B implies rn (A) 6= rn (B) for some n.

3

rn (A) = rm (B) implies n = m and rk (A) = rk (B) for all k < n.

A.2 (Finitization) There is a quasi-ordering ≤fin on AR such that1{a ∈ AR : a ≤fin b} is finite for all b ∈ AR,

2

A ≤ B iff (∀n)(∃m) rn (A) ≤fin rm (B),3∀a, b, c ∈ AR [a < b ∧ b ≤fin c → ∃d < c a ≤fin d].

DobrinenRamsey theory in forcing

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11 / 44The Axioms A.1 - A.4
A.1 (Sequencing)1r0 (A) = ∅ for all A ∈ R.

2

A 6= B implies rn (A) 6= rn (B) for some n.

3

rn (A) = rm (B) implies n = m and rk (A) = rk (B) for all k < n.

A.2 (Finitization) There is a quasi-ordering ≤fin on AR such that1{a ∈ AR : a ≤fin b} is finite for all b ∈ AR,

2

A ≤ B iff (∀n)(∃m) rn (A) ≤fin rm (B),3∀a, b, c ∈ AR [a < b ∧ b ≤fin c → ∃d < c a ≤fin d].

depthB (a) is the least n (if it exists) such that a ≤fin rn (B).
If depthB (a) = n, then [depthB (a), B] denotes [rn (B), B].
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A.3 (Amalgamation)
1
2

If depthB (a) < ∞ then [a, A] 6= ∅ for all A ∈ [depthB (a), B].
A ≤ B and [a, A] 6= ∅ imply that there is A0 ∈ [depthB (a), B] suchthat ∅ =6 [a, A0 ] ⊆ [a, A].

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A.3 (Amalgamation)
1
2

If depthB (a) < ∞ then [a, A] 6= ∅ for all A ∈ [depthB (a), B].
A ≤ B and [a, A] 6= ∅ imply that there is A0 ∈ [depthB (a), B] suchthat ∅ =6 [a, A0 ] ⊆ [a, A].

For a ∈ AR, |a| denotes the k such that a ∈ ARk .
If n > |a|, then rn [a, A] = {rn (X ) : X ∈ [a, A]}.

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A.3 (Amalgamation)
1
2

If depthB (a) < ∞ then [a, A] 6= ∅ for all A ∈ [depthB (a), B].
A ≤ B and [a, A] 6= ∅ imply that there is A0 ∈ [depthB (a), B] suchthat ∅ =6 [a, A0 ] ⊆ [a, A].

For a ∈ AR, |a| denotes the k such that a ∈ ARk .
If n > |a|, then rn [a, A] = {rn (X ) : X ∈ [a, A]}.

A.4 (Pigeonhole) Given B ∈ R and a ∈ AR with depthB (a) < ∞, thenfor any O ⊆ r|a|+1 [a, B], there is A ∈ [depthB (a), B] such thatr|a|+1 [a, A] ⊆ O or r|a|+1 [a, A] ⊆ Oc .

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Abstract Ellentuck Theorem
Def. X ⊆ R is Ramsey iff for each ∅ =6 [a, A], there is a B ∈ [a, A] suchthat either [a, B] ⊆ X or [a, B] ∩ X = ∅.

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Abstract Ellentuck Theorem
Def. X ⊆ R is Ramsey iff for each ∅ =6 [a, A], there is a B ∈ [a, A] suchthat either [a, B] ⊆ X or [a, B] ∩ X = ∅.
Def. (Todorcevic) A triple (R, ≤, r ) is a topological Ramsey space ifevery subset of R with the Baire property is Ramsey, and if every
meager subset of R is Ramsey null.

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Abstract Ellentuck Theorem
Def. X ⊆ R is Ramsey iff for each ∅ =6 [a, A], there is a B ∈ [a, A] suchthat either [a, B] ⊆ X or [a, B] ∩ X = ∅.
Def. (Todorcevic) A triple (R, ≤, r ) is a topological Ramsey space ifevery subset of R with the Baire property is Ramsey, and if every
meager subset of R is Ramsey null.
Building on prior work of Carlson and Simpson, Todorcevic proved an
abstract version of Ellentuck’s Theorem.

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Abstract Ellentuck Theorem
Def. X ⊆ R is Ramsey iff for each ∅ =6 [a, A], there is a B ∈ [a, A] suchthat either [a, B] ⊆ X or [a, B] ∩ X = ∅.
Def. (Todorcevic) A triple (R, ≤, r ) is a topological Ramsey space ifevery subset of R with the Baire property is Ramsey, and if every
meager subset of R is Ramsey null.
Building on prior work of Carlson and Simpson, Todorcevic proved an
abstract version of Ellentuck’s Theorem.
Abstract Ellentuck Thm. (Todorcevic) If (R, ≤, r ) satisfies Axioms
A.1 - A.4 and R is closed (in ARN ), then (R, ≤, r ) is a topologicalRamsey space.
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Topological Ramsey Spaces are useful for
Finding exact Tukey and Rudin-Keisler structures in βω.

DobrinenRamsey theory in forcing

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14 / 44

Topological Ramsey Spaces are useful for
Finding exact Tukey and Rudin-Keisler structures in βω.
Proving complete combinatorics for ultrafilters satisfying some weakpartition properties, and for some other structures.

DobrinenRamsey theory in forcing

University of Denver

14 / 44

Topological Ramsey Spaces are useful for
Finding exact Tukey and Rudin-Keisler structures in βω.
Proving complete combinatorics for ultrafilters satisfying some weakpartition properties, and for some other structures.
Proving finite Ramsey degrees for universal relational structures.

DobrinenRamsey theory in forcing

University of Denver

14 / 44

Topological Ramsey Spaces are useful for
Finding exact Tukey and Rudin-Keisler structures in βω.
Proving complete combinatorics for ultrafilters satisfying some weakpartition properties, and for some other structures.
Proving finite Ramsey degrees for universal relational structures.
Showing certain ultrafilters are preserved by side-by-side Sacks forcings.

DobrinenRamsey theory in forcing

University of Denver

14 / 44

Topological Ramsey Spaces are useful for
Finding exact Tukey and Rudin-Keisler structures in βω.
Proving complete combinatorics for ultrafilters satisfying some weakpartition properties, and for some other structures.
Proving finite Ramsey degrees for universal relational structures.
Showing certain ultrafilters are preserved by side-by-side Sacks forcings.
Solving problems in Banach spaces.

DobrinenRamsey theory in forcing

University of Denver

14 / 44

Topological Ramsey Spaces are useful for
Finding exact Tukey and Rudin-Keisler structures in βω.
Proving complete combinatorics for ultrafilters satisfying some weakpartition properties, and for some other structures.
Proving finite Ramsey degrees for universal relational structures.
Showing certain ultrafilters are preserved by side-by-side Sacks forcings.
Solving problems in Banach spaces.
Streamlining proofs and creating general frameworks for certain classes ofstructures and results.

DobrinenRamsey theory in forcing

University of Denver

14 / 44

Topological Ramsey Spaces are useful for
Finding exact Tukey and Rudin-Keisler structures in βω.
Proving complete combinatorics for ultrafilters satisfying some weakpartition properties, and for some other structures.
Proving finite Ramsey degrees for universal relational structures.
Showing certain ultrafilters are preserved by side-by-side Sacks forcings.
Solving problems in Banach spaces.
Streamlining proofs and creating general frameworks for certain classes ofstructures and results.
Motivating new Ramsey theorems.
DobrinenRamsey theory in forcing

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Standard Examples of Topological Ramsey Spaces
1

Ellentuck space2

Carlson-Simpson space of equivalence relations on ω with infinitely
many equivalence classes (dual Ramsey)

3Pröml-Voigt spaces of parameter words and ascending parameter
words

4

Milliken space of block sequences FINk

5

Carlson’s space of infinite dimensional vector spaces F N where F is a
finite field.

[∞]

DobrinenRamsey theory in forcing

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15 / 44

Standard Examples of Topological Ramsey Spaces
1

Ellentuck space2

Carlson-Simpson space of equivalence relations on ω with infinitely
many equivalence classes (dual Ramsey)

3Pröml-Voigt spaces of parameter words and ascending parameter
words

4

Milliken space of block sequences FINk

5

Carlson’s space of infinite dimensional vector spaces F N where F is a
finite field.

[∞]

D. and Mijares have an example schema which encompasses (2) - (5) as
special cases.

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Standard Examples of Topological Ramsey Spaces
1

Ellentuck space2

Carlson-Simpson space of equivalence relations on ω with infinitely
many equivalence classes (dual Ramsey)

3Pröml-Voigt spaces of parameter words and ascending parameter
words

4

Milliken space of block sequences FINk

5

Carlson’s space of infinite dimensional vector spaces F N where F is a
finite field.

[∞]

D. and Mijares have an example schema which encompasses (2) - (5) as
special cases.
All of these spaces (except the Ellentuck space) have ≤ essentially given
by a composition operator and are surjective spaces.
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Inner Topological Ramsey Spaces
Several classes of σ-closed forcings adding new ultrafilters have beenshown to contain dense subsets which form topological Ramsey spaces.

DobrinenRamsey theory in forcing

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Inner Topological Ramsey Spaces
Several classes of σ-closed forcings adding new ultrafilters have beenshown to contain dense subsets which form topological Ramsey spaces.
Like the Ellentuck space, the partial orderings ≤ are given by
structure-preserving injections.

DobrinenRamsey theory in forcing

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16 / 44

Inner Topological Ramsey Spaces
Several classes of σ-closed forcings adding new ultrafilters have beenshown to contain dense subsets which form topological Ramsey spaces.
Like the Ellentuck space, the partial orderings ≤ are given by
structure-preserving injections.
The motivation for this was to find exact Tukey and Rudin-Keisler
structures of ultrafilters. We will cover that tomorrow.

DobrinenRamsey theory in forcing

University of Denver

16 / 44

Inner Topological Ramsey Spaces
Several classes of σ-closed forcings adding new ultrafilters have beenshown to contain dense subsets which form topological Ramsey spaces.
Like the Ellentuck space, the partial orderings ≤ are given by
structure-preserving injections.
The motivation for this was to find exact Tukey and Rudin-Keisler
structures of ultrafilters. We will cover that tomorrow.
We will survey several of these topological Ramsey spaces and how they
were constructed from known partial orderings.

DobrinenRamsey theory in forcing

University of Denver

16 / 44

Inner Topological Ramsey Spaces
Several classes of σ-closed forcings adding new ultrafilters have beenshown to contain dense subsets which form topological Ramsey spaces.
Like the Ellentuck space, the partial orderings ≤ are given by
structure-preserving injections.
The motivation for this was to find exact Tukey and Rudin-Keisler
structures of ultrafilters. We will cover that tomorrow.
We will survey several of these topological Ramsey spaces and how they
were constructed from known partial orderings. First, a word about
ultrafilters forced by topological Ramsey spaces.

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Ultrafilters forced by topological Ramsey spaces

Mijares showed that every topological Ramsey space has an induced
σ-closed partial ordering of almost reduction ≤∗ , similarly to ⊆∗ on the
Ellentuck space.

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Ultrafilters forced by topological Ramsey spaces

Mijares showed that every topological Ramsey space has an induced
σ-closed partial ordering of almost reduction ≤∗ , similarly to ⊆∗ on the
Ellentuck space.
(R, ≤∗ ) forces a maximal generic filter on R.

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Ultrafilters forced by topological Ramsey spaces

Mijares showed that every topological Ramsey space has an induced
σ-closed partial ordering of almost reduction ≤∗ , similarly to ⊆∗ on the
Ellentuck space.
(R, ≤∗ ) forces a maximal generic filter on R.
The generic ultrafilter usually induces an ultrafilter on the base set AR1 .

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Selective Coideals and Complete Combinatorics
Given a topological Ramsey space (R, ≤, r ), a coideal U ⊆ R is selective
if for each A ∈ U and any collection (Aa )a∈AR|A of members of U  A,
there is a U ∈ U which diagonalizes (Aa )a∈AR|A .

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18 / 44

Selective Coideals and Complete Combinatorics
Given a topological Ramsey space (R, ≤, r ), a coideal U ⊆ R is selective
if for each A ∈ U and any collection (Aa )a∈AR|A of members of U  A,
there is a U ∈ U which diagonalizes (Aa )a∈AR|A .
Forcing with (R, ≤∗ ) adds a selective ultrafilter or coideal U on R.

DobrinenRamsey theory in forcing

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18 / 44

Selective Coideals and Complete Combinatorics
Given a topological Ramsey space (R, ≤, r ), a coideal U ⊆ R is selective
if for each A ∈ U and any collection (Aa )a∈AR|A of members of U  A,
there is a U ∈ U which diagonalizes (Aa )a∈AR|A .
Forcing with (R, ≤∗ ) adds a selective ultrafilter or coideal U on R.
Thm. (Mijares) each (R, ≤) is forcing equivalent to forcing first with(R, ≤∗ ) to obtain a generic ultrafilter U, and then forcing with the
σ-closed localized version (RU , ≤) where the tails are in U.

DobrinenRamsey theory in forcing

University of Denver

18 / 44

Selective Coideals and Complete Combinatorics
Given a topological Ramsey space (R, ≤, r ), a coideal U ⊆ R is selective
if for each A ∈ U and any collection (Aa )a∈AR|A of members of U  A,
there is a U ∈ U which diagonalizes (Aa )a∈AR|A .
Forcing with (R, ≤∗ ) adds a selective ultrafilter or coideal U on R.
Thm. (Mijares) each (R, ≤) is forcing equivalent to forcing first with(R, ≤∗ ) to obtain a generic ultrafilter U, and then forcing with the
σ-closed localized version (RU , ≤) where the tails are in U.
Thm. (Di Prisco/Mijares/Nieto) In the presence of a supercompact
cardinal, every selective coideal U ⊆ R is generic for (R, ≤∗ ) over L(R).
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Laflamme’s forcing to add a weakly Ramsey ultrafilter
P1 = ([ω]ω , ≤1 ).

DobrinenRamsey theory in forcing

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Laflamme’s forcing to add a weakly Ramsey ultrafilter
P1 = ([ω]ω , ≤1 ). For X , Y ∈ [ω]ω , enumerating them in increasingorder and in blocks of increasing size as
X = hx11 , x12 , x22 , x13 , x23 , x33 , . . .i and Y = hy11 , y12 , y22 , y13 , y23 , y33 , . . .i,m } ⊆ {x n , . . . x n }.
then Y ≤1 X iff ∀m ∃n such that {y1m , . . . , ymn1

DobrinenRamsey theory in forcing

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Laflamme’s forcing to add a weakly Ramsey ultrafilter
P1 = ([ω]ω , ≤1 ). For X , Y ∈ [ω]ω , enumerating them in increasingorder and in blocks of increasing size as
X = hx11 , x12 , x22 , x13 , x23 , x33 , . . .i and Y = hy11 , y12 , y22 , y13 , y23 , y33 , . . .i,m } ⊆ {x n , . . . x n }.
then Y ≤1 X iff ∀m ∃n such that {y1m , . . . , ymn1Note: Y ≤1 X =⇒ Y ⊆ X .

DobrinenRamsey theory in forcing

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Laflamme’s forcing to add a weakly Ramsey ultrafilter
P1 = ([ω]ω , ≤1 ). For X , Y ∈ [ω]ω , enumerating them in increasingorder and in blocks of increasing size as
X = hx11 , x12 , x22 , x13 , x23 , x33 , . . .i and Y = hy11 , y12 , y22 , y13 , y23 , y33 , . . .i,m } ⊆ {x n , . . . x n }.
then Y ≤1 X iff ∀m ∃n such that {y1m , . . . , ymn1Note: Y ≤1 X =⇒ Y ⊆ X .
Y ≤∗1 X iff ∀∞ n, the n-th block of Y is contained in some block of X .

DobrinenRamsey theory in forcing

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Laflamme’s forcing to add a weakly Ramsey ultrafilter
P1 = ([ω]ω , ≤1 ). For X , Y ∈ [ω]ω , enumerating them in increasingorder and in blocks of increasing size as
X = hx11 , x12 , x22 , x13 , x23 , x33 , . . .i and Y = hy11 , y12 , y22 , y13 , y23 , y33 , . . .i,m } ⊆ {x n , . . . x n }.
then Y ≤1 X iff ∀m ∃n such that {y1m , . . . , ymn1Note: Y ≤1 X =⇒ Y ⊆ X .
Y ≤∗1 X iff ∀∞ n, the n-th block of Y is contained in some block of X .
So (P1 , ≤∗1 ) is like ([ω]ω , ⊆∗ ) except the partial ordering is more restrictive.

DobrinenRamsey theory in forcing

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Thm. (Laflamme) (P1 , ≤∗1 ) forces a weakly Ramsey ultrafilter.
U is weakly Ramsey if for each finitary coloring c of [ω]2 , there is a
U ∈ U for which c takes on at most two colors on [U]2 .
U → (U)2k,2 .

DobrinenRamsey theory in forcing

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Thm. (Laflamme) (P1 , ≤∗1 ) forces a weakly Ramsey ultrafilter.
U is weakly Ramsey if for each finitary coloring c of [ω]2 , there is a
U ∈ U for which c takes on at most two colors on [U]2 .
U → (U)2k,2 .

If κ is a Mahlo cardinal and G is Levy(κ)-generic over V , then any
ultrafilter U on ω in V [G ] which is not Ramsey but is rapid andsatisfies RP(k) for all k is generic over HOD(R)V [G ] (the original formof ‘complete combinatorics’).

DobrinenRamsey theory in forcing

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Laflamme’s forcing (P1 , ≤1 ). Example: Y ≤1 X

0

1 3

4 5 6

8 10 11 12 13 15 17 19 20 24 27 29 31 37 42
Figure: X ∈ [ω]ωDobrinenRamsey theory in forcing

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Laflamme’s forcing (P1 , ≤1 ). Example: Y ≤1 X

0

1 3

4 5 6

8 10 11 12 13 15 17 19 20 24 27 29 31 37 42
Figure: X ∈ [ω]ω3

13 15 17 19 20

27 29 31 37

Figure: Y ∈ [ω]ω & Y ≤1 X

DobrinenRamsey theory in forcing

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21 / 44The topological Ramsey space dense in (P1 , ≤1 )0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0

1

2

3

4

5

Figure: The maximum member of R1 .

DobrinenRamsey theory in forcing

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22 / 44The topological Ramsey space dense in (P1 , ≤1 )0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0

1

2
2 3

3
5 6

4
8 9

5
16 17 18 19

Y
1

2
2

3
6

9

5
16 17

19

X
1

3

5

Figure: Two members X and Y of R1 with X ≤ Y .
DobrinenRamsey theory in forcing

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23 / 44A subtree not in R1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0

1

2

3

4

2

5
15 16

18 19 20Z
1

5
Figure: Z 6∈ R1

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Thm. (D./Todorcevic) (R1 , ≤) is a topological Ramsey space and isdense below any member of P1 .

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Thm. (D./Todorcevic) (R1 , ≤) is a topological Ramsey space and isdense below any member of P1 .

The generic filter forced by (R1 , ≤∗ ) induces an ultrafilter on AR1 .
Call it U1 . U1 is weakly Ramsey: U1 → (U1 )2k,2 .

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Thm. (D./Todorcevic) (R1 , ≤) is a topological Ramsey space and isdense below any member of P1 .

The generic filter forced by (R1 , ≤∗ ) induces an ultrafilter on AR1 .
Call it U1 . U1 is weakly Ramsey: U1 → (U1 )2k,2 .

Prop. For each n ≥ 2, U1 → (U1 )nk,2n−1 .
This is stated in [Laflamme 89]. An elegant proof using topologicalRamsey space methods is given in Navarro Flores’ Masters Thesis.

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Thm. (D./Todorcevic) (R1 , ≤) is a topological Ramsey space and isdense below any member of P1 .

The generic filter forced by (R1 , ≤∗ ) induces an ultrafilter on AR1 .
Call it U1 . U1 is weakly Ramsey: U1 → (U1 )2k,2 .

Prop. For each n ≥ 2, U1 → (U1 )nk,2n−1 .
This is stated in [Laflamme 89]. An elegant proof using topologicalRamsey space methods is given in Navarro Flores’ Masters Thesis.
Exercise. Prove A.4 for the space R1 .

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Thm. (D./Todorcevic) (R1 , ≤) is a topological Ramsey space and isdense below any member of P1 .

The generic filter forced by (R1 , ≤∗ ) induces an ultrafilter on AR1 .
Call it U1 . U1 is weakly Ramsey: U1 → (U1 )2k,2 .

Prop. For each n ≥ 2, U1 → (U1 )nk,2n−1 .
This is stated in [Laflamme 89]. An elegant proof using topologicalRamsey space methods is given in Navarro Flores’ Masters Thesis.
Exercise. Prove A.4 for the space R1 .
Ramsey spaces Rα dense inside Laflamme’s forcings Pα , α < ω1 , werealso constructed in [D./Todorcevic 15].
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25 / 44The n-square forcing of Blass
A subset of ω × ω of the form s × t is an n-square if |s| = |t| = n.
X ⊆ ω × ω is in Pn−square iff for each n < ω, X contains and n × n-square.
For X , Y ∈ Pn−square , Y ≤ X iff Y ⊆ X .

DobrinenRamsey theory in forcing

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26 / 44The n-square forcing of Blass
A subset of ω × ω of the form s × t is an n-square if |s| = |t| = n.
X ⊆ ω × ω is in Pn−square iff for each n < ω, X contains and n × n-square.
For X , Y ∈ Pn−square , Y ≤ X iff Y ⊆ X .

Thm. (Blass) Pn−square forces a p-point with two Rudin-Keisler
incomparable predecessors.

DobrinenRamsey theory in forcing

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26 / 44The n-square forcing of Blass
A subset of ω × ω of the form s × t is an n-square if |s| = |t| = n.
X ⊆ ω × ω is in Pn−square iff for each n < ω, X contains and n × n-square.
For X , Y ∈ Pn−square , Y ≤ X iff Y ⊆ X .

Thm. (Blass) Pn−square forces a p-point with two Rudin-Keisler
incomparable predecessors.

Def. U is a p-point if each sequence X0 ⊇∗ X1 ⊇∗ . . . of members of U
has a pseudointersection U ∈ U; i.e. U ⊆∗ Xi for all i.

DobrinenRamsey theory in forcing

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26 / 44The Ramsey space H2 dense in the n-square forcing

9
8
7
6
5
4
3
2
1
0

14
13
12
11
10
10 11 12 13 14

6 7 8 9

3 4 5

1 2

0
Figure: The maximum member of H2 .

DobrinenRamsey theory in forcing

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27 / 44The Ramsey space H2 dense in the n-square forcing

9
8
7
6
5
4
3
2
1
0

14
13
12
11
10
10 11 12 13 14

6 7 8 9

3 4 5

1 2

0
Figure: A member of X in H2 .

DobrinenRamsey theory in forcing

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28 / 44The Ramsey space H2 dense in the n-square forcing

9
8
7
6
5
4
3
2
1
0

14
13
12
11
10
10 11 12 13 14

6 7 8 9

3 4 5

1 2

0

Figure: Two members Y and X of H2 with Y ≤ X
DobrinenRamsey theory in forcing

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29 / 44The Hypercube topological Ramsey spaces
Thm. (D./Trujillo) (H2 , ≤, r ) is a topological Ramsey space.

DobrinenRamsey theory in forcing

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30 / 44The Hypercube topological Ramsey spaces
Thm. (D./Trujillo) (H2 , ≤, r ) is a topological Ramsey space.
Let V2 denote the ultrafilter forced by (H2 , ≤). Then V2 → (V2 )2k,6 .

DobrinenRamsey theory in forcing

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30 / 44The Hypercube topological Ramsey spaces
Thm. (D./Trujillo) (H2 , ≤, r ) is a topological Ramsey space.
Let V2 denote the ultrafilter forced by (H2 , ≤). Then V2 → (V2 )2k,6 .
Rem 1. H2 behaves like a product of two copies of R1 ; each copy of
R1 is recovered by projection maps.

DobrinenRamsey theory in forcing

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30 / 44The Hypercube topological Ramsey spaces
Thm. (D./Trujillo) (H2 , ≤, r ) is a topological Ramsey space.
Let V2 denote the ultrafilter forced by (H2 , ≤). Then V2 → (V2 )2k,6 .
Rem 1. H2 behaves like a product of two copies of R1 ; each copy of
R1 is recovered by projection maps.
Rem 2. Higher dimensional hypercube spaces were constructed in[D./Mijares/Trujillo] including a space where the dimension of the n-th
block is n + 1.

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Arrow Ultrafilters of Baumgartner and Taylor
Def. An ultrafilter U is k-arrow if U → (U, k)2 .

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Arrow Ultrafilters of Baumgartner and Taylor
Def. An ultrafilter U is k-arrow if U → (U, k)2 .
Thm. (Baumgartner/Taylor) There are forcings which constructultrafilters Wk+1 which are k-arrow but not (k + 1)-arrow ultrafilters,
for all k ≥ 2.

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Arrow Ultrafilters of Baumgartner and Taylor
Def. An ultrafilter U is k-arrow if U → (U, k)2 .
Thm. (Baumgartner/Taylor) There are forcings which constructultrafilters Wk+1 which are k-arrow but not (k + 1)-arrow ultrafilters,
for all k ≥ 2.
Thm. (D./Mijares/Trujillo) For each k ≥ 2, there is a topologicalRamsey space Ak+1 which is dense in the Baumgartner-Taylor partial
order forcing a k-arrow, not (k + 1)-arrow ultrafilter.

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Arrow Ultrafilters of Baumgartner and Taylor
Def. An ultrafilter U is k-arrow if U → (U, k)2 .
Thm. (Baumgartner/Taylor) There are forcings which constructultrafilters Wk+1 which are k-arrow but not (k + 1)-arrow ultrafilters,
for all k ≥ 2.
Thm. (D./Mijares/Trujillo) For each k ≥ 2, there is a topologicalRamsey space Ak+1 which is dense in the Baumgartner-Taylor partial
order forcing a k-arrow, not (k + 1)-arrow ultrafilter.
Exercise. Prove A.4 for A3 .

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31 / 44The topological Ramsey space A3
Let K3 be the Fraı̈sse class of finite ordered triangle-free graphs.

DobrinenRamsey theory in forcing

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32 / 44The topological Ramsey space A3
Let K3 be the Fraı̈sse class of finite ordered triangle-free graphs.

Block structure for A3···
Block 0

1

2

3

Figure: A member X of A3

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32 / 44The topological Ramsey space A3
Let K3 be the Fraı̈sse class of finite ordered triangle-free graphs.

Block structure for A3···
Block 0

1

2

3

Figure: A member X of A3

block n + 2 → (block n + 1)block n

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32 / 44The topological Ramsey space A3
Let K3 be the Fraı̈sse class of finite ordered triangle-free graphs.

Block structure for A3···
Block 0

1

2

3

Figure: Example: Two members Y and X of A3 with Y ≤ X

block n + 2 → (block n + 1)block n

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Ramsey degrees for these forced ultrafilters (in [DMT])The space (A3 , ≤∗ ) forces a 2-arrow but not 3-arrow ultrafilter, W3 , thesame one as Baumgarter and Taylor.

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Ramsey degrees for these forced ultrafilters (in [DMT])The space (A3 , ≤∗ ) forces a 2-arrow but not 3-arrow ultrafilter, W3 , thesame one as Baumgarter and Taylor. The Ramsey degrees are
W3 → (W3 )2k,3
W3 → (W3 )3k,12W3 → (W3 )4k,35

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Ramsey degrees for these forced ultrafilters (in [DMT])The space (A3 , ≤∗ ) forces a 2-arrow but not 3-arrow ultrafilter, W3 , thesame one as Baumgarter and Taylor. The Ramsey degrees are
W3 → (W3 )2k,3
W3 → (W3 )3k,12W3 → (W3 )4k,35(1)

More generally, any product K of finitely many Fraı̈ssé classes of finite
ordered relational structures with the Ramsey property can be used tocompose the block structures of members of a topological Ramsey space.

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Ramsey degrees for these forced ultrafilters (in [DMT])The space (A3 , ≤∗ ) forces a 2-arrow but not 3-arrow ultrafilter, W3 , thesame one as Baumgarter and Taylor. The Ramsey degrees are
W3 → (W3 )2k,3
W3 → (W3 )3k,12W3 → (W3 )4k,35(1)

More generally, any product K of finitely many Fraı̈ssé classes of finite
ordered relational structures with the Ramsey property can be used tocompose the block structures of members of a topological Ramsey space.
The topological Ramsey space structure allows one to find formulas for thenumbers d such that UK → (UK )mk,d , for such forced ultrafilters.
DobrinenRamsey theory in forcing

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Topological Ramsey spaces dense in creature forcings

Observation (Todorcevic). There are strong connections between
creature forcings and topological Ramsey spaces deserving of asystematic study.

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Topological Ramsey spaces dense in creature forcings

Observation (Todorcevic). There are strong connections between
creature forcings and topological Ramsey spaces deserving of asystematic study.
Question. Which creature forcings are essentially topological Ramsey
spaces?

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Topological Ramsey spaces dense in creature forcings

Observation (Todorcevic). There are strong connections between
creature forcings and topological Ramsey spaces deserving of asystematic study.
Question. Which creature forcings are essentially topological Ramsey
spaces?
In [D. TopApp 16], we proved that three of the examples of purecandidates for creature forcings given in [Roslanowski/Shelah 13]
contain dense subsets which are topological Ramsey spaces.

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Let H be any function with dom(H) = ω such that H(n) is a finite
non-empty set for each n < ω.

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Let H be any function with dom(H) = ω such that H(n) is a finite
non-empty set for each n < ω.
[ Y
FH =
H(n).
n∈u
u∈FIN

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Let H be any function with dom(H) = ω such that H(n) is a finite
non-empty set for each n < ω.
[ Y
FH =
H(n).
n∈u
u∈FIN
pure candidates are certain infinite sequences t̄ of creatures (finite
structures). pos(t̄) is an infinite subset of FH induced by t̄.

DobrinenRamsey theory in forcing

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Let H be any function with dom(H) = ω such that H(n) is a finite
non-empty set for each n < ω.
[ Y
FH =
H(n).
n∈u
u∈FIN
pure candidates are certain infinite sequences t̄ of creatures (finite
structures). pos(t̄) is an infinite subset of FH induced by t̄.
Thm. (Roslanowski/Shelah) (CH) There is an ultrafilter U on base set
FH generated by {pos(t̄α ) : α < ω1 } for a decreasing sequence of purecandidates ht̄α : α < ω1 i satisfying the partition theorem:
For any t̄ such that pos(t̄ ) ∈ U and any partition of pos(t̄ ) into finitely
many pieces, there is a pure candidate s̄ ≤ t̄ such that pos(s̄) is
contained in one piece of the partition and pos(s̄) ∈ U.
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Remark 1. This generalizes the construction of a stable-ordered union
ultrafilter on FIN using Hindman’s Theorem.

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Remark 1. This generalizes the construction of a stable-ordered union
ultrafilter on FIN using Hindman’s Theorem.
Remark 2. The proofs in [RS] use the Galvin-Glazer method extended
to certain classes of creature forcings.

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Thm. (D.) Three examples of sets of pure candidates in [RS] containdense subsets forming topological Ramsey spaces.

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Thm. (D.) Three examples of sets of pure candidates in [RS] containdense subsets forming topological Ramsey spaces.
For two of these spaces, the pigeonhole principles rely on the following
product tree Ramsey theorem.

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38 / 44New Product Tree Ramsey Theorem
LetY

Kj = K0 × · · · × Kp−1 × [Kp ]k × Kp+1 × · · · × Kn .

j∈n+1,pDobrinenRamsey theory in forcing

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39 / 44New Product Tree Ramsey Theorem
LetY

Kj = K0 × · · · × Kp−1 × [Kp ]k × Kp+1 × · · · × Kn .

j∈n+1,pThm. (D.) Given k ≥ 1, a sequence of positive integers (m0 , m1 , . . . ),sets Kj , j < ω such that |Kj | ≥ j + 1, and a coloring[ [ Yc:
Kj → 2,n<ω p≤n j∈n+1,pDobrinenRamsey theory in forcing

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39 / 44New Product Tree Ramsey Theorem
LetY

Kj = K0 × · · · × Kp−1 × [Kp ]k × Kp+1 × · · · × Kn .

j∈n+1,pThm. (D.) Given k ≥ 1, a sequence of positive integers (m0 , m1 , . . . ),sets Kj , j < ω such that |Kj | ≥ j + 1, and a coloring[ [ Yc:
Kj → 2,n<ω p≤n j∈n+1,pthere are infinite sets L, N ⊆ ω such that l0 ≤ n0 < l1 ≤ n1 < . . . , andthere are Hj ⊆ Kj , j < ω, such that |Hli | = mi for each i < ω, |Hj | = 1
for each j ∈ ω \ L, and c is constant on
[
[
Y
Hj .
n∈N l∈L∩(n+1) j∈n+1,lDobrinenRamsey theory in forcing

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39 / 44The proof built on the following Product Tree Theorem.

DobrinenRamsey theory in forcing

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40 / 44The proof built on the following Product Tree Theorem.
Thm. (Di Prisco/Llopis/Todorcevic) There is an R : (N+ )<ω → N+such that for every infinite sequence (mj )j<ω of positive integers and forevery coloring[Yc:
R(m0 , . . . , mj ) → 2,n<ω j≤nthere exist Hj ⊆ R(m0 , . . Q. , mj ), |Hj | = mj , for j < ω, such that c isconstant on the product j≤n Hj for infinitely many n < ω.

DobrinenRamsey theory in forcing

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40 / 44The proof built on the following Product Tree Theorem.
Thm. (Di Prisco/Llopis/Todorcevic) There is an R : (N+ )<ω → N+such that for every infinite sequence (mj )j<ω of positive integers and forevery coloring[Yc:
R(m0 , . . . , mj ) → 2,n<ω j≤nthere exist Hj ⊆ R(m0 , . . Q. , mj ), |Hj | = mj , for j < ω, such that c isconstant on the product j≤n Hj for infinitely many n < ω.
Remark. The difference is that we need sets of size k to be able tomove up and down indices of the product.

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Conclusion for Day 1

Topological Ramsey spaces provide a unifying framework for many
ultrafilters satisfying partition properties and, moreover, yield ultrafilterswith complete combinatorics. They also motivate new Ramsey theorems.

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Conclusion for Day 1

Topological Ramsey spaces provide a unifying framework for many
ultrafilters satisfying partition properties and, moreover, yield ultrafilterswith complete combinatorics. They also motivate new Ramsey theorems.
With further work (tomorrow), we obtain:1

New canonical equivalence relations on fronts and barriers.

DobrinenRamsey theory in forcing

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Conclusion for Day 1

Topological Ramsey spaces provide a unifying framework for many
ultrafilters satisfying partition properties and, moreover, yield ultrafilterswith complete combinatorics. They also motivate new Ramsey theorems.
With further work (tomorrow), we obtain:1

New canonical equivalence relations on fronts and barriers.

2

Exact Rudin-Keisler and Tukey structures as well as the structure of
the Rudin-Keisler classes inside the Tukey types.

DobrinenRamsey theory in forcing

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41 / 44References for Day 1
Baumgartner/Taylor, Partition Theorems and Ultrafilters, TAMS (1978).
Blass, The Rudin-Keisler ordering of P-points, TAMS (1973).
Carlson/Simpson, Topological Ramsey theory Springer (1990).
Di Prisco/Llopis/Todorcevic, Parametrized partitions of products of finite
sets, Combinatorica (2004).
Di Prisco/Mijares/Nieto, Local Ramsey theory. An abstract approach(submitted).
Dobrinen, Creature forcing and topological Ramsey spaces, Topology and
Its Appl. (2016)
Dobrinen/Mijares, Topological Ramsey spaces and metrically Baire sets,
Jour. Combinatorial Theory, Series A (2015).
DobrinenRamsey theory in forcing

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42 / 44References for Day 1
Dobrinen/Mijares/Trujillo, Topological Ramsey spaces from Fraı̈ssé classes
and initial Tukey structures, AFML (to appear).
Dobrinen/Todorcevic, Ramsey-Classification Theorems and theirapplications in the Tukey theory of ultrafilters, Parts 1 and 2, TAMS
(2014, 2015).
Ellentuck, A new proof that analytic sets are Ramsey JSL (1974).
Galvin/Prikry, Borel sets and Ramsey’s Theorem JSL (1973).
Laflamme, Forcing with filters and complete combinatorics APAL (1989).
Mathias, Happy families Ann. Math. Logic (1977).
Mijares, A notion of selective ultrafilter corresponding to topologicalRamsey spaces, MLQ (2007).
DobrinenRamsey theory in forcing

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43 / 44References for Day 1
Nash-Williams, On well-quasi-ordering transfinite sequences, Proc.
Cambridge Phil. Soc. (1965).
Ramsey, On a problem of formal logic, Proc. London MathematicalSociety (1929).
Roslanowski/Shelah, Partition theorems from creatures and idempotent
ultrafilters Ann. Combinatorics (2013).
Silver, Some applications of model theory in set theory, Ann.
Mathematical Logic (1971).
Todorcevic, Introduction to Ramsey spaces, Princeton University Press,(2010).
Trujillo, PhD Thesis, Topological Ramsey spaces, associated ultrafilters,
and their applications to the Tukey theory of ultrafilters and Dedekind cutsof nonstandard arithmetic, (2014).
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