The strength of Ramsey Theorem for coloring
relatively large sets
Lorenzo Carlucci
(joint work with Konrad Zdanowski)
Department of Computer Science
University of Rome “La Sapienza”
April 2012
Workshop on Proof Theory and Modal Logic
University of Barcelona
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
1 / 31
Outline
1
Motivation and Background
Ramsey Theorem for large sets
RT(!ω) and Ramsey’s Theorem
Known facts about Ramsey Theorems
2
Lower Bounds on RT(!ω)
Weak Lower Bound on RT(!ω)
Strong Lower Bounds on RT(!ω)
3
Upper Bounds on RT(!ω)
4
Conclusion and prospects
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
2 / 31
Outline
1
Motivation and Background
Ramsey Theorem for large sets
RT(!ω) and Ramsey’s Theorem
Known facts about Ramsey Theorems
2
Lower Bounds on RT(!ω)
Weak Lower Bound on RT(!ω)
Strong Lower Bounds on RT(!ω)
3
Upper Bounds on RT(!ω)
4
Conclusion and prospects
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
3 / 31
Overview
We analyze the effective and proof-theoretic strength of an infinitary
Ramsey-type theorem due to Pudlák and Rödl and independently to
Farmaki.
The main result is that the theorem is equivalent over Computable
Mathematics (RCA0 ) to closure under the ω-th Turing jump.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
4 / 31
Large Sets
A set S = {s0 , s1 , . . . , sn } is
I
I
large if n ≥ s0 , and is
exactly large if n = s0 .
The notion of large set was introduced by Paris and Harrington
and is the key notion in the Paris-Harrington Large Ramsey
Theorem.
We denote by [X ]!ω the set of exactly large subsets of X .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
5 / 31
Large Sets
A set S = {s0 , s1 , . . . , sn } is
I
I
large if n ≥ s0 , and is
exactly large if n = s0 .
The notion of large set was introduced by Paris and Harrington
and is the key notion in the Paris-Harrington Large Ramsey
Theorem.
We denote by [X ]!ω the set of exactly large subsets of X .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
5 / 31
Large Sets
A set S = {s0 , s1 , . . . , sn } is
I
I
large if n ≥ s0 , and is
exactly large if n = s0 .
The notion of large set was introduced by Paris and Harrington
and is the key notion in the Paris-Harrington Large Ramsey
Theorem.
We denote by [X ]!ω the set of exactly large subsets of X .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
5 / 31
Large Sets
A set S = {s0 , s1 , . . . , sn } is
I
I
large if n ≥ s0 , and is
exactly large if n = s0 .
The notion of large set was introduced by Paris and Harrington
and is the key notion in the Paris-Harrington Large Ramsey
Theorem.
We denote by [X ]!ω the set of exactly large subsets of X .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
5 / 31
Large Sets
A set S = {s0 , s1 , . . . , sn } is
I
I
large if n ≥ s0 , and is
exactly large if n = s0 .
The notion of large set was introduced by Paris and Harrington
and is the key notion in the Paris-Harrington Large Ramsey
Theorem.
We denote by [X ]!ω the set of exactly large subsets of X .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
5 / 31
Large Sets
A set S = {s0 , s1 , . . . , sn } is
I
I
large if n ≥ s0 , and is
exactly large if n = s0 .
The notion of large set was introduced by Paris and Harrington
and is the key notion in the Paris-Harrington Large Ramsey
Theorem.
We denote by [X ]!ω the set of exactly large subsets of X .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
5 / 31
Coloring Large Sets
The theorem we study is the following.
Theorem
For every infinite M ⊆ N, for every coloring C of the exactly large
subsets of N in two colors there exists an infinite homogeneous set
H ⊆ M.
Alternative formulations:
1
Pudlák-Rödl: uniform families.
2
Farmaki: (thin) Schreier families (Banach Space Theory).
Schreier family:
{s = {n1 , . . . , nk } : n1 ≥ k }
Thin Schreier family: n1 = k .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
6 / 31
Coloring Large Sets
The theorem we study is the following.
Theorem
For every infinite M ⊆ N, for every coloring C of the exactly large
subsets of N in two colors there exists an infinite homogeneous set
H ⊆ M.
Alternative formulations:
1
Pudlák-Rödl: uniform families.
2
Farmaki: (thin) Schreier families (Banach Space Theory).
Schreier family:
{s = {n1 , . . . , nk } : n1 ≥ k }
Thin Schreier family: n1 = k .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
6 / 31
Coloring Large Sets
The theorem we study is the following.
Theorem
For every infinite M ⊆ N, for every coloring C of the exactly large
subsets of N in two colors there exists an infinite homogeneous set
H ⊆ M.
Alternative formulations:
1
Pudlák-Rödl: uniform families.
2
Farmaki: (thin) Schreier families (Banach Space Theory).
Schreier family:
{s = {n1 , . . . , nk } : n1 ≥ k }
Thin Schreier family: n1 = k .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
6 / 31
A Combinatorial Proof of RT(!ω)
Let M be an infinite subset of N, let C : [N]!ω → 2. We build an infinite
homogeneous subset L ⊆ M for C in stages. Let
Ca : [N \ {1, . . . , a}]a → 2 be defined by
Ca (x1 , . . . , xa ) = C(a, x1 , . . . , xa ).
We define a sequence {(ai , Xi )}i∈N such that
a0 = min(M),
Xi+1 ⊆ Xi ⊆ M,
Xi is an infinite and Cai –homogeneous and ai < min(Xi ),
ai+1 = min Xi .
At the i-th step of the construction use Ramsey’s Theorem for coloring
ai –tuples from the infinite set Xi−1 (where X−1 = M).
Finally apply the Infinite Pigeonhole Principle to the sequence {ai }i∈N
to get an infinite C–homogeneous set.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
7 / 31
Outline
1
Motivation and Background
Ramsey Theorem for large sets
RT(!ω) and Ramsey’s Theorem
Known facts about Ramsey Theorems
2
Lower Bounds on RT(!ω)
Weak Lower Bound on RT(!ω)
Strong Lower Bounds on RT(!ω)
3
Upper Bounds on RT(!ω)
4
Conclusion and prospects
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
8 / 31
RT(!ω) and ∀nRTn2
Proposition
Over RCA0 , RT(!ω) implies ∀nRTn2 .
Let n ≥ 1 and C : [N]n → 2 be given. Define C 0 : [N]!ω → 2 as follows.
Let S = {s0 , . . . , sm } be an exactly large set (then m = s0 ).
(
C(s0 , . . . , sn−1 ) if s0 ≥ n,
C 0 (s) :=
0
otherwise.
1
Let H be an infinite C 0 -homogeneous set of colour i ∈ {0, 1}.
2
Let H 0 = H ∩ [n, ∞), and let S ∈ [H 0 ]n . Then min(S) ≥ n.
3
Let S 0 be any exactly large set extending S in H 0 .
4
Then C(S) = C 0 (S 0 ) = i. Thus H 0 is C-homogeneous of color i.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
9 / 31
RT(!ω) and ∀nRTn2
Proposition
Over RCA0 , RT(!ω) implies ∀nRTn2 .
Let n ≥ 1 and C : [N]n → 2 be given. Define C 0 : [N]!ω → 2 as follows.
Let S = {s0 , . . . , sm } be an exactly large set (then m = s0 ).
(
C(s0 , . . . , sn−1 ) if s0 ≥ n,
C 0 (s) :=
0
otherwise.
1
Let H be an infinite C 0 -homogeneous set of colour i ∈ {0, 1}.
2
Let H 0 = H ∩ [n, ∞), and let S ∈ [H 0 ]n . Then min(S) ≥ n.
3
Let S 0 be any exactly large set extending S in H 0 .
4
Then C(S) = C 0 (S 0 ) = i. Thus H 0 is C-homogeneous of color i.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
9 / 31
RT(!ω) and ∀nRTn2
Proposition
Over RCA0 , RT(!ω) implies ∀nRTn2 .
Let n ≥ 1 and C : [N]n → 2 be given. Define C 0 : [N]!ω → 2 as follows.
Let S = {s0 , . . . , sm } be an exactly large set (then m = s0 ).
(
C(s0 , . . . , sn−1 ) if s0 ≥ n,
C 0 (s) :=
0
otherwise.
1
Let H be an infinite C 0 -homogeneous set of colour i ∈ {0, 1}.
2
Let H 0 = H ∩ [n, ∞), and let S ∈ [H 0 ]n . Then min(S) ≥ n.
3
Let S 0 be any exactly large set extending S in H 0 .
4
Then C(S) = C 0 (S 0 ) = i. Thus H 0 is C-homogeneous of color i.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
9 / 31
RT(!ω) and ∀nRTn2
Proposition
Over RCA0 , RT(!ω) implies ∀nRTn2 .
Let n ≥ 1 and C : [N]n → 2 be given. Define C 0 : [N]!ω → 2 as follows.
Let S = {s0 , . . . , sm } be an exactly large set (then m = s0 ).
(
C(s0 , . . . , sn−1 ) if s0 ≥ n,
C 0 (s) :=
0
otherwise.
1
Let H be an infinite C 0 -homogeneous set of colour i ∈ {0, 1}.
2
Let H 0 = H ∩ [n, ∞), and let S ∈ [H 0 ]n . Then min(S) ≥ n.
3
Let S 0 be any exactly large set extending S in H 0 .
4
Then C(S) = C 0 (S 0 ) = i. Thus H 0 is C-homogeneous of color i.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
9 / 31
RT(!ω) and ∀nRTn2
Proposition
Over RCA0 , RT(!ω) implies ∀nRTn2 .
Let n ≥ 1 and C : [N]n → 2 be given. Define C 0 : [N]!ω → 2 as follows.
Let S = {s0 , . . . , sm } be an exactly large set (then m = s0 ).
(
C(s0 , . . . , sn−1 ) if s0 ≥ n,
C 0 (s) :=
0
otherwise.
1
Let H be an infinite C 0 -homogeneous set of colour i ∈ {0, 1}.
2
Let H 0 = H ∩ [n, ∞), and let S ∈ [H 0 ]n . Then min(S) ≥ n.
3
Let S 0 be any exactly large set extending S in H 0 .
4
Then C(S) = C 0 (S 0 ) = i. Thus H 0 is C-homogeneous of color i.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
9 / 31
Outline
1
Motivation and Background
Ramsey Theorem for large sets
RT(!ω) and Ramsey’s Theorem
Known facts about Ramsey Theorems
2
Lower Bounds on RT(!ω)
Weak Lower Bound on RT(!ω)
Strong Lower Bounds on RT(!ω)
3
Upper Bounds on RT(!ω)
4
Conclusion and prospects
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
10 / 31
Computability results
Theorem (Jockusch 1972)
1
2
3
For each n ≥ 2 there exists a computable coloring C : [N]n → 2
with no infinite homogeneous set in Σ0n .
For each n, for each computable coloring C : [N]n → 2, there
exists an infinite C-homogeneous set in Π0n .
For each n ≥ 2 there exists a computable coloring C : [N]n → 2 all
of whose homogeneous sets compute 0(n−2) .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
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Reverse Mathematics results
Theorem (Simpson)
The following are equivalent over RCA0 .
1
RT3 ,
2
RTn for any n ∈ N, n ≥ 3,
3
∀X ∃Y (Y = X 0 ).
Thus RCA0 + RT3 ≡ ACA0 .
Theorem (McAloon 1985)
The following are equivalent over RCA0 .
1
∀nRTn ,
2
∀n∀X ∃Y (Y = X (n) ).
Thus RCA0 + ∀nRTn ≡ ACA00 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
12 / 31
Outline
1
Motivation and Background
Ramsey Theorem for large sets
RT(!ω) and Ramsey’s Theorem
Known facts about Ramsey Theorems
2
Lower Bounds on RT(!ω)
Weak Lower Bound on RT(!ω)
Strong Lower Bounds on RT(!ω)
3
Upper Bounds on RT(!ω)
4
Conclusion and prospects
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
13 / 31
A weak Lower Bound
Theorem
There exists a computable coloring C : [N]!ω → 2 such that any infinite
homogeneous set for C is not Σ0i , for any i ∈ N.
Recall that there exists sets that are incomparable with all 0(i) , i ≥ 1.
Proposition
There exists a computable sequence of X -computable functions
enX : [N]n+2 → {0, 1} such that for any n ≥ 0, for every i ∈ N,
1
enKi is Ki -computable, and
2
enKi computes a coloring with no homogeneous set in Σ0i+n+2 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
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Proof of Theorem from Proposition
Define C : [N]!ω → 2 as follows.
C(s0 , s1 , . . . , ss0 ) = esK00−2 (s1 , . . . , ss0 ).
If Y is ihs for C then for every a ∈ Y , Y ∩ {x ∈ N : x ≥ a} is ihs for
K0
. Then it is not Σ0a by Proposition.
ea−2
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Ramsey Theorem for large sets
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15 / 31
Proof of Proposition
Ingredients:
1
2
Jockusch’s result: There exists a computable e : [N]2 → 2 without
ihs in Σ02 .
Relativizes to: There exists an X -computable eX : [N]2 → 2
without ihs in Σ0i+2 if X is Σ0i -complete.
3
Schoenfield’s Limit Lemma: functions computable in K are
lim-computable.
4
Generalizes to: If B is c.e. in A and f is computable in B then f is
A-lim-computable.
5
Uniform formulation: there exists a function g X (i, e, x, s) such that,
if B = WiA and f = {e}B then
f (x) = lim g A (i, e, x, s)
s→∞
.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
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Proof of Proposition
Ingredients:
1
2
Jockusch’s result: There exists a computable e : [N]2 → 2 without
ihs in Σ02 .
Relativizes to: There exists an X -computable eX : [N]2 → 2
without ihs in Σ0i+2 if X is Σ0i -complete.
3
Schoenfield’s Limit Lemma: functions computable in K are
lim-computable.
4
Generalizes to: If B is c.e. in A and f is computable in B then f is
A-lim-computable.
5
Uniform formulation: there exists a function g X (i, e, x, s) such that,
if B = WiA and f = {e}B then
f (x) = lim g A (i, e, x, s)
s→∞
.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
16 / 31
Proof of Proposition
Ingredients:
1
2
Jockusch’s result: There exists a computable e : [N]2 → 2 without
ihs in Σ02 .
Relativizes to: There exists an X -computable eX : [N]2 → 2
without ihs in Σ0i+2 if X is Σ0i -complete.
3
Schoenfield’s Limit Lemma: functions computable in K are
lim-computable.
4
Generalizes to: If B is c.e. in A and f is computable in B then f is
A-lim-computable.
5
Uniform formulation: there exists a function g X (i, e, x, s) such that,
if B = WiA and f = {e}B then
f (x) = lim g A (i, e, x, s)
s→∞
.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
16 / 31
Proof of Proposition
Ingredients:
1
2
Jockusch’s result: There exists a computable e : [N]2 → 2 without
ihs in Σ02 .
Relativizes to: There exists an X -computable eX : [N]2 → 2
without ihs in Σ0i+2 if X is Σ0i -complete.
3
Schoenfield’s Limit Lemma: functions computable in K are
lim-computable.
4
Generalizes to: If B is c.e. in A and f is computable in B then f is
A-lim-computable.
5
Uniform formulation: there exists a function g X (i, e, x, s) such that,
if B = WiA and f = {e}B then
f (x) = lim g A (i, e, x, s)
s→∞
.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
16 / 31
Proof of Proposition
Ingredients:
1
2
Jockusch’s result: There exists a computable e : [N]2 → 2 without
ihs in Σ02 .
Relativizes to: There exists an X -computable eX : [N]2 → 2
without ihs in Σ0i+2 if X is Σ0i -complete.
3
Schoenfield’s Limit Lemma: functions computable in K are
lim-computable.
4
Generalizes to: If B is c.e. in A and f is computable in B then f is
A-lim-computable.
5
Uniform formulation: there exists a function g X (i, e, x, s) such that,
if B = WiA and f = {e}B then
f (x) = lim g A (i, e, x, s)
s→∞
.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
16 / 31
Proof of Proposition
Ingredients:
1
2
Jockusch’s result: There exists a computable e : [N]2 → 2 without
ihs in Σ02 .
Relativizes to: There exists an X -computable eX : [N]2 → 2
without ihs in Σ0i+2 if X is Σ0i -complete.
3
Schoenfield’s Limit Lemma: functions computable in K are
lim-computable.
4
Generalizes to: If B is c.e. in A and f is computable in B then f is
A-lim-computable.
5
Uniform formulation: there exists a function g X (i, e, x, s) such that,
if B = WiA and f = {e}B then
f (x) = lim g A (i, e, x, s)
s→∞
.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
16 / 31
Proof of Proposition (continued)
1
2
3
Fix {Ki }i∈N s.t. K0 = ∅, Ki+1 is Σ0i+1 -complete.
Base: result of Jockusch’s, relativized.
X : [N]n+3 → {0, 1}. Ensure
Step: define en+1
1
2
4
K
Ki
⇒ homset for en i+1 .
If X = Ki then: homset for en+1
X
Index for en+1
obtained effectively from index for enX .
By Shoenfield’s Lemma we have that
K
lim g Ki (en , x1 , . . . , xn+2 , s) = en i+1 (x1 , . . . , xn+2 ).
s→∞
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Ramsey Theorem for large sets
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Proof of Proposition (continued)
1
2
3
Fix {Ki }i∈N s.t. K0 = ∅, Ki+1 is Σ0i+1 -complete.
Base: result of Jockusch’s, relativized.
X : [N]n+3 → {0, 1}. Ensure
Step: define en+1
1
2
4
K
Ki
⇒ homset for en i+1 .
If X = Ki then: homset for en+1
X
Index for en+1
obtained effectively from index for enX .
By Shoenfield’s Lemma we have that
K
lim g Ki (en , x1 , . . . , xn+2 , s) = en i+1 (x1 , . . . , xn+2 ).
s→∞
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
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Proof of Proposition (continued)
1
2
3
Fix {Ki }i∈N s.t. K0 = ∅, Ki+1 is Σ0i+1 -complete.
Base: result of Jockusch’s, relativized.
X : [N]n+3 → {0, 1}. Ensure
Step: define en+1
1
2
4
K
Ki
⇒ homset for en i+1 .
If X = Ki then: homset for en+1
X
Index for en+1
obtained effectively from index for enX .
By Shoenfield’s Lemma we have that
K
lim g Ki (en , x1 , . . . , xn+2 , s) = en i+1 (x1 , . . . , xn+2 ).
s→∞
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
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Proof of Proposition (continued)
1
2
3
Fix {Ki }i∈N s.t. K0 = ∅, Ki+1 is Σ0i+1 -complete.
Base: result of Jockusch’s, relativized.
X : [N]n+3 → {0, 1}. Ensure
Step: define en+1
1
2
4
K
Ki
⇒ homset for en i+1 .
If X = Ki then: homset for en+1
X
Index for en+1
obtained effectively from index for enX .
By Shoenfield’s Lemma we have that
K
lim g Ki (en , x1 , . . . , xn+2 , s) = en i+1 (x1 , . . . , xn+2 ).
s→∞
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
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Proof of Proposition (continued)
1
2
3
Fix {Ki }i∈N s.t. K0 = ∅, Ki+1 is Σ0i+1 -complete.
Base: result of Jockusch’s, relativized.
X : [N]n+3 → {0, 1}. Ensure
Step: define en+1
1
2
4
K
Ki
⇒ homset for en i+1 .
If X = Ki then: homset for en+1
X
Index for en+1
obtained effectively from index for enX .
By Shoenfield’s Lemma we have that
K
lim g Ki (en , x1 , . . . , xn+2 , s) = en i+1 (x1 , . . . , xn+2 ).
s→∞
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Ramsey Theorem for large sets
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X
Define en+1
as follows.
X
en+1
(x1 , . . . , xn+2 , s) := g X (en , x1 , . . . , xn+2 , s).
Ki
If Y is an infinite homset for en+1
colored b ∈ {0, 1}, then any tuple
K
(x1 , . . . , xn+2 ) ∈ [Y ]n+2 has to be colored b by en i+1
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Ramsey Theorem for large sets
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Outline
1
Motivation and Background
Ramsey Theorem for large sets
RT(!ω) and Ramsey’s Theorem
Known facts about Ramsey Theorems
2
Lower Bounds on RT(!ω)
Weak Lower Bound on RT(!ω)
Strong Lower Bounds on RT(!ω)
3
Upper Bounds on RT(!ω)
4
Conclusion and prospects
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
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Strong Lower Bounds
Theorem
For each set A there exists a computable in A coloring Cω : [N]!ω → 2
such that all infinite homogeneous sets for Cω compute A(ω) .
Proposition
There exists computable in A colorings Cn : [N]n+1 → {0, 1}, for n ∈ N
and n ≥ 2, and TMs Mn (x, y ) s.t., for any n ≥ 2, the following three
points hold.
1
All infinite homogeneous sets for Cn have color 1.
2
If X is an ihs for Cn then for any a1 < · · · < an+1 ∈ X then
Mn (x, (a1 , . . . , an+1 )) decides A(n−1) for machines with indices
less than or equal to a1 .
3
Machines Mn are total. (If their inputs are not from an ihs for Cn
then we have no guarantee on the correctness of their output).
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Ramsey Theorem for large sets
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Strong Lower Bounds
Theorem
For each set A there exists a computable in A coloring Cω : [N]!ω → 2
such that all infinite homogeneous sets for Cω compute A(ω) .
Proposition
There exists computable in A colorings Cn : [N]n+1 → {0, 1}, for n ∈ N
and n ≥ 2, and TMs Mn (x, y ) s.t., for any n ≥ 2, the following three
points hold.
1
All infinite homogeneous sets for Cn have color 1.
2
If X is an ihs for Cn then for any a1 < · · · < an+1 ∈ X then
Mn (x, (a1 , . . . , an+1 )) decides A(n−1) for machines with indices
less than or equal to a1 .
3
Machines Mn are total. (If their inputs are not from an ihs for Cn
then we have no guarantee on the correctness of their output).
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
20 / 31
Strong Lower Bounds
Theorem
For each set A there exists a computable in A coloring Cω : [N]!ω → 2
such that all infinite homogeneous sets for Cω compute A(ω) .
Proposition
There exists computable in A colorings Cn : [N]n+1 → {0, 1}, for n ∈ N
and n ≥ 2, and TMs Mn (x, y ) s.t., for any n ≥ 2, the following three
points hold.
1
All infinite homogeneous sets for Cn have color 1.
2
If X is an ihs for Cn then for any a1 < · · · < an+1 ∈ X then
Mn (x, (a1 , . . . , an+1 )) decides A(n−1) for machines with indices
less than or equal to a1 .
3
Machines Mn are total. (If their inputs are not from an ihs for Cn
then we have no guarantee on the correctness of their output).
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
20 / 31
Strong Lower Bounds
Theorem
For each set A there exists a computable in A coloring Cω : [N]!ω → 2
such that all infinite homogeneous sets for Cω compute A(ω) .
Proposition
There exists computable in A colorings Cn : [N]n+1 → {0, 1}, for n ∈ N
and n ≥ 2, and TMs Mn (x, y ) s.t., for any n ≥ 2, the following three
points hold.
1
All infinite homogeneous sets for Cn have color 1.
2
If X is an ihs for Cn then for any a1 < · · · < an+1 ∈ X then
Mn (x, (a1 , . . . , an+1 )) decides A(n−1) for machines with indices
less than or equal to a1 .
3
Machines Mn are total. (If their inputs are not from an ihs for Cn
then we have no guarantee on the correctness of their output).
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
20 / 31
Proof of Theorem from Proposition
Define
Cω (a1 , . . . , aa1 +1 ) = Ca1 −1 (a2 , . . . , aa1 +1 ).
Claim: Any ihs H for Cω computes Aω = {(i, j) : j ∈ Ai }.
For any h ∈ H, Cω coincides with Ch−1 on large sets with
minimum h and rest in H − = H − {1, . . . , h}.
Thus, Mh−1 (e, σ) decides A(h−2) up to min(σ) if σ is picked in H − .
Given (i, j) find a1 ∈ H such that i, j ≤ (a1 − 2).
(j ∈ Ai ) iff fi,(a1 −2) (j) ∈ A(a1 −2) (fi,(a1 −2) reducing Ai to Aa1 −2 ).
Find (a1 − 1) values σ in H above fi,(a1 −2) (j).
Output M(a1 −1) (fi,(a1 −2) (j), σ).
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
21 / 31
Proof of Theorem from Proposition
Define
Cω (a1 , . . . , aa1 +1 ) = Ca1 −1 (a2 , . . . , aa1 +1 ).
Claim: Any ihs H for Cω computes Aω = {(i, j) : j ∈ Ai }.
For any h ∈ H, Cω coincides with Ch−1 on large sets with
minimum h and rest in H − = H − {1, . . . , h}.
Thus, Mh−1 (e, σ) decides A(h−2) up to min(σ) if σ is picked in H − .
Given (i, j) find a1 ∈ H such that i, j ≤ (a1 − 2).
(j ∈ Ai ) iff fi,(a1 −2) (j) ∈ A(a1 −2) (fi,(a1 −2) reducing Ai to Aa1 −2 ).
Find (a1 − 1) values σ in H above fi,(a1 −2) (j).
Output M(a1 −1) (fi,(a1 −2) (j), σ).
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
21 / 31
Proof of Theorem from Proposition
Define
Cω (a1 , . . . , aa1 +1 ) = Ca1 −1 (a2 , . . . , aa1 +1 ).
Claim: Any ihs H for Cω computes Aω = {(i, j) : j ∈ Ai }.
For any h ∈ H, Cω coincides with Ch−1 on large sets with
minimum h and rest in H − = H − {1, . . . , h}.
Thus, Mh−1 (e, σ) decides A(h−2) up to min(σ) if σ is picked in H − .
Given (i, j) find a1 ∈ H such that i, j ≤ (a1 − 2).
(j ∈ Ai ) iff fi,(a1 −2) (j) ∈ A(a1 −2) (fi,(a1 −2) reducing Ai to Aa1 −2 ).
Find (a1 − 1) values σ in H above fi,(a1 −2) (j).
Output M(a1 −1) (fi,(a1 −2) (j), σ).
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
21 / 31
Proof of Theorem from Proposition
Define
Cω (a1 , . . . , aa1 +1 ) = Ca1 −1 (a2 , . . . , aa1 +1 ).
Claim: Any ihs H for Cω computes Aω = {(i, j) : j ∈ Ai }.
For any h ∈ H, Cω coincides with Ch−1 on large sets with
minimum h and rest in H − = H − {1, . . . , h}.
Thus, Mh−1 (e, σ) decides A(h−2) up to min(σ) if σ is picked in H − .
Given (i, j) find a1 ∈ H such that i, j ≤ (a1 − 2).
(j ∈ Ai ) iff fi,(a1 −2) (j) ∈ A(a1 −2) (fi,(a1 −2) reducing Ai to Aa1 −2 ).
Find (a1 − 1) values σ in H above fi,(a1 −2) (j).
Output M(a1 −1) (fi,(a1 −2) (j), σ).
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
21 / 31
Proof of Theorem from Proposition
Define
Cω (a1 , . . . , aa1 +1 ) = Ca1 −1 (a2 , . . . , aa1 +1 ).
Claim: Any ihs H for Cω computes Aω = {(i, j) : j ∈ Ai }.
For any h ∈ H, Cω coincides with Ch−1 on large sets with
minimum h and rest in H − = H − {1, . . . , h}.
Thus, Mh−1 (e, σ) decides A(h−2) up to min(σ) if σ is picked in H − .
Given (i, j) find a1 ∈ H such that i, j ≤ (a1 − 2).
(j ∈ Ai ) iff fi,(a1 −2) (j) ∈ A(a1 −2) (fi,(a1 −2) reducing Ai to Aa1 −2 ).
Find (a1 − 1) values σ in H above fi,(a1 −2) (j).
Output M(a1 −1) (fi,(a1 −2) (j), σ).
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
21 / 31
Proof of Theorem from Proposition
Define
Cω (a1 , . . . , aa1 +1 ) = Ca1 −1 (a2 , . . . , aa1 +1 ).
Claim: Any ihs H for Cω computes Aω = {(i, j) : j ∈ Ai }.
For any h ∈ H, Cω coincides with Ch−1 on large sets with
minimum h and rest in H − = H − {1, . . . , h}.
Thus, Mh−1 (e, σ) decides A(h−2) up to min(σ) if σ is picked in H − .
Given (i, j) find a1 ∈ H such that i, j ≤ (a1 − 2).
(j ∈ Ai ) iff fi,(a1 −2) (j) ∈ A(a1 −2) (fi,(a1 −2) reducing Ai to Aa1 −2 ).
Find (a1 − 1) values σ in H above fi,(a1 −2) (j).
Output M(a1 −1) (fi,(a1 −2) (j), σ).
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
21 / 31
Proof of Theorem from Proposition
Define
Cω (a1 , . . . , aa1 +1 ) = Ca1 −1 (a2 , . . . , aa1 +1 ).
Claim: Any ihs H for Cω computes Aω = {(i, j) : j ∈ Ai }.
For any h ∈ H, Cω coincides with Ch−1 on large sets with
minimum h and rest in H − = H − {1, . . . , h}.
Thus, Mh−1 (e, σ) decides A(h−2) up to min(σ) if σ is picked in H − .
Given (i, j) find a1 ∈ H such that i, j ≤ (a1 − 2).
(j ∈ Ai ) iff fi,(a1 −2) (j) ∈ A(a1 −2) (fi,(a1 −2) reducing Ai to Aa1 −2 ).
Find (a1 − 1) values σ in H above fi,(a1 −2) (j).
Output M(a1 −1) (fi,(a1 −2) (j), σ).
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
21 / 31
Proof of Proposition
For this proof let
X 0 = {e : {e}X (0)↓}.
We define C2 as
(
C2 (k , y , z) =
1
2
1 if ∀e ≤ k ({e}Ay (0)↓ ⇔ {e}Az (0)↓)
0 otherwise.
Any infinite homset has color 1.
We can build machine M2 (e, (k , b, b0 )) deciding A0 up to k if
b0 > b > k are from an infinite homset for C2 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
22 / 31
Proof of Proposition, continued
We set Cn+1 (a1 , . . . , an+2 ) =


1 if {a1 , . . . , an+2 } is Cn –homogeneous and




∀e ≤ a1 ({e}Ya2 (0)↓ ⇔ {e}Ya3 (0)↓), where

Y = {i ≤ a2 : Mn (i, (a2 , . . . , an+2 )) accepts,}



0 otherwise.
We are approximating the condition
(n−1)
∀e ≤ a1 ({e}Aa2
(n−1)
(0)↓ ⇔ {e}Aa3
(0)↓)
Using homegeneity and infinity we can get
(n−1)
∀e ≤ a1 ({e}Ya2 (0)↓ ⇔ {e}A
(0)↓),
for suitable a1 , a2 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
23 / 31
Reverse Mathematics Corollary
Theorem
RT(!ω) implies ∀X ∃Y (Y = X (ω) ) over RCA0 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
24 / 31
Alternative Proof of Theorem
We use the following Proposition, which can be proved by adapting an
argument from a recent paper by Dzhafarov and Hirst (2011).
Proposition
For every set X there exists a computable in X coloring C X : [N]!ω → 2
such that if H ⊆ 2N is an infinite homogeneous set for C then H
computes X (n−1) for every 2n ∈ H.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
25 / 31
Proof of Theorem from Proposition
Recall {0i : i ∈ N} has no lub.
Enderton-Putnam result: If I is infinite such that (∀i ∈ I)(X i ≤T Y ),
then X ω ≤T Y 2 .
Prove Theorem as follows.
1
2
3
X computable, C X as in Proposition, H ihs for C X .
Then for infinitely many i ∈ N, X i ≤T H.
Then X ω ≤T H 2 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
26 / 31
Proof of Theorem from Proposition
Recall {0i : i ∈ N} has no lub.
Enderton-Putnam result: If I is infinite such that (∀i ∈ I)(X i ≤T Y ),
then X ω ≤T Y 2 .
Prove Theorem as follows.
1
2
3
X computable, C X as in Proposition, H ihs for C X .
Then for infinitely many i ∈ N, X i ≤T H.
Then X ω ≤T H 2 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
26 / 31
Proof of Theorem from Proposition
Recall {0i : i ∈ N} has no lub.
Enderton-Putnam result: If I is infinite such that (∀i ∈ I)(X i ≤T Y ),
then X ω ≤T Y 2 .
Prove Theorem as follows.
1
2
3
X computable, C X as in Proposition, H ihs for C X .
Then for infinitely many i ∈ N, X i ≤T H.
Then X ω ≤T H 2 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
26 / 31
Proof of Theorem from Proposition
Recall {0i : i ∈ N} has no lub.
Enderton-Putnam result: If I is infinite such that (∀i ∈ I)(X i ≤T Y ),
then X ω ≤T Y 2 .
Prove Theorem as follows.
1
2
3
X computable, C X as in Proposition, H ihs for C X .
Then for infinitely many i ∈ N, X i ≤T H.
Then X ω ≤T H 2 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
26 / 31
Proof of Theorem from Proposition
Recall {0i : i ∈ N} has no lub.
Enderton-Putnam result: If I is infinite such that (∀i ∈ I)(X i ≤T Y ),
then X ω ≤T Y 2 .
Prove Theorem as follows.
1
2
3
X computable, C X as in Proposition, H ihs for C X .
Then for infinitely many i ∈ N, X i ≤T H.
Then X ω ≤T H 2 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
26 / 31
Proof of Theorem from Proposition
Recall {0i : i ∈ N} has no lub.
Enderton-Putnam result: If I is infinite such that (∀i ∈ I)(X i ≤T Y ),
then X ω ≤T Y 2 .
Prove Theorem as follows.
1
2
3
X computable, C X as in Proposition, H ihs for C X .
Then for infinitely many i ∈ N, X i ≤T H.
Then X ω ≤T H 2 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
26 / 31
Proof of Theorem from Proposition
Recall {0i : i ∈ N} has no lub.
Enderton-Putnam result: If I is infinite such that (∀i ∈ I)(X i ≤T Y ),
then X ω ≤T Y 2 .
Prove Theorem as follows.
1
2
3
X computable, C X as in Proposition, H ihs for C X .
Then for infinitely many i ∈ N, X i ≤T H.
Then X ω ≤T H 2 .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
26 / 31
Proof of Proposition
Y = X 0 ⇔ ∀x∀e(hx, ei ∈ Y ↔ ∃s({e}Xs (x)↓)).
For any set X and integer s define
X
Xs0 = {hm, ei : (∃t < s)m ∈ We,t
}.
For integers u1 , . . . , un and s define X (0) = X , and
(n+1)
(n)
Xun ,...,u1 ,s = (Xun ,...,u1 )0s .
Let A = {a0 , . . . , a2n+1 } be exactly large. Let C X (A) = 1 if there exist
1 ≤ i ≤ n and ∃(e, m) < an−i such that
(i)
(i)
¬((m, e) ∈ Xan ,...,an−i+1 ↔ (m, e) ∈ Xa2n ,...,a2n−i+1 )
and C X (A) = 0 otherwise.
Let H ⊆ [2, ∞) ∩ 2N be an ihs for C X . We claim that
1
The color of C X on [H]!ω is 0.
2
For every h ∈ H, X (n−1) is definable by recursive comprehension
from H, where h = 2n.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
27 / 31
A proof in ACA+
0
Theorem
∀X ∃Y (Y = X (ω) ) implies RT(!ω) over RCA0 .
Idea of proof:
Turn the induction in the combinatorial proof of RT(!ω) into a
first-order induction.
Replace the sets Xi by Turing machines with oracles from C (a) ,
(for a an element of a model of RCA0 ) constructed in a uniform
way and computing the sets Xi .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
28 / 31
A proof in ACA+
0
Theorem
∀X ∃Y (Y = X (ω) ) implies RT(!ω) over RCA0 .
Idea of proof:
Turn the induction in the combinatorial proof of RT(!ω) into a
first-order induction.
Replace the sets Xi by Turing machines with oracles from C (a) ,
(for a an element of a model of RCA0 ) constructed in a uniform
way and computing the sets Xi .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
28 / 31
A proof in ACA+
0
Theorem
∀X ∃Y (Y = X (ω) ) implies RT(!ω) over RCA0 .
Idea of proof:
Turn the induction in the combinatorial proof of RT(!ω) into a
first-order induction.
Replace the sets Xi by Turing machines with oracles from C (a) ,
(for a an element of a model of RCA0 ) constructed in a uniform
way and computing the sets Xi .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
28 / 31
We want the following first-order induction: for each n there exists a
sequence {(ai , fai ) : i ≤ n} such that:
a0 = 2,
rg(fai+1 ) ⊆ rg(fai ) ⊆ M,
rg(fai ) is infinite and Cai -homogeneous,
ai+1 = min(rg(fai ) ∩ {x ∈ N : x > ai }.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
29 / 31
Lemma
Let a ≥ 1. Let C : [U]a → 2. One can find effectively a machine fa with
oracle (C ⊕ U)(2a) such that fa computes a C–homogeneous set.
1
2
Base: f1 . Ask the Π02 (C ⊕ U) oracle whether
∀n∃k ≥ n(C(k ) = 0 ∧ U(k )). If yes, then compute the set
C(x) = 0 ∧ U(x), else compute the set C(x) = 1 ∧ U(x).
Step: fa+1 .
1
2
3
4
5
Build Erdős–Rado tree Ta for C : [U]a+1 → 2 (computably in C ⊕ U).
Obtain index for a machine computing the leftmost infinite path P of
Ta using a Π02 (C ⊕ U)–complete oracle.
The color of any (a + 1)–tuple from P does not depend on the last
element of the tuple. Induces C 0 coloring of a-tuples.
Use fa (inductive hypothesis) with oracle (C 0 ⊕ P)(2a) . Any infinite
C 0 –homogeneous subset of P computed by fa is also
C–homogeneous.
Since P is recursive in Π02 (C ⊕ U), the complexity of the oracle is
(C ⊕ U)(2(a+1)) .
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
30 / 31
Conclusion and prospects
1
Other results:
1
2
3
2
Analogous results for Regressive Ramsey Theorem on large sets.
Characterization in terms of ω truth-predicates over PA.
Alternative proof by reduction to well-ordering preservation
principles.
Further research:
1
2
Prove generalization for RT(!α) for α ∈ ω CK .
Extract finite forms.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
31 / 31
Conclusion and prospects
1
Other results:
1
2
3
2
Analogous results for Regressive Ramsey Theorem on large sets.
Characterization in terms of ω truth-predicates over PA.
Alternative proof by reduction to well-ordering preservation
principles.
Further research:
1
2
Prove generalization for RT(!α) for α ∈ ω CK .
Extract finite forms.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
31 / 31
Conclusion and prospects
1
Other results:
1
2
3
2
Analogous results for Regressive Ramsey Theorem on large sets.
Characterization in terms of ω truth-predicates over PA.
Alternative proof by reduction to well-ordering preservation
principles.
Further research:
1
2
Prove generalization for RT(!α) for α ∈ ω CK .
Extract finite forms.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
31 / 31
Conclusion and prospects
1
Other results:
1
2
3
2
Analogous results for Regressive Ramsey Theorem on large sets.
Characterization in terms of ω truth-predicates over PA.
Alternative proof by reduction to well-ordering preservation
principles.
Further research:
1
2
Prove generalization for RT(!α) for α ∈ ω CK .
Extract finite forms.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
31 / 31
Conclusion and prospects
1
Other results:
1
2
3
2
Analogous results for Regressive Ramsey Theorem on large sets.
Characterization in terms of ω truth-predicates over PA.
Alternative proof by reduction to well-ordering preservation
principles.
Further research:
1
2
Prove generalization for RT(!α) for α ∈ ω CK .
Extract finite forms.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
31 / 31
Conclusion and prospects
1
Other results:
1
2
3
2
Analogous results for Regressive Ramsey Theorem on large sets.
Characterization in terms of ω truth-predicates over PA.
Alternative proof by reduction to well-ordering preservation
principles.
Further research:
1
2
Prove generalization for RT(!α) for α ∈ ω CK .
Extract finite forms.
Lorenzo Carlucci (Rome I)
Ramsey Theorem for large sets
Barcelona, April 2012
31 / 31