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An abstract approach to finite Ramsey theory and a self-dual Ramsey theorem S lawomir Solecki University of Illinois at UrbanaChampaign May 2011 Outline Outline of Topics Self-dual Ramsey theorem 1 Algebraic notions 2 Abstract

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An abstract approach to finite Ramsey theory and a self-dual Ramsey theorem

S lawomir Solecki

University of Illinois at Urbana–Champaign

May 2011

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Outline

Outline of Topics

1

Self-dual Ramsey theorem

2

Algebraic notions

3

Abstract pigeonhole and main theorem

4

Localizing and propagating pigeonhole

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 2 / 42

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Outline

We give a general approach to finite Ramsey theory

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 3 / 42

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Outline

We give a general approach to finite Ramsey theory, formulate within it an abstract pigeonhole principle and an abstract Ramsey theorem

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 3 / 42

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Outline

We give a general approach to finite Ramsey theory, formulate within it 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 May 2011 3 / 42

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Outline

We give a general approach to finite Ramsey theory, formulate within it an abstract pigeonhole principle and an abstract Ramsey theorem, and prove that the pigeonhole principle implies the Ramsey theorem. 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 May 2011 3 / 42

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Outline

We give a general approach to finite Ramsey theory, formulate within it an abstract pigeonhole principle and an abstract Ramsey theorem, and prove that the pigeonhole principle implies the Ramsey theorem. The classical Ramsey theorem, the Hales–Jewett theorem, the Graham–Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and a new self-dual Ramsey theorem

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 3 / 42

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Outline

We give a general approach to finite Ramsey theory, formulate within it an abstract pigeonhole principle and an abstract Ramsey theorem, and prove that the pigeonhole principle implies the Ramsey theorem. The classical Ramsey theorem, the Hales–Jewett theorem, the Graham–Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and a new self-dual Ramsey theorem are all

  • btained as iterative applications of the general result.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 3 / 42

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Outline

We give a general approach to finite Ramsey theory, formulate within it an abstract pigeonhole principle and an abstract Ramsey theorem, and prove that the pigeonhole principle implies the Ramsey theorem. The classical Ramsey theorem, the Hales–Jewett theorem, the Graham–Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and a new self-dual Ramsey theorem are all

  • btained as iterative applications of the general result.

Plan:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 3 / 42

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Outline

We give a general approach to finite Ramsey theory, formulate within it an abstract pigeonhole principle and an abstract Ramsey theorem, and prove that the pigeonhole principle implies the Ramsey theorem. The classical Ramsey theorem, the Hales–Jewett theorem, the Graham–Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and a new self-dual Ramsey theorem are all

  • btained as iterative applications of the general result.

Plan: present the concrete self-dual Ramsey result

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 3 / 42

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Outline

We give a general approach to finite Ramsey theory, formulate within it an abstract pigeonhole principle and an abstract Ramsey theorem, and prove that the pigeonhole principle implies the Ramsey theorem. The classical Ramsey theorem, the Hales–Jewett theorem, the Graham–Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and a new self-dual Ramsey theorem are all

  • btained as iterative applications of the general result.

Plan: present the concrete self-dual Ramsey result, present the abstract approach

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 3 / 42

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Outline

We give a general approach to finite Ramsey theory, formulate within it an abstract pigeonhole principle and an abstract Ramsey theorem, and prove that the pigeonhole principle implies the Ramsey theorem. The classical Ramsey theorem, the Hales–Jewett theorem, the Graham–Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and a new self-dual Ramsey theorem are all

  • btained as iterative applications of the general result.

Plan: present the concrete self-dual Ramsey result, present the abstract approach, and roughly outline how it is applied to get the above results.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 3 / 42

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Outline

Some features of the approach:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 4 / 42

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Outline

Some features of the approach:

  • 1. a new self-dual Ramsey theorem;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 4 / 42

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Outline

Some features of the approach:

  • 1. a new self-dual Ramsey theorem;
  • 2. the Hales–Jewett theorem has a natural proof that gives Shelah’s

primitive recursive bounds for the parameters involved in it;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 4 / 42

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Outline

Some features of the approach:

  • 1. a new self-dual Ramsey theorem;
  • 2. the Hales–Jewett theorem has a natural proof that gives Shelah’s

primitive recursive bounds for the parameters involved in it;

  • 3. the Graham–Rothschild theorem for partitions is proved directly

without proving it first for parameter sets; however, the parameter set generalization can also be obtained from the abstract result;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 4 / 42

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Outline

Some features of the approach:

  • 1. a new self-dual Ramsey theorem;
  • 2. the Hales–Jewett theorem has a natural proof that gives Shelah’s

primitive recursive bounds for the parameters involved in it;

  • 3. the Graham–Rothschild theorem for partitions is proved directly

without proving it first for parameter sets; however, the parameter set generalization can also be obtained from the abstract result;

  • 4. 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 self-dual Ramsey theorem four.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 4 / 42

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Self-dual Ramsey theorem

Self-dual Ramsey theorem

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 5 / 42

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Self-dual Ramsey theorem

Definition A function i : [K] → [L] is an increasing injection if it is injective and preimages of initial segments of [L] are initial segments of [K].

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 6 / 42

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Self-dual Ramsey theorem

Definition A function i : [K] → [L] is an increasing injection if it is injective and preimages of initial segments of [L] are initial segments of [K]. i is an increasing injection iff i(x) ≥ 1 + maxy<x i(y) for all x ∈ [K].

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 6 / 42

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Self-dual Ramsey theorem

Definition A function i : [K] → [L] is an increasing injection if it is injective and preimages of initial segments of [L] are initial segments of [K]. i is an increasing injection iff i(x) ≥ 1 + maxy<x i(y) for all x ∈ [K]. Theorem (Ramsey) Given K, L and d > 0 there exists M such that for each d-coloring of all increasing injections [K] → [M] there exists an increasing injection j0 : [L] → [M] such that {j0 ◦ i : i : [K] → [L] an increasing injection} is monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 6 / 42

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Self-dual Ramsey theorem

Definition A function s : [L] → [K] is a rigid surjection if it is surjective and images

  • f initial segments of [L] are initial segments of [K].

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 7 / 42

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Self-dual Ramsey theorem

Definition A function s : [L] → [K] is a rigid surjection if it is surjective and images

  • f initial segments of [L] are initial segments of [K].

s is a rigid surjection iff s(x) ≤ 1 + maxy<x s(y) for each x ∈ [L].

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 7 / 42

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Self-dual Ramsey theorem

Definition A function s : [L] → [K] is a rigid surjection if it is surjective and images

  • f initial segments of [L] are initial segments of [K].

s is a rigid surjection iff s(x) ≤ 1 + maxy<x s(y) for each x ∈ [L]. Theorem (Graham–Rothschild) Given K, L and d > 0 there exists M such that for each d-coloring of all rigid surjections [M] → [K] there exists a rigid surjection t0 : [M] → [L] such that {s ◦ t0 : s : [L] → [K] a rigid surjection} is monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 7 / 42

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Self-dual Ramsey theorem

Definition A pair (s, i) is a connection between L and K if s : [L] → [K], i : [K] → [L] and for each x ∈ [K] s(i(x)) = x and ∀y < i(x) s(y) ≤ x.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 8 / 42

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Self-dual Ramsey theorem

Definition A pair (s, i) is a connection between L and K if s : [L] → [K], i : [K] → [L] and for each x ∈ [K] s(i(x)) = x and ∀y < i(x) s(y) ≤ x. So, i is a left inverse of s and at each x ∈ [K] the value i(x) is picked only from those elements of s−1(x) that are “visible from x,”

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 8 / 42

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Self-dual Ramsey theorem

Definition A pair (s, i) is a connection between L and K if s : [L] → [K], i : [K] → [L] and for each x ∈ [K] s(i(x)) = x and ∀y < i(x) s(y) ≤ x. So, i is a left inverse of s and at each x ∈ [K] the value i(x) is picked only from those elements of s−1(x) that are “visible from x,” that is, from those y′ ∈ s−1(x) for which s ↾ ({y : y < y′}) ≤ x.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 8 / 42

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Self-dual Ramsey theorem

Definition A pair (s, i) is a connection between L and K if s : [L] → [K], i : [K] → [L] and for each x ∈ [K] s(i(x)) = x and ∀y < i(x) s(y) ≤ x. So, i is a left inverse of s and at each x ∈ [K] the value i(x) is picked only from those elements of s−1(x) that are “visible from x,” that is, from those y′ ∈ s−1(x) for which s ↾ ({y : y < y′}) ≤ x. We write (s, i): [L] ↔ [K].

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 8 / 42

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Self-dual Ramsey theorem

Given connections (s, i): [L] ↔ [K] and (t, j): [M] ↔ [L], define (s, i) · (t, j): [M] ↔ [K] as (s ◦ t, j ◦ i).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 9 / 42

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Self-dual Ramsey theorem

Given connections (s, i): [L] ↔ [K] and (t, j): [M] ↔ [L], define (s, i) · (t, j): [M] ↔ [K] as (s ◦ t, j ◦ i). Theorem (S.) For natural numbers K, L and d > 0 there exists M such that for each d-coloring of all connections between M and K there is (t0, j0): [M] ↔ [L] such that {(s, i) · (t0, j0): (s, i): [L] ↔ [K]} is monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 9 / 42

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Algebraic notions

Algebraic notions

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Algebraic notions

Abstract Ramsey statement:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42

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Algebraic notions

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

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42

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Algebraic notions

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

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42

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Algebraic notions

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

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42

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Algebraic notions

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

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42

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Algebraic notions

Abstract Ramsey statement: given S find F for which F × S ∋ (f , x) → f . x ∈ F . S is defined; color F . S; find f ∈ F with f . S monochromatic. Algebraic approach: multiplication/action

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42

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Algebraic notions

Abstract Ramsey statement: given S find F for which F × S ∋ (f , x) → f . x ∈ F . S is defined; color F . S; find f ∈ F with f . S monochromatic. Algebraic approach: multiplication/action, lifting them to sets

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42

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Algebraic notions

Abstract Ramsey statement: given S find F for which F × S ∋ (f , x) → f . x ∈ F . S is defined; color F . S; find f ∈ F with f . S monochromatic. Algebraic approach: multiplication/action, lifting them to sets, truncation operator.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 11 / 42

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Algebraic notions

Multiplicative part

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 12 / 42

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Algebraic notions

Definition A local actoid

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42

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Algebraic notions

Definition A local actoid consists of two sets A and Z,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42

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Algebraic notions

Definition A local actoid consists of two sets A and Z, a partial binary function from A × A to A: (a, b) → a · b,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42

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Algebraic notions

Definition A local actoid consists of two sets A and Z, a partial binary function from A × A to A: (a, b) → a · b, and a partial binary function from A × Z to Z: (a, z) → a . z

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42

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Algebraic notions

Definition A local actoid consists of two sets A and Z, a partial binary function from A × A to A: (a, b) → a · b, and a partial binary function from A × Z to Z: (a, z) → a . z such that for a, b ∈ A and z ∈ Z if a . (b . z) and (a · b) . z are both defined,

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42

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Algebraic notions

Definition A local actoid consists of two sets A and Z, a partial binary function from A × A to A: (a, b) → a · b, and a partial binary function from A × Z to Z: (a, z) → a . z such that for a, b ∈ A and z ∈ Z if a . (b . z) and (a · b) . z are both defined, then a . (b . z) = (a · b) . z.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 13 / 42

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Algebraic notions

Definition A local actoid (A, Z) is called an actoid if for all a, b ∈ A and z ∈ Z, if a . (b . z) is defined, then so is (a · b) . z.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 14 / 42

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Algebraic notions

Definition A local actoid (A, Z) is called an actoid if for all a, b ∈ A and z ∈ Z, if a . (b . z) is defined, then so is (a · b) . z. Note: for a, b, z as above, one has a . (b . z) = (a · b) . z.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 14 / 42

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Algebraic notions

Example.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 15 / 42

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Algebraic notions

Example. s : [L] → [K], t : [N] → [M] rigid surjections

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 15 / 42

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Algebraic notions

Example. s : [L] → [K], t : [N] → [M] rigid surjections The canonical composition of s and t, denoted by s ◦ t, is defined if L ≤ M.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 15 / 42

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Example. s : [L] → [K], t : [N] → [M] rigid surjections The canonical composition of s and t, denoted by s ◦ t, is defined if L ≤ M. In this case, let s ◦ t be the composition of s with t restricted to the largest initial segment of [N] on which this composition is defined.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 15 / 42

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Algebraic notions

Example. s : [L] → [K], t : [N] → [M] rigid surjections The canonical composition of s and t, denoted by s ◦ t, is defined if L ≤ M. In this case, let s ◦ t be the composition of s with t restricted to the largest initial segment of [N] on which this composition is defined. Then s ◦ t : [N0] → [K] is a rigid surjection for some N0 ≤ N.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 15 / 42

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Algebraic notions

A0 = Z0 = rigid surjections

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 16 / 42

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Algebraic notions

A0 = Z0 = rigid surjections For s, t ∈ A0 = Z0, let t · s = t . s = s ◦ t whenever s ◦ t is defined.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 16 / 42

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Algebraic notions

A0 = Z0 = rigid surjections For s, t ∈ A0 = Z0, let t · s = t . s = s ◦ t whenever s ◦ t is defined. (A0, Z0) is a local actoid.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 16 / 42

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Algebraic notions

Lifting multiplication and action to sets

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 17 / 42

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Algebraic notions

Each local actoid (A, Z) induces operations on subsets.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 18 / 42

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Algebraic notions

Each local actoid (A, Z) induces operations 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 May 2011 18 / 42

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Algebraic notions

Each local actoid (A, Z) induces operations 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}. For F ⊆ A and S ⊆ Z, F . S is defined if f . x is defined for all f ∈ F and x ∈ S, and we let F . S = {f . x : f ∈ F, x ∈ S}.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 18 / 42

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Algebraic notions

Definition (A, Z) a local actoid.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42

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Algebraic notions

Definition (A, Z) a local actoid. Let F be a family of subsets of A and S a family of subsets of Z.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42

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Algebraic notions

Definition (A, Z) a local actoid. Let F be a family of subsets of A and S a family of subsets of Z. We have partial functions from F × F to F and from F × S to S: (F, G) → F • G and (F, S) → F • S.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42

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Algebraic notions

Definition (A, Z) a local actoid. Let F be a family of subsets of A and S a family of subsets of Z. We have partial functions from F × F to F and from F × S to S: (F, G) → F • G and (F, S) → F • S. We say that (F, S) with these two operations is a local actoid of sets

  • ver (A, Z) provided that

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42

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Algebraic notions

Definition (A, Z) a local actoid. Let F be a family of subsets of A and S a family of subsets of Z. We have partial functions from F × F to F and from F × S to S: (F, G) → F • G and (F, S) → F • S. We say that (F, S) with these two operations is a local actoid of sets

  • ver (A, Z) 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 May 2011 19 / 42

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Algebraic notions

Definition (A, Z) a local actoid. Let F be a family of subsets of A and S a family of subsets of Z. We have partial functions from F × F to F and from F × S to S: (F, G) → F • G and (F, S) → F • S. We say that (F, S) with these two operations is a local actoid of sets

  • ver (A, Z) provided that whenever F • G is defined, then so is F · G and

F • G = F · G, and whenever F • S is defined, then so is F . S and F • S = F . S.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 19 / 42

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Algebraic notions

Example.(ctd)

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 20 / 42

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Algebraic notions

Example.(ctd) (A0, Z0) the local actoid defined earlier

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 20 / 42

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Algebraic notions

Example.(ctd) (A0, Z0) the local actoid defined earlier F0 = S0 consist of sets of the form FL,K = SL,K = {s ∈ A0 = Z0 : s : [L] → [K]}, for L ≥ K > 0.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 20 / 42

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Algebraic notions

FN,M • FL,K defined if and only if M = L and FN,L • FL,K = FN,K.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 21 / 42

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Algebraic notions

FN,M • FL,K defined if and only if M = L and FN,L • FL,K = FN,K. FN,M • SL,K defined if and only if M = L and FN,L • SL,K = SN,K.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 21 / 42

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Algebraic notions

FN,M • FL,K defined if and only if M = L and FN,L • FL,K = FN,K. FN,M • SL,K defined if and only if M = L and FN,L • SL,K = SN,K. (F0, S0) with these operations is an actoid of sets over (A0, Z0).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 21 / 42

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Algebraic notions

FN,M • FL,K defined if and only if M = L and FN,L • FL,K = FN,K. FN,M • SL,K defined if and only if M = L and FN,L • SL,K = SN,K. (F0, S0) with these operations is an actoid of sets over (A0, Z0). Note that FN,M · FL,K and FN,M . SL,K are defined if only M ≥ L.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 21 / 42

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Algebraic notions

Truncation added

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Algebraic notions

Definition A background is a local actoid (A, Z)

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 23 / 42

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Algebraic notions

Definition A background is a local actoid (A, Z) together with a unary function ∂ : Z → Z such that

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 23 / 42

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Algebraic notions

Definition A background is a local actoid (A, Z) together with a unary function ∂ : Z → Z such that for a ∈ A and z ∈ Z, if a . z is defined, then a . ∂z is defined and a . ∂z = ∂(a . z).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 23 / 42

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Algebraic notions

Definition A background is a local actoid (A, Z) together with a unary function ∂ : Z → Z such that for a ∈ A and z ∈ Z, if a . z is defined, then a . ∂z is defined and a . ∂z = ∂(a . z). ∂ is call a truncation.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 23 / 42

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Algebraic notions

Definition A background is a local actoid (A, Z) together with a unary function ∂ : Z → Z such that for a ∈ A and z ∈ Z, if a . z is defined, then a . ∂z is defined and a . ∂z = ∂(a . z). ∂ is call a truncation. It is a type of a restriction operator.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 23 / 42

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Algebraic notions

Notation: for a background (A, Z) with a truncation ∂ and for S ⊆ Z, let ∂S = {∂x : x ∈ S} and, more generally, for t ∈ N ∂tS = {∂tx : x ∈ S}.

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Algebraic notions

Example.(ctd)

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Algebraic notions

Example.(ctd) s : [L] → [K] a rigid surjection

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42

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Algebraic notions

Example.(ctd) s : [L] → [K] a rigid surjection If K > 0, then L > 0, and let L0 = min{y ∈ [L]: s(y) = K}.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42

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Algebraic notions

Example.(ctd) s : [L] → [K] a rigid surjection If K > 0, then L > 0, and let L0 = min{y ∈ [L]: s(y) = K}. Define ∂0s = s ↾ [L0 − 1].

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42

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Algebraic notions

Example.(ctd) s : [L] → [K] a rigid surjection If K > 0, then L > 0, and let L0 = min{y ∈ [L]: s(y) = K}. Define ∂0s = s ↾ [L0 − 1]. If K = 0, then L = 0 and s is the empty function and we let ∂0∅ = ∅.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42

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Algebraic notions

Example.(ctd) s : [L] → [K] a rigid surjection If K > 0, then L > 0, and let L0 = min{y ∈ [L]: s(y) = K}. Define ∂0s = s ↾ [L0 − 1]. If K = 0, then L = 0 and s is the empty function and we let ∂0∅ = ∅. ∂0 is a truncation forgetting the largest value.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 25 / 42

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Algebraic notions

(A0, Z0) the local actoid defined earlier;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 26 / 42

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Algebraic notions

(A0, Z0) the local actoid defined earlier; for s ∈ Z0, take ∂0s as the truncation.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 26 / 42

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Algebraic notions

(A0, Z0) the local actoid defined earlier; for s ∈ Z0, take ∂0s as the truncation. (A0, Z0) with ∂0 is a background.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 26 / 42

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Abstract pigeonhole and main theorem

Abstract pigeonhole and main theorem

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Abstract pigeonhole and main theorem

(F, S) a local actoid of sets over a background, S ∈ S

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 28 / 42

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Abstract pigeonhole and main theorem

(F, S) a local actoid of sets over a background, S ∈ S Recall the abstract Ramsey statement:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 28 / 42

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Abstract pigeonhole and main theorem

(F, S) a local actoid of sets over a background, S ∈ S Recall the abstract Ramsey statement: find F ∈ F for which F • S is defined; color F • S; find f ∈ F with f . S monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 28 / 42

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Abstract pigeonhole and main theorem

We consider the equivalence relation ∼ on S given by x1 ∼ x2 ⇐ ⇒ ∂x1 = ∂x2.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 29 / 42

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Abstract pigeonhole and main theorem

We consider the equivalence relation ∼ on S given by x1 ∼ x2 ⇐ ⇒ ∂x1 = ∂x2. We are looking for a principle of the form: there is F ∈ F such that for each coloring of F • S there is f ∈ F with multiplication by f stabilizing the coloring on equivalence classes of ∼.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 29 / 42

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Abstract pigeonhole and main theorem

Definition Let (F, S) be an actoid of sets over a background (A, Z). We call (F, S) a pigeonhole actoid if

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 30 / 42

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Abstract pigeonhole and main theorem

Definition Let (F, S) be an actoid of sets over a background (A, Z). We call (F, S) a pigeonhole actoid if (ph) for every d > 0 and S ∈ S there exists F ∈ F such that F • S is defined and for each d-coloring c of F . S there exists f ∈ F such that for all x1, x2 ∈ S we have ∂x1 = ∂x2 = ⇒ c(f .x1) = c(f .x2).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 30 / 42

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Abstract pigeonhole and main theorem

Definition Let (F, S) be an actoid of sets over a background (A, Z). We call (F, S) a pigeonhole actoid if (ph) for every t ≥ 0, d > 0 and S ∈ S there exists F ∈ F such that F • S is defined and for each d-coloring c of F . ∂tS there exists f ∈ F such that for all x1, x2 ∈ ∂tS we have ∂x1 = ∂x2 = ⇒ c(f .x1) = c(f .x2).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 30 / 42

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Abstract pigeonhole and main theorem

Definition Let (F, S) be an actoid of sets over a background (A, Z). We call (F, S) a pigeonhole actoid if (ph) for every t ≥ 0, d > 0 and S ∈ S there exists F ∈ F such that F • S is defined and for each d-coloring c of F . ∂tS there exists f ∈ F such that for all x1, x2 ∈ ∂tS we have ∂x1 = ∂x2 = ⇒ c(f .x1) = c(f .x2). (ph): multiplication by f fixes color on equivalence classes of the equivalence relation on ∂tS given by ∂x1 = ∂x2.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 30 / 42

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Abstract pigeonhole and main theorem

Definition A family I of subsets of Z for a background (A, Z) is called vanishing if for every S ∈ I there is t ∈ N such that ∂tS consists of at most one element.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 31 / 42

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Abstract pigeonhole and main theorem

Definition A family I of subsets of Z for a background (A, Z) is called vanishing if for every S ∈ I there is t ∈ N such that ∂tS consists of at most one element. Theorem (S.) Let (F, S) be a pigeonhole actoid. Assume S is vanishing. Then for every d > 0 and S ∈ S there exists F ∈ F such that F • S is defined and for each d-coloring of F • S there exists f ∈ F for which f .S is monochromatic.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 31 / 42

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Localizing and propagating pigeonhole

Localizing and propagating pigeonhole

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Localizing and propagating pigeonhole

Localizing property (ph)

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Localizing and propagating pigeonhole

A quasi-order on a local actoid:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 34 / 42

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Localizing and propagating pigeonhole

A quasi-order on a local actoid: (A, Z) a local actoid

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 34 / 42

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Localizing and propagating pigeonhole

A quasi-order on a local actoid: (A, Z) a local actoid Z carries a natural quasi-order: the largest binary relation ≤A,Z on Z such that

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 34 / 42

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Localizing and propagating pigeonhole

A quasi-order on a local actoid: (A, Z) a local actoid Z carries a natural quasi-order: the largest binary relation ≤A,Z on Z such that for x, y ∈ Z x ≤A,Z y ⇒ ∀a ∈ A (if a.y is defined, then a.x is defined and a.x ≤A,Z a.y)

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 34 / 42

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Localizing and propagating pigeonhole

Localizing (ph):

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Localizing and propagating pigeonhole

Localizing (ph): In (ph), we color F.(∂tS) and are asked to find f ∈ F making the coloring constant on each equivalence class of the equivalence relation on ∂tS given by ∂x1 = ∂x2.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 35 / 42

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Localizing and propagating pigeonhole

Localizing (ph): In (ph), we color F.(∂tS) and are asked to find f ∈ F making the coloring constant on each equivalence class of the equivalence relation on ∂tS given by ∂x1 = ∂x2. Space of invariants of the equivalence relation is ∂t+1S: ∂tS ∋ x → ∂x ∈ ∂t+1S.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 35 / 42

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Localizing and propagating pigeonhole

Localizing (ph): In (ph), we color F.(∂tS) and are asked to find f ∈ F making the coloring constant on each equivalence class of the equivalence relation on ∂tS given by ∂x1 = ∂x2. Space of invariants of the equivalence relation is ∂t+1S: ∂tS ∋ x → ∂x ∈ ∂t+1S. Each equivalence class is determined by y ∈ ∂t+1S.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 35 / 42

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Localizing and propagating pigeonhole

Localizing (ph): In (ph), we color F.(∂tS) and are asked to find f ∈ F making the coloring constant on each equivalence class of the equivalence relation on ∂tS given by ∂x1 = ∂x2. Space of invariants of the equivalence relation is ∂t+1S: ∂tS ∋ x → ∂x ∈ ∂t+1S. Each equivalence class is determined by y ∈ ∂t+1S. Notation: (∂tS)y

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 35 / 42

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Localizing and propagating pigeonhole

Localization: require making the coloring constant by multiplication by f ∈ F on a fixed equivalence class (∂tS)y for some y ∈ ∂t+1S; prove it implies full (ph).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 36 / 42

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Localizing and propagating pigeonhole

Localization: require making the coloring constant by multiplication by f ∈ F on a fixed equivalence class (∂tS)y for some y ∈ ∂t+1S; prove it implies full (ph). Price: need to keep a prescribed behavior of f on a part of the space of invariants ∂t+1S containing y.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 36 / 42

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Localizing and propagating pigeonhole

(F, S) an actoid of sets over a background (A, Z). We consider the following criterion:

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 37 / 42

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Localizing and propagating pigeonhole

(F, S) an actoid of sets over a background (A, Z). We consider the following criterion: (ph−) for d > 0, S ∈ S, and y ∈ ∂S, there is F ∈ F such that F • S is defined and for every d-coloring of F.( S)y there is f ∈ F such that f .( S)y is monochromatic

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 37 / 42

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Localizing and propagating pigeonhole

(F, S) an actoid of sets over a background (A, Z). We consider the following criterion: (ph−) for d > 0, S ∈ S, and y ∈ ∂S, there is F ∈ F and a ∈ A such that F • S is defined, a.y is defined, and for every d-coloring of F.( S)y there is f ∈ F such that f .( S)y is monochromatic and f extends a on ∂S.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 37 / 42

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Localizing and propagating pigeonhole

(F, S) an actoid of sets over a background (A, Z). We consider the following criterion: (ph−) for t ≥ 0, d > 0, S ∈ S, and y ∈ ∂∂tS, there is F ∈ F and a ∈ A such that F • S is defined, a.y is defined, and for every d-coloring of F.(∂tS)y there is f ∈ F such that f .(∂tS)y is monochromatic and f extends a on ∂∂tS.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 37 / 42

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Localizing and propagating pigeonhole

(F, S) an actoid of sets over a background (A, Z). We consider the following criterion: (ph−) for t ≥ 0, d > 0, S ∈ S, and y ∈ ∂t+1S, there is F ∈ F and a ∈ A such that F • S is defined, a.y is defined, and for every d-coloring of F.(∂tS)y there is f ∈ F such that f .(∂tS)y is monochromatic and f extends a on ∂t+1S.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 37 / 42

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Localizing and propagating pigeonhole

Theorem (S.) Let (F, S) be an actoid of sets over a background (A, Z). Assume that S consists of finite sets and that ≤A,Z is quasi-linear when restricted to S for each S ∈ S. If (F, S) fulfills criterion (ph−), then it is a pigeonhole actoid.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 38 / 42

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Localizing and propagating pigeonhole

Example.(ctd)

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Localizing and propagating pigeonhole

Example.(ctd) The actiod of sets (F0, S0) over (A0, Z0) fulfills the assumptions of the above theorem, in particular, it has (ph−).

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 39 / 42

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Localizing and propagating pigeonhole

Example.(ctd) The actiod of sets (F0, S0) over (A0, Z0) fulfills the assumptions of the above theorem, in particular, it has (ph−). So it is a pigeonhole actoid of sets.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 39 / 42

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Localizing and propagating pigeonhole

Example.(ctd) The actiod of sets (F0, S0) over (A0, Z0) fulfills the assumptions of the above theorem, in particular, it has (ph−). So it is a pigeonhole actoid of

  • sets. The main theorem applied to it gives the Graham–Rothschild

theorem for partitions.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 39 / 42

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Localizing and propagating pigeonhole

Propagating property (ph)

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Localizing and propagating pigeonhole

There are two results:

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Localizing and propagating pigeonhole

There are two results: the first result involves the notion of finite product of actoids of sets;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 41 / 42

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Localizing and propagating pigeonhole

There are two results: the first result involves the notion of finite product of actoids of sets; result: finite products of actoids of sets are pigeonhole assuming the factors are pigeonhole;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 41 / 42

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Localizing and propagating pigeonhole

There are two results: the first result involves the notion of finite product of actoids of sets; result: finite products of actoids of sets are pigeonhole assuming the factors are pigeonhole; proof: uses the main theorem;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 41 / 42

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Localizing and propagating pigeonhole

There are two results: the first result involves the notion of finite product of actoids of sets; result: finite products of actoids of sets are pigeonhole assuming the factors are pigeonhole; proof: uses the main theorem; the second result involves the notion of interpretability of sets from an actoid of sets in other actoids of sets;

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 41 / 42

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Localizing and propagating pigeonhole

There are two results: the first result involves the notion of finite product of actoids of sets; result: finite products of actoids of sets are pigeonhole assuming the factors are pigeonhole; proof: uses the main theorem; the second result involves the notion of interpretability of sets from an actoid of sets in other actoids of sets; result: if each set from an actoid of sets is interpretable in some pigeonhole actoid of sets, then the actoid of sets is pigeonhole.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 41 / 42

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Localizing and propagating pigeonhole

One obtains all the theorems mentioned in the introduction by repeated applications of the main theorem with the aid of the localization and propagation results.

S lawomir Solecki (University of Illinois) Abstract approach to Ramsey May 2011 42 / 42