An abstract approach to finite Ramsey theory with applications S - - PowerPoint PPT Presentation

An abstract approach to finite Ramsey theory with applications

S lawomir Solecki

University of Illinois at Urbana–Champaign

January 2013

Outline

Outline of Topics

1

Introduction

2

Algebraic notions

3

Abstract Ramsey and abstract pigeonhole statements

4

The theorem

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 2 / 43

Introduction

Introduction

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 3 / 43

Introduction

Ramsey theory has become important in topological dynamics of Polish, not necessarily locally compact, groups, particularly after close connections between Ramsey theory and extreme amenability were found by Kechris–Pestov–Todorcevic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 4 / 43

Introduction

Ramsey theory has become important in topological dynamics of Polish, not necessarily locally compact, groups, particularly after close connections between Ramsey theory and extreme amenability were found by Kechris–Pestov–Todorcevic. Finite Ramsey theory splits into unstructured (induction) and structural (amalgamation) Ramsey theory.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 4 / 43

Introduction

Ramsey theory has become important in topological dynamics of Polish, not necessarily locally compact, groups, particularly after close connections between Ramsey theory and extreme amenability were found by Kechris–Pestov–Todorcevic. Finite Ramsey theory splits into unstructured (induction) and structural (amalgamation) Ramsey theory. I will outline an approach that recovers most of the unstructured Ramsey theory and makes it possible to prove new results.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 4 / 43

Introduction

For a natural number n, let [n] = {1, . . . , n}; in particular, [0] = ∅.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 5 / 43

Introduction

For a natural number n, let [n] = {1, . . . , n}; in particular, [0] = ∅. The classical Ramsey theorem:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 5 / 43

Introduction

For a natural number n, let [n] = {1, . . . , n}; in particular, [0] = ∅. The classical Ramsey theorem: Given d and k ≤ l,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 5 / 43

Introduction

For a natural number n, let [n] = {1, . . . , n}; in particular, [0] = ∅. The classical Ramsey theorem: Given d and k ≤ l, there exists m such that

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 5 / 43

Introduction

For a natural number n, let [n] = {1, . . . , n}; in particular, [0] = ∅. The classical Ramsey theorem: Given d and k ≤ l, there exists m such that for each d-coloring, that is, a coloring with d colors, of all k element subsets of [m],

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 5 / 43

Introduction

For a natural number n, let [n] = {1, . . . , n}; in particular, [0] = ∅. The classical Ramsey theorem: Given d and k ≤ l, there exists m such that for each d-coloring, that is, a coloring with d colors, of all k element subsets of [m], there exists an l element subset z of [m] such that

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 5 / 43

Introduction

For a natural number n, let [n] = {1, . . . , n}; in particular, [0] = ∅. The classical Ramsey theorem: Given d and k ≤ l, there exists m such that for each d-coloring, that is, a coloring with d colors, of all k element subsets of [m], there exists an l element subset z of [m] such that all k element subsets of z have the same color.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 5 / 43

Introduction

We reveal the formal algebraic structure underlying finite pure Ramsey theorems.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 6 / 43

Introduction

We reveal the formal algebraic structure underlying finite pure Ramsey theorems. We formulate within this approach an abstract pigeonhole principle and an abstract Ramsey theorem, and prove that the pigeonhole principle implies the Ramsey theorem.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 6 / 43

Introduction

The following theorems are instances of the general result:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 7 / 43

Introduction

The following theorems are instances of the general result: — the classical Ramsey theorem,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 7 / 43

Introduction

The following theorems are instances of the general result: — the classical Ramsey theorem, — the Hales–Jewett theorem,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 7 / 43

Introduction

The following theorems are instances of the general result: — the classical Ramsey theorem, — the Hales–Jewett theorem, — the Graham–Rothschild theorem,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 7 / 43

Introduction

The following theorems are instances of the general result: — the classical Ramsey theorem, — the Hales–Jewett theorem, — the Graham–Rothschild theorem, — the versions of these results for partial rigid surjections due to Voigt,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 7 / 43

Introduction

The following theorems are instances of the general result: — the classical Ramsey theorem, — the Hales–Jewett theorem, — the Graham–Rothschild theorem, — the versions of these results for partial rigid surjections due to Voigt, — a new self-dual Ramsey theorem,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 7 / 43

Introduction

The following theorems are instances of the general result: — the classical Ramsey theorem, — the Hales–Jewett theorem, — the Graham–Rothschild theorem, — the versions of these results for partial rigid surjections due to Voigt, — a new self-dual Ramsey theorem, — the Milliken Ramsey theorem for finite trees,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 7 / 43

Introduction

The following theorems are instances of the general result: — the classical Ramsey theorem, — the Hales–Jewett theorem, — the Graham–Rothschild theorem, — the versions of these results for partial rigid surjections due to Voigt, — a new self-dual Ramsey theorem, — the Milliken Ramsey theorem for finite trees, — a new common generalization of Deuber’s and Jasi´ nski’s Ramsey theorems for finite trees,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 7 / 43

Introduction

The following theorems are instances of the general result: — the classical Ramsey theorem, — the Hales–Jewett theorem, — the Graham–Rothschild theorem, — the versions of these results for partial rigid surjections due to Voigt, — a new self-dual Ramsey theorem, — the Milliken Ramsey theorem for finite trees, — a new common generalization of Deuber’s and Jasi´ nski’s Ramsey theorems for finite trees, — Spencer’s generalization of the Graham–Rothschild theorem and the Ramsey theorem for affine subspaces (Min Zhao).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 7 / 43

Introduction

The following theorems are instances of the general result: — the classical Ramsey theorem, — the Hales–Jewett theorem, — the Graham–Rothschild theorem, — the versions of these results for partial rigid surjections due to Voigt, — a new self-dual Ramsey theorem, — the Milliken Ramsey theorem for finite trees, — a new common generalization of Deuber’s and Jasi´ nski’s Ramsey theorems for finite trees, — Spencer’s generalization of the Graham–Rothschild theorem and the Ramsey theorem for affine subspaces (Min Zhao).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 8 / 43

Introduction

Some consequences of the approach:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 9 / 43

Introduction

Some consequences of the approach: — new concrete Ramsey results;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 9 / 43

Introduction

Some consequences of the approach: — new concrete Ramsey results; — a hierarchy of the Ramsey results according to the number of times the abstract Ramsey theorem is applied in their proofs

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 9 / 43

Introduction

Some consequences of the approach: — new concrete Ramsey results; — a hierarchy of the Ramsey results according to the number of times the abstract Ramsey theorem is applied in their proofs: the classical Ramsey theorem requires one application, the Hales–Jewett theorem requires two, the Graham–Rothschild theorem three, and the new results four;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 9 / 43

Introduction

Some consequences of the approach: — new concrete Ramsey results; — a hierarchy of the Ramsey results according to the number of times the abstract Ramsey theorem is applied in their proofs: the classical Ramsey theorem requires one application, the Hales–Jewett theorem requires two, the Graham–Rothschild theorem three, and the new results four; — a possibility of classifying concrete Ramsey theorems.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 9 / 43

Introduction

The self-dual Ramsey theorem:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction

The self-dual Ramsey theorem: R a partition of [n], C a subset of [n], m ∈ N.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction

The self-dual Ramsey theorem: R a partition of [n], C a subset of [n], m ∈ N. (R, C) is an m-connection if R and C have m elements each and, upon listing R as R1, . . . , Rm with min Ri < min Ri+1 and C as c1, . . . , cm with ci < ci+1

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction

The self-dual Ramsey theorem: R a partition of [n], C a subset of [n], m ∈ N. (R, C) is an m-connection if R and C have m elements each and, upon listing R as R1, . . . , Rm with min Ri < min Ri+1 and C as c1, . . . , cm with ci < ci+1, we have ci ∈ Ri for i ≤ m and ci < min Ri+1 for i < m.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction

The self-dual Ramsey theorem: R a partition of [n], C a subset of [n], m ∈ N. (R, C) is an m-connection if R and C have m elements each and, upon listing R as R1, . . . , Rm with min Ri < min Ri+1 and C as c1, . . . , cm with ci < ci+1, we have ci ∈ Ri for i ≤ m and ci < min Ri+1 for i < m. An l-connection (Q, B) is an l-subconnection of an m-connection (R, C)

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction

The self-dual Ramsey theorem: R a partition of [n], C a subset of [n], m ∈ N. (R, C) is an m-connection if R and C have m elements each and, upon listing R as R1, . . . , Rm with min Ri < min Ri+1 and C as c1, . . . , cm with ci < ci+1, we have ci ∈ Ri for i ≤ m and ci < min Ri+1 for i < m. An l-connection (Q, B) is an l-subconnection of an m-connection (R, C) if R is a coarser partition than Q and B ⊆ C.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 10 / 43

Introduction

Theorem (S.) Let d > 0. For k, l ∈ N there exists m ∈ N such that for each d-coloring

• f all k-subconnections of an m-connection (R, C)

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 11 / 43

Introduction

Theorem (S.) Let d > 0. For k, l ∈ N there exists m ∈ N such that for each d-coloring

• f all k-subconnections of an m-connection (R, C) there exists an

l-subconnection (Q, B) of (R, C) such that all k-subconnections of (Q, B) get the same color.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 11 / 43

Algebraic notions

Algebraic notions

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 12 / 43

Algebraic notions

Abstract Ramsey statement:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 13 / 43

Algebraic notions

Abstract Ramsey statement: given P find F for which F × P ∋ (f , x) → f . x ∈ F . P is defined;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 13 / 43

Algebraic notions

Abstract Ramsey statement: given P find F for which F × P ∋ (f , x) → f . x ∈ F . P is defined; color F . P;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 13 / 43

Algebraic notions

Abstract Ramsey statement: given P find F for which F × P ∋ (f , x) → f . x ∈ F . P is defined; color F . P; find f0 ∈ F with f0 . P monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 13 / 43

Algebraic notions

Restatement of the classical Ramsey theorem:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions

Restatement of the classical Ramsey theorem: Identify p element subsets of [q] with increasing injections from [p] to [q].

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions

Restatement of the classical Ramsey theorem: Identify p element subsets of [q] with increasing injections from [p] to [q]. Fix natural numbers d and k ≤ l.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions

Restatement of the classical Ramsey theorem: Identify p element subsets of [q] with increasing injections from [p] to [q]. Fix natural numbers d and k ≤ l. Let P be the set of all increasing injections from [k] to [l].

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions

Restatement of the classical Ramsey theorem: Identify p element subsets of [q] with increasing injections from [p] to [q]. Fix natural numbers d and k ≤ l. Let P be the set of all increasing injections from [k] to [l]. There is an m such that

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions

Restatement of the classical Ramsey theorem: Identify p element subsets of [q] with increasing injections from [p] to [q]. Fix natural numbers d and k ≤ l. Let P be the set of all increasing injections from [k] to [l]. There is an m such that for the set F of all increasing injections from [l] to [m], if we d-color the set {f ◦ x : f ∈ F, x ∈ P} = all increasing injections from [k] to [m],

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions

Restatement of the classical Ramsey theorem: Identify p element subsets of [q] with increasing injections from [p] to [q]. Fix natural numbers d and k ≤ l. Let P be the set of all increasing injections from [k] to [l]. There is an m such that for the set F of all increasing injections from [l] to [m], if we d-color the set {f ◦ x : f ∈ F, x ∈ P} = all increasing injections from [k] to [m], then there exists f0 ∈ F such that {f0 ◦ x : x ∈ P} is monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 14 / 43

Algebraic notions

Normed backgrounds

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 15 / 43

Algebraic notions

Let (A, ·, ∂, | · |, o) be such that

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 16 / 43

Algebraic notions

Let (A, ·, ∂, | · |, o) be such that — · is a partial function from A × A to A (multiplication);

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 16 / 43

Algebraic notions

Let (A, ·, ∂, | · |, o) be such that — · is a partial function from A × A to A (multiplication); — ∂ is a function from A to A (truncation);

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 16 / 43

Algebraic notions

Let (A, ·, ∂, | · |, o) be such that — · is a partial function from A × A to A (multiplication); — ∂ is a function from A to A (truncation); — | · | is a function from A to a linearly ordered set (L, ≤) (norm);

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 16 / 43

Algebraic notions

Let (A, ·, ∂, | · |, o) be such that — · is a partial function from A × A to A (multiplication); — ∂ is a function from A to A (truncation); — | · | is a function from A to a linearly ordered set (L, ≤) (norm); — o is an element of A (zero).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 16 / 43

Algebraic notions

Such a structure with associative multiplication is called a normed background provided that for a, b, c ∈ A:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 17 / 43

Algebraic notions

Such a structure with associative multiplication is called a normed background provided that for a, b, c ∈ A: (i) if a · b and a · ∂b are defined, then ∂(a · b) = a · ∂b;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 17 / 43

Algebraic notions

Such a structure with associative multiplication is called a normed background provided that for a, b, c ∈ A: (i) if a · b and a · ∂b are defined, then ∂(a · b) = a · ∂b; (ii) |∂a| ≤ |a|;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 17 / 43

Algebraic notions

Such a structure with associative multiplication is called a normed background provided that for a, b, c ∈ A: (i) if a · b and a · ∂b are defined, then ∂(a · b) = a · ∂b; (ii) |∂a| ≤ |a|; (iii) if |b| ≤ |c| and a · c is defined, then so is a · b and |a · b| ≤ |a · c|;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 17 / 43

Algebraic notions

Such a structure with associative multiplication is called a normed background provided that for a, b, c ∈ A: (i) if a · b and a · ∂b are defined, then ∂(a · b) = a · ∂b; (ii) |∂a| ≤ |a|; (iii) if |b| ≤ |c| and a · c is defined, then so is a · b and |a · b| ≤ |a · c|; (iv) there is t = ta ∈ N with ∂ta = o, and ∂o = o.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 17 / 43

Algebraic notions

Example.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 18 / 43

Algebraic notions

Example. A = the set of all strictly increasing functions from [k] = {1, . . . , k} to N \ {0}.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 18 / 43

Algebraic notions

Example. A = the set of all strictly increasing functions from [k] = {1, . . . , k} to N \ {0}. For a, b ∈ A with a: [k] → N \ {0} and b: [l] → N \ {0},

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 18 / 43

Algebraic notions

Example. A = the set of all strictly increasing functions from [k] = {1, . . . , k} to N \ {0}. For a, b ∈ A with a: [k] → N \ {0} and b: [l] → N \ {0}, a · b defined when [k] contains the image of b and a · b = a ◦ b.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 18 / 43

Algebraic notions

For a ∈ A with a: [k] → N \ {0}, let

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions

For a ∈ A with a: [k] → N \ {0}, let ∂a = a ↾ [k − 1].

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions

For a ∈ A with a: [k] → N \ {0}, let ∂a = a ↾ [k − 1]. Define | · |: A → N, where N is taken with its natural linear order. For a ∈ A with a: [k] → N \ {0}, let

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions

For a ∈ A with a: [k] → N \ {0}, let ∂a = a ↾ [k − 1]. Define | · |: A → N, where N is taken with its natural linear order. For a ∈ A with a: [k] → N \ {0}, let |a| =

• a(k),

if k > 0; 0, if k = 0.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions

For a ∈ A with a: [k] → N \ {0}, let ∂a = a ↾ [k − 1]. Define | · |: A → N, where N is taken with its natural linear order. For a ∈ A with a: [k] → N \ {0}, let |a| =

• a(k),

if k > 0; 0, if k = 0. Let o ∈ A be the empty function.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions

For a ∈ A with a: [k] → N \ {0}, let ∂a = a ↾ [k − 1]. Define | · |: A → N, where N is taken with its natural linear order. For a ∈ A with a: [k] → N \ {0}, let |a| =

• a(k),

if k > 0; 0, if k = 0. Let o ∈ A be the empty function. A with the above defined operations is a normed background.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 19 / 43

Algebraic notions

Lifting multiplication to sets

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 20 / 43

Algebraic notions

Each normed background A induces multiplication on subsets.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 21 / 43

Algebraic notions

Each normed background A induces multiplication on subsets. For F, G ⊆ A, F · G is defined if f · g is defined for all f ∈ F and g ∈ G, and we let F · G = {f · g : f ∈ F, g ∈ G}.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 21 / 43

Algebraic notions

Definition A a normed background. Let F be a family of subsets of A.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 22 / 43

Algebraic notions

Definition A a normed background. Let F be a family of subsets of A. Assume we have a partial function from F × F to F: (F, G) → F • G.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 22 / 43

Algebraic notions

Definition A a normed background. Let F be a family of subsets of A. Assume we have a partial function from F × F to F: (F, G) → F • G. We say that F with this operation is a family over A provided that

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 22 / 43

Algebraic notions

Definition A a normed background. Let F be a family of subsets of A. Assume we have a partial function from F × F to F: (F, G) → F • G. We say that F with this operation is a family over A provided that whenever F • G is defined, then so is F · G and F • G = F · G.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 22 / 43

Algebraic notions

Example.(ctd)

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 23 / 43

Algebraic notions

Example.(ctd) For k, l ∈ N with k ≤ l, let l k

• = the set of all increasing functions from [k] to [l].

l

k

• can be identified with the set of all k element subsets of [l].

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 23 / 43

Algebraic notions

Example.(ctd) For k, l ∈ N with k ≤ l, let l k

• = the set of all increasing functions from [k] to [l].

l

k

• can be identified with the set of all k element subsets of [l].

Let F = all l

k

• with k ≤ l.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 23 / 43

Algebraic notions

Example.(ctd) For k, l ∈ N with k ≤ l, let l k

• = the set of all increasing functions from [k] to [l].

l

k

• can be identified with the set of all k element subsets of [l].

Let F = all l

k

• with k ≤ l.

Declare n

m

• l

k

• n F to be defined when m = l and

n l

• l

k

• =

n k

• .

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 23 / 43

Algebraic notions

Example.(ctd) For k, l ∈ N with k ≤ l, let l k

• = the set of all increasing functions from [k] to [l].

l

k

• can be identified with the set of all k element subsets of [l].

Let F = all l

k

• with k ≤ l.

Declare n

m

• l

k

• n F to be defined when m = l and

n l

• l

k

• =

n k

• .

Clear: n

l

• l

k

• =

n

l

• ·

l

k

• .

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 23 / 43

Abstract Ramsey and abstract pigeonhole statements

Abstract Ramsey and abstract pigeonhole statements

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 24 / 43

Abstract Ramsey and abstract pigeonhole statements

Ramsey statement

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 25 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background The following condition is our Ramsey statement:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background The following condition is our Ramsey statement: (R) given d > 0, for each P ∈ F,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background The following condition is our Ramsey statement: (R) given d > 0, for each P ∈ F, there is F ∈ F such that

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background The following condition is our Ramsey statement: (R) given d > 0, for each P ∈ F, there is F ∈ F such that F • P is defined and

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background The following condition is our Ramsey statement: (R) given d > 0, for each P ∈ F, there is F ∈ F such that F • P is defined and for every d-coloring of F • P

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background The following condition is our Ramsey statement: (R) given d > 0, for each P ∈ F, there is F ∈ F such that F • P is defined and for every d-coloring of F • P there is f ∈ F such that f · P is monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 26 / 43

Abstract Ramsey and abstract pigeonhole statements

Example.(ctd)

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements

Example.(ctd) Condition (R):

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements

Example.(ctd) Condition (R): given d > 0 and k ≤ l

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements

Example.(ctd) Condition (R): given d > 0 and k ≤ l there exists m ≥ l

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements

Example.(ctd) Condition (R): given d > 0 and k ≤ l there exists m ≥ l such that for each d-coloring of m

l

• l

k

• =

m

k

• S

lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements

Example.(ctd) Condition (R): given d > 0 and k ≤ l there exists m ≥ l such that for each d-coloring of m

l

• l

k

• =

m

k

• there exists a ∈

m

l

• S

lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements

Example.(ctd) Condition (R): given d > 0 and k ≤ l there exists m ≥ l such that for each d-coloring of m

l

• l

k

• =

m

k

• there exists a ∈

m

l

• such that

{a ◦ x : x ∈ l k

• } is monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements

Example.(ctd) Condition (R): given d > 0 and k ≤ l there exists m ≥ l such that for each d-coloring of m

l

• l

k

• =

m

k

• there exists a ∈

m

l

• such that

{a ◦ x : x ∈ l k

• } is monochromatic.

This is the classical Ramsey theorem.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 27 / 43

Abstract Ramsey and abstract pigeonhole statements

Pigeonhole statement

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 28 / 43

Abstract Ramsey and abstract pigeonhole statements

a ∈ A can be viewed as a partial function from A to A defined on {x ∈ A: a · x defined}. a is a restriction of b ∈ A if b extends a as a partial function

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 29 / 43

Abstract Ramsey and abstract pigeonhole statements

a ∈ A can be viewed as a partial function from A to A defined on {x ∈ A: a · x defined}. a is a restriction of b ∈ A if b extends a as a partial function For F ⊆ A and a ∈ A, let

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 29 / 43

Abstract Ramsey and abstract pigeonhole statements

a ∈ A can be viewed as a partial function from A to A defined on {x ∈ A: a · x defined}. a is a restriction of b ∈ A if b extends a as a partial function For F ⊆ A and a ∈ A, let F a = {f ∈ F : a is a restriction of f }.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 29 / 43

Abstract Ramsey and abstract pigeonhole statements

For x ∈ A, ∂x is a form of restriction: for each a ∈ A if a · x is defined, then so is a · ∂x.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 30 / 43

Abstract Ramsey and abstract pigeonhole statements

For x ∈ A, ∂x is a form of restriction: for each a ∈ A if a · x is defined, then so is a · ∂x. For P ⊆ A and y ∈ A, let

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 30 / 43

Abstract Ramsey and abstract pigeonhole statements

For x ∈ A, ∂x is a form of restriction: for each a ∈ A if a · x is defined, then so is a · ∂x. For P ⊆ A and y ∈ A, let Py = {x ∈ P : y = ∂x}.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 30 / 43

Abstract Ramsey and abstract pigeonhole statements

In (R), we color F · P and are asked to find f ∈ F making the coloring constant on f · P.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 31 / 43

Abstract Ramsey and abstract pigeonhole statements

In (R), we color F · P and are asked to find f ∈ F making the coloring constant on f · P. Consider the equivalence relation on P given by ∂x1 = ∂x2, whose equivalence classes are Py for y ∈ ∂P.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 31 / 43

Abstract Ramsey and abstract pigeonhole statements

In (R), we color F · P and are asked to find f ∈ F making the coloring constant on f · P. Consider the equivalence relation on P given by ∂x1 = ∂x2, whose equivalence classes are Py for y ∈ ∂P. Require making the coloring constant on f · Py for a fixed y ∈ ∂P.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 31 / 43

Abstract Ramsey and abstract pigeonhole statements

In (R), we color F · P and are asked to find f ∈ F making the coloring constant on f · P. Consider the equivalence relation on P given by ∂x1 = ∂x2, whose equivalence classes are Py for y ∈ ∂P. Require making the coloring constant on f · Py for a fixed y ∈ ∂P. Price:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 31 / 43

Abstract Ramsey and abstract pigeonhole statements

In (R), we color F · P and are asked to find f ∈ F making the coloring constant on f · P. Consider the equivalence relation on P given by ∂x1 = ∂x2, whose equivalence classes are Py for y ∈ ∂P. Require making the coloring constant on f · Py for a fixed y ∈ ∂P. Price: make f behave as prescribed by some a ∈ A on a part of A containing y.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 31 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background A.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 32 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background A. We consider the following criterion:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 32 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background A. We consider the following criterion: (P) for d > 0, P ∈ F, and y ∈ ∂P, there is F ∈ F such that F • P is defined, and for every d-coloring of F . Py there is f ∈ F such that f . Py is monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 32 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background A. We consider the following criterion: (P) for d > 0, P ∈ F, and y ∈ ∂P, there is F ∈ F and a ∈ A such that F • P is defined, a . y is defined and for every d-coloring of F . Py there is f ∈ F such that f . Py is monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 32 / 43

Abstract Ramsey and abstract pigeonhole statements

F a family over a normed background A. We consider the following criterion: (P) for d > 0, P ∈ F, and y ∈ ∂P, there is F ∈ F and a ∈ A such that F • P is defined, a . y is defined and for every d-coloring of F . Py there is f ∈ F a such that f . Py is monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 32 / 43

Abstract Ramsey and abstract pigeonhole statements

Example.(ctd) Condition (P): follows from the standard pigeonhole principle.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 33 / 43

Abstract Ramsey and abstract pigeonhole statements

Example.(ctd) Condition (P): follows from the standard pigeonhole principle. k k − 1 x y id l′ l′ l f m [l] [m] [k]

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 33 / 43

Abstract Ramsey and abstract pigeonhole statements

Aim: prove a theorem showing that (P) implies (R).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 34 / 43

The theorem

The theorem

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 35 / 43

The theorem

Additional conditions

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 36 / 43

The theorem

(A) if F ∈ F, then ∂F ∈ F;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 37 / 43

The theorem

(A) if F ∈ F, then ∂F ∈ F; (B) if F, G ∈ F and F • ∂G is defined

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 37 / 43

The theorem

(A) if F ∈ F, then ∂F ∈ F; (B) if F, G ∈ F and F • ∂G is defined, then there is F ′ ∈ F such that F ′ • G is defined

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 37 / 43

The theorem

(A) if F ∈ F, then ∂F ∈ F; (B) if F, G ∈ F and F • ∂G is defined, then there is F ′ ∈ F such that F ′ • G is defined and for each f ∈ F there is f ′ ∈ F ′ extending f .

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 37 / 43

The theorem

(∗) if F, G, H ∈ F and F • (G • H) is defined

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 38 / 43

The theorem

(∗) if F, G, H ∈ F and F • (G • H) is defined, then so is (F • G) • H.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 38 / 43

The theorem

Statement of the theorem

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 39 / 43

The theorem

Theorem (S.) Let F be a family over a normed background fulfilling (A), (B) and (∗). Assume each set in F is finite. Then (P) implies (R).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 40 / 43

The theorem

Example.(ctd)

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 41 / 43

The theorem

Example.(ctd) Conditions (A) and (B).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 41 / 43

The theorem

Example.(ctd) Conditions (A) and (B). Since ∂ n

m

• =

n−1

m−1

• , F is closed under ∂, so (A) holds.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 41 / 43

The theorem

Example.(ctd) Conditions (A) and (B). Since ∂ n

m

• =

n−1

m−1

• , F is closed under ∂, so (A) holds.

To check (B), let F = n

m

• and G =

l

k

• and assume

F • ∂G = n m

• l − 1

k − 1

• is defined.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 41 / 43

The theorem

Example.(ctd) Conditions (A) and (B). Since ∂ n

m

• =

n−1

m−1

• , F is closed under ∂, so (A) holds.

To check (B), let F = n

m

• and G =

l

k

• and assume

F • ∂G = n m

• l − 1

k − 1

• is defined. Then m = l − 1. Take F ′ =

n+1

l

• to witness (B).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 41 / 43

The theorem

Example.(ctd) Conditions (A) and (B). Since ∂ n

m

• =

n−1

m−1

• , F is closed under ∂, so (A) holds.

To check (B), let F = n

m

• and G =

l

k

• and assume

F • ∂G = n m

• l − 1

k − 1

• is defined. Then m = l − 1. Take F ′ =

n+1

l

• to witness (B).

It works since n+1

l

• l

k

• is defined and each element of

n

l−1

• is extended

by an element of n+1

l

• .

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 41 / 43

The theorem

Condition (∗).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 42 / 43

The theorem

Condition (∗). If q p

• (

n m

• l

k

• )

is defined, then m = l and p = n,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 42 / 43

The theorem

Condition (∗). If q p

• (

n m

• l

k

• )

is defined, then m = l and p = n, but then ( q p

• n

m

• ) •

l k

• is defined.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 42 / 43

The theorem

Conclusion:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 43 / 43

The theorem

Conclusion: the classical Ramsey theorem holds.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey January 2013 43 / 43