Damir D. Dzhafarov University of Chicago 20 August 2009 · Damir D. DzhafarovStable Ramsey’s theorem and measure. Ramsey’s theorem Moral Complete disorder is impossible: in any

Stable Ramsey’s theorem and measure

Damir D. DzhafarovUniversity of Chicago

University of Hawai‘i at Mānoa20 August 2009

Damir D. Dzhafarov Stable Ramsey’s theorem and measure

Ramsey’s theorem

Definition

Fix n, k ∈ N and let X be an infinite set.

1 [X ]n = {Y ⊂ X : |Y | = n}.2 A k-coloring of exponent n is a map f : [X ]n → k = {0, . . . , k − 1}.3 A set H is homogeneous for f if it is infinite and f � [H]n is constant.

Ramsey’s theorem

Fix n, k ∈ N. Every coloring f : [N]n → k has a homogeneous set.

Ramsey’s theorem

Definition

Fix n, k ∈ N and let X be an infinite set.

1 [X ]n = {Y ⊂ X : |Y | = n}.2 A k-coloring of exponent n is a map f : [X ]n → k = {0, . . . , k − 1}.3 A set H is homogeneous for f if it is infinite and f � [H]n is constant.

Ramsey’s theorem

Fix n, k ∈ N. Every coloring f : [N]n → k has a homogeneous set.

Ramsey’s theorem

1 20 3 4 5 . . .

A coloring [N]1 → 2.

Ramsey’s theorem

1 3 4 . . .

H = {1, 3, 4, 6, 9, 10, 11, 14, 15, . . .} is homogeneous.

Ramsey’s theorem

00123

1 2 3 4 5

45

. . .

A coloring [N]2 → 2.

Ramsey’s theorem

1

3

1 3 4

4

. . .

H = {1, 3, 4, 6, 9, 10, 11, 14, 15, . . .} is homogeneous.

Ramsey’s theorem

Corollary (Chain Antichain Condition)

Every partial ordering of N has an infinite chain or infinite antichain.

Proof.

Let � be a partial ordering of N. Define f : [N]2 → 2 by

f ({x , y}) ={

0 if x and y are comparable under �1 otherwise

.

Apply Ramsey’s theorem to get a homogeneous set H for f .

If f has value 0 on [H]2 then H is a chain under �, and if f has value 1 on[H]2 then H is an antichain.

Ramsey’s theorem

Corollary (Chain Antichain Condition)

Every partial ordering of N has an infinite chain or infinite antichain.

Proof.

Let � be a partial ordering of N. Define f : [N]2 → 2 by

f ({x , y}) ={

0 if x and y are comparable under �1 otherwise

.

Apply Ramsey’s theorem to get a homogeneous set H for f .

If f has value 0 on [H]2 then H is a chain under �, and if f has value 1 on[H]2 then H is an antichain.

Ramsey’s theorem

Moral

Complete disorder is impossible: in any configuration or arrangement ofobjects, however complicated or disorganized, some amount of structureand regularity is necessary.

Computability theory

Okay, but how easy is it to describe this regularity?

Computability-theoretic approach: let’s try to gauge the complexity,with respect to some hierarchy, of homogeneous sets.

We restrict to computable colorings, i.e., f : [N]n → k for which thereis an algorithm to compute f ({x1, . . . , xn}) given x1, . . . , xn ∈ N.

Theorem (Specker, 1969)

There exists a comp. f : [N]2 → 2 with no computable homogeneous set.

Definition (Turing jump)

Let X be a set.

1 The (Turing) jump of X is the set X ′ defined as

{(e, x) : the eth program with access to X halts on input x}.

2 For n ≥ 1, the (n + 1) st jump of X is the jump of the nth.

Turing hierarchy (≤T )

. . .

Theorem (Jockusch, 1972)

For each n ≥ 3, there exists a computable f : [N]n → 2 such that H ≥T ∅′(in fact, H ≥T ∅(n−2)) for every homogeneous set H of f .

This reduces the investigation to the n = 2 case.

Theorem (Seetapun, 1995)

For every noncomputable set C , every computable f : [N]2 → k has ahomogeneous set H �T C .

Definition (Low and low2 sets)

A set X is

1 low if X ′ ≤T ∅′;2 low2 if X

′′ ≤T ∅′′.

. . .

A low set.

. . .

A low set.

. . .

A low set.

. . .

A low set.

. . .

A low2 set.

. . .

A low2 set.

. . .

A low2 set.

. . .

Lowness hierarchy.

. . .

low sets

Lowness hierarchy.

. . .

low2 sets

Lowness hierarchy.

There exists a comp. f : [N]2 → 2 with no low hom. set (or even ≤T ∅′).

Theorem (Cholak, Jockusch, and Slaman, 2001)

Every computable f : [N]2 → k has a low2 homogeneous set H.

Question (Cholak, Jockusch, and Slaman, 2001)

Can cone avoidance be added to this result?

Theorem (Dzhafarov and Jockusch, 2009)

For every noncomputable set C , every computable f : [N]2 → k has a low2homogeneous set that does not compute C .

Stable Ramsey’s theorem

Definition

A coloring f : [N]2 → k is stable if for all x , limy→∞ f ({x , y}) exists.

Every computable stable f : [N]2 → k has a homogeneous set H ≤T ∅′.

Theorem (Downey, Hirschfeldt, Lempp, and Solomon, 2001)

There exists a computable stable f : [ω]2 → 2 with no low hom. set.

Definition

. . .

low sets

space

Theorem (Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, 2008)

Every computable stable f : [ω]2 → 2 has an homogeneous set H

Theorem (Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, 2008)

Every computable stable f : [ω]2 → 2 has an homogeneous set H

∆02 measure

Is there a way of telling which of the above results are “typical”?

Measure-theoretic approach: define a notion of measure or nullity onthe class of sets computable in ∅′.

∆02 measure

Is there a way of telling which of the above results are “typical”?

Measure-theoretic approach: define a notion of measure or nullity onthe class of sets computable in ∅′.

∆02 measure

Definition (Martingales)

1 A martingale is a function M : {finite binary strings} → Q≥0 suchthat for every string σ,

M(σ) =M(σ0) + M(σ1)

2.

2 M succeeds on a set X if

lim supn→∞

M(X � n) =∞.

∆02 measure

Definition (Martingales)

1 A martingale is a function M : {finite binary strings} → Q≥0 suchthat for every string σ,

M(σ) =M(σ0) + M(σ1)

2.

2 M succeeds on a set X if

lim supn→∞

M(X � n) =∞.

∆02 measure

Theorem (Ville, 1939)

A class C of reals has Lebesgue measure zero if and only if there is amartingale that succeeds on every X ∈ C .

Definition (∆02 nullity)

A class C of reals has ∆02 measure zero if there is a martingale Mcomputable in ∅′ that succeeds on every X ∈ C .

∆02 measure

Theorem (Ville, 1939)

A class C of reals has Lebesgue measure zero if and only if there is amartingale that succeeds on every X ∈ C .

Definition (∆02 nullity)

A class C of reals has ∆02 measure zero if there is a martingale Mcomputable in ∅′ that succeeds on every X ∈ C .

∆02 measure

Reasonable notion of measure on the class of sets computable in ∅′:The class of all sets computable in ∅′ does not have ∆02 measure zero.For each set X computable in ∅′, the singleton {X} has ∆02 measurezero.

Gives us a notion of measure on the class of computable stablecolorings.

∆02 measure

Reasonable notion of measure on the class of sets computable in ∅′:The class of all sets computable in ∅′ does not have ∆02 measure zero.For each set X computable in ∅′, the singleton {X} has ∆02 measurezero.

Gives us a notion of measure on the class of computable stablecolorings.

∆02 measure

There exists a computable stable f : [ω]2 → 2 with no low homogeneousset.

Theorem (Hirschfeldt and Terwijn, 2008)

The class of computable stable colorings with no low homogeneous set has∆02 measure zero.

∆02 measure

There exists a computable stable f : [ω]2 → 2 with no low homogeneousset.

Theorem (Hirschfeldt and Terwijn, 2008)

The class of computable stable colorings with no low homogeneous set has∆02 measure zero.

∆02 measure

Theorem (Mileti, 2005)

There is no single set X

∆02 measure

There is no single set X

∆02 measure

. . .

space

∆02 measure

. . .

Computable stablecolorings with a hom.set H ≤T ∅.

space

∆02 measure

. . .

space

∆02 measure

. . .

Computable stablecolorings with a hom.set H ≤T X .

space

∆02 measure

There is no single low2 set that computes a homogeneous set for everycomputable stable f : [ω]2 → 2.

Theorem (Dzhafarov, submitted)

There is a set X ≤T 0′′ such that the class of computable stable coloringshaving a hom. set H ≤T X does not have ∆02 measure zero but is notequal to the class of all computable stable colorings.

∆02 measure

There is no single low2 set that computes a homogeneous set for everycomputable stable f : [ω]2 → 2.

Theorem (Dzhafarov, submitted)

There is a set X ≤T 0′′ such that the class of computable stable coloringshaving a hom. set H ≤T X does not have ∆02 measure zero but is notequal to the class of all computable stable colorings.

∆02 measure

. . .

space

∆02 measure

. . .

space

∆02 measure

. . .

Computable stablecolorings with a hom.set H ≤T X .

space

∆02 measure

. . .