Damir D. Dzhafarov University of Chicago 20 August 2009 · Damir D. DzhafarovStable Ramsey’s...
Transcript of Damir D. Dzhafarov University of Chicago 20 August 2009 · Damir D. DzhafarovStable Ramsey’s...
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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
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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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Ramsey’s theorem
1 20 3 4 5 . . .
A coloring [N]1 → 2.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Ramsey’s theorem
1 3 4 . . .
H = {1, 3, 4, 6, 9, 10, 11, 14, 15, . . .} is homogeneous.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Ramsey’s theorem
00123
1 2 3 4 5
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A coloring [N]2 → 2.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Ramsey’s theorem
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1 3 4
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H = {1, 3, 4, 6, 9, 10, 11, 14, 15, . . .} is homogeneous.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
Theorem (Specker, 1969)
There exists a comp. f : [N]2 → 2 with no computable homogeneous set.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
Turing hierarchy (≤T )
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
Turing hierarchy (≤T )
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
. . .
Turing hierarchy (≤T )
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
. . .
Turing hierarchy (≤T )
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
. . .
Turing hierarchy (≤T )
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
. . .
Turing hierarchy (≤T )
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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. . .
Turing hierarchy (≤T )
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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 .
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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 .
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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 .
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
Definition (Low and low2 sets)
A set X is
1 low if X ′ ≤T ∅′;2 low2 if X
′′ ≤T ∅′′.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
. . .
A low set.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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A low set.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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A low set.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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A low set.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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A low2 set.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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A low2 set.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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A low2 set.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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Lowness hierarchy.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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low sets
Lowness hierarchy.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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low2 sets
Lowness hierarchy.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
Theorem (Jockusch, 1972)
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?
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
Theorem (Jockusch, 1972)
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?
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
Theorem (Jockusch, 1972)
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?
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Computability theory
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 .
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Stable Ramsey’s theorem
Definition
A coloring f : [N]2 → k is stable if for all x , limy→∞ f ({x , y}) exists.
Theorem (Jockusch, 1972)
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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Stable Ramsey’s theorem
Definition
A coloring f : [N]2 → k is stable if for all x , limy→∞ f ({x , y}) exists.
Theorem (Jockusch, 1972)
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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Stable Ramsey’s theorem
Definition
A coloring f : [N]2 → k is stable if for all x , limy→∞ f ({x , y}) exists.
Theorem (Jockusch, 1972)
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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Stable Ramsey’s theorem
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low sets
space
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Stable Ramsey’s theorem
Theorem (Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, 2008)
Every computable stable f : [ω]2 → 2 has an homogeneous set H
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Stable Ramsey’s theorem
Theorem (Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, 2008)
Every computable stable f : [ω]2 → 2 has an homogeneous set H
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∆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 ∅′.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆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 ∅′.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆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) =∞.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆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) =∞.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆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 .
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆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 .
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
Theorem (Downey, Hirschfeldt, Lempp, and Solomon, 2001)
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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
Theorem (Downey, Hirschfeldt, Lempp, and Solomon, 2001)
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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
Theorem (Mileti, 2005)
There is no single set X
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∆02 measure
Theorem (Mileti, 2005)
There is no single set X
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∆02 measure
. . .
space
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
. . .
Computable stablecolorings with a hom.set H ≤T ∅.
space
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
. . .
space
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
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Computable stablecolorings with a hom.set H ≤T X .
space
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
Theorem (Mileti, 2005)
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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
Theorem (Mileti, 2005)
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.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
. . .
space
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
. . .
space
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
. . .
Computable stablecolorings with a hom.set H ≤T X .
space
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
. . .
low2 sets
Open Question: Can X be chosen low2?
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
. . .
low2 sets
Open Question: Can X be chosen low2?
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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∆02 measure
. . .
low2 sets
Open Question: Can X be chosen low2?
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Questions
Is there a notion of measure that would allow for a similar analysis ofRamsey’s theorem proper?
What is the situation if instead of ∆02 measure we use an effectivenotion of dimension?
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Questions
Is there a notion of measure that would allow for a similar analysis ofRamsey’s theorem proper?
What is the situation if instead of ∆02 measure we use an effectivenotion of dimension?
Damir D. Dzhafarov Stable Ramsey’s theorem and measure
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Thank you for your attention.
Damir D. Dzhafarov Stable Ramsey’s theorem and measure