Degrees bounding principles and universal instances in reverse mathematics
aa r X i v : . [ m a t h . L O ] N ov DEGREES BOUNDING PRINCIPLESAND UNIVERSAL INSTANCES IN REVERSE MATHEMATICS
LUDOVIC PATEYA
BSTRACT . A Turing degree d bounds a principle P of reverse mathematics if everycomputable instance of P has a d -computable solution. P admits a universal instance if there exists a computable instance such that every solution bounds P . We prove thatthe stable version of the ascending descending sequence principle ( SADS ) as well asthe stable version of the thin set theorem for pairs (
STS ( ) ) do not admit a bound oflow degree. Therefore no principle between Ramsey’s theorem for pairs ( RT ) and SADS or STS ( ) admit a universal instance. We construct a low degree boundingthe Erd˝os Moser theorem ( EM ) , thereby showing that previous argument does nothold for EM . Finally, we prove that the only ∆ degree bounding a stable version ofthe rainbow Ramsey theorem for pairs ( SRRT ) is ′ . Hence no principle betweenthe stable Ramsey theorem for pairs ( SRT ) and SRRT admit a universal instance.In particular the stable version of the Erd˝os Moser theorem does not admit one. Itremains unknown whether EM admits a universal instance.
1. I
NTRODUCTION
Reverse mathematics is a program whose goal is to classify theorems in functionof their computational strength, within the framework of subsystems of second orderarithmetic. Proofs are done relatively to a very weak system (
RCA ) meant to capture computational mathematics . RCA is composed of basic Peano axioms, ∆ compre-hension and Σ induction schemes. See [ ] for a good introductory book. Most ofstatements in reverse mathematics are of the form ∀ X (Φ( X ) → ∃ Y Ψ( X , Y )) where Φ and Ψ are arithmetical formulas.A set X such that Φ( X ) holds is called an instance of P and a set Y such that Ψ( X , Y ) holds is a solution to X . We can see relations between two instances X , X of a state-ment P as a mass problem consisting of computing a solution to X given any solutionto X . Definition 1.1
Given a statement P , a degree d is P -bounding ( d ≫ P ; ) if every com-putable instance X of P has a d -computable solution. A statement P admits a universalinstance if it has a computable instance X such that every solution to X bounds P .The notation d ≫ ; historically means that the degree d is PA and therefore is equiv-alent to d ≫ WKL ; where WKL is the weak König’s lemma principle, i.e. König’slemma restricted to subtrees of 2 <ω . It is well-known that WKL admits a universalinstance – e.g. take the Π class of completions of Peano arithmetics –. A few princi-ples have been proven to admit universal instances – WKL [ ] , König’s lemma ( KL ) Date : July 26, 2018. [ ] , the Ramsey-type weak weak König’s lemma ( RWWKL ) [ ] , the finite intersec-tion property ( FIP ) [ ] , the omitting partial type theorem ( OPT ) [ ] , or even therainbow Ramsey theorem for pairs ( RRT ) [ ] – but most of principles do not admitone. An important notion for proving such a result is computable reducibility. Definition 1.2
A statement P is computably reducible to a statement Q (written P ≤ c Q ) if for every instance X of P there exists an instance Y of Q computable from X suchthat each solution to Y computes relative to X a solution to X .Mileti proved in [ ] that the stable Ramsey theorem for pairs ( SRT ) admits nobound of low degree. Therefore every statement P having an ω -model with onlylow sets, and such that SRT ≤ c P , admits no universal instance. In particular noneof Ramsey’s theorem for pairs ( RT ), SRT and the Ramsey-type weak König’s lemmarelative to ; ′ ( RWKL [ ; ′ ] ) admit a universal instance. Independently, Hirschfeldt &Shore proved in [ ] that the stable ascending descending sequence principle ( SADS )admits no bound of low degree. Hence none of
SADS and the stable chain antichainprinciple (
SCAC ) admit a universal instance.We generalize both results by proving that
SADS admits not bound of low de-gree, proving therefore that if a statement P has an ω -model with only low sets and SADS ≤ c P then P admits no universal instance. We also extend the result to state-ments to which the stable thin set theorem for pairs ( STS ( ) ) computably reduces.Hence we deduce that none of the ascending descending sequence principle ( ADS ),the chain antichain principle (
CAC ), the thin set theorem for pairs ( TS ( ) ), the freeset theorem for pairs ( FS ( ) ) and their stable versions admit a universal instance.We generalize the result to arbitrary tuples and prove that none of RT n2 , FS ( n ) , TS ( n ) and their stable versions admit a universal instance for n ≥
2. The questionremains open for the rainbow Ramsey theorem for n -tuples ( RRT n2 ) with n ≥
3. Weconstruct a low degree bounding the Erd˝os Moser theorem ( EM ) , thereby showingthat previous argument does not hold for EM .Mileti proved in [ ] that the only ∆ degree bounding SRT is ′ . Using the factthat every ∆ set has an infinite incomplete ∆ subset in either it or its complement [ ] , we obtain another proof that SRT admits no universal instance. We extendthis result by proving that the only ∆ degree bounding a stable version version of therainbow Ramsey theorem for pairs ( SRRT ) is ′ . Hence none of the statements P satisfying SRRT ≤ c P ≤ c SRT admit a universal instance. In particular we deducethat neither SRRT nor the stable version of the Erd˝os Moser theorem ( SEM ) admitsa universal instance.1.1.
Notations.
Formulas . The notation ( ∀ ∞ s ) ϕ ( s ) means that ϕ ( s ) holds for all butfinitely many s , i.e. is translated to ( ∃ s )( ∀ s ≥ s ) ϕ ( s ) . Given two sets X and Y , wedenote by X ⊆ ∗ Y the statement ( ∀ ∞ s ∈ X )[ s ∈ Y ] . Accordingly, X = ∗ Y means thatboth X ⊆ ∗ Y and Y ⊆ ∗ X hold, i.e. X and Y differ by finitely many elements. Turing functional and lowness . We fix an effective enumeration of all Turing func-tionals Φ , Φ , . . . We denote by Φ e , s the partial approximation of the Turing functional Φ e at stage s . Given a set X , we denote by X ′ the jump of X and by X ( n ) the n th jumpof X . A set X is low n over Y if ( X ⊕ Y ) ( n ) ≤ Y ( n ) . A set is low n if it is low n over ; . A low n -ness index of a set X low n over Y is a Turing index e such that Φ Y ( n ) e = ( X ⊕ Y ) ( n ) . EGREES BOUNDING PRINCIPLES AND UNIVERSAL INSTANCES 3
Mathias forcing . Given two sets E and F , we denote by E < F the formula ( ∀ x ∈ E )( ∀ y ∈ F ) x < y . A Mathias condition is a pair ( F , X ) where F is a finite set, X is aninfinite set and F < X . A condition ( ˜ F , ˜ X ) extends ( F , X ) (written ( ˜ F , ˜ X ) ≤ ( F , X ) ) if F ⊆ ˜ F , ˜ X ⊆ X and ˜ F r F ⊂ X . A set G satisfies a Mathias condition ( F , X ) if F ⊂ G and G r F ⊆ X . 2. D EGREES BOUNDING COHESIVENESS
A standard proof of Ramsey’s theorem for pairs consists of reducing an arbitrarycoloring of pairs into a stable one using the cohesiveness principle. The understandingof the links between cohesiveness and stability is a very active subject of research inreverse mathematics [
4, 13, 5 ] . Definition 2.1 (Cohesiveness) An infinite set C is ~ R -cohesive for a sequence of sets R , R , . . . if for each i ∈ ω , C ⊆ ∗ R i or C ⊆ ∗ R i . A set C is cohesive (resp. r-cohesive ) ifit is ~ R -cohesive where ~ R is an enumeration of all c.e. (resp. computable) sets. COH isthe statement “Every uniform sequence of sets ~ R has an ~ R -cohesive set.”Jockusch & al. proved in [ ] the existence of a low cohesive set. Degrees bound-ing COH are quite well understood and admit a simple characterization:
Theorem 2.2 (Jockusch & Stephan [ ] ) Fix an n ∈ ω .1. For every set C such that C ′ ≫ ; ′ , C ≫ COH ; .2. There exists a uniformly ; ( n ) -computable sequence of sets ~ R such that for every ~ R -cohesive set C , ( C ⊕ ; ( n ) ) ′ ≫ ; ( n + ) .In particular, taking a set P ≫ ; ′ low over ; ′ and a set C such that C ′ = T P whoseexistence is ensured by Friedberg’s jump inversion theorem, we obtain a low degreebounding COH . The canonical ; ( n ) -computable sequence of sets ~ R whose existence isclaimed in clause 2 of Theorem 2.2 is R e = { s : Φ ; ( n + ) s e , s ( e ) ↓ = } Every ~ R -cohesive set C computes a function f ( · , · ) such that lim s ∈ C f ( e , s ) exists foreach e ∈ ω and lim s ∈ C f ( e , s ) = Φ ; ( n + ) e ( e ) for each Turing index e such that Φ ; ( n + ) e ( e ) ↓ .By a relativized version of Schoenfield’s limit lemma, ( C ⊕ ; ( n ) ) ′ computes the function˜ f ( x ) = lim s ∈ C f ( x , s ) and is therefore of PA degree relative to ; ( n + ) . Corollary 2.3
COH admits a universal instance.
Proof.
The uniformly computable sequence of sets ~ R such that the jump of every ~ R -cohesive set is of PA degree relative to ; ′ is a universal instance by previous theorem. (cid:3) Wang proved in [ ] that for every set P ≫ ; ′′ and every uniformly ; ′ -computablesequence of sets ~ R , there exists an ~ R -cohesive set C such that C ′′ ≤ T C ⊕ ; ′′ ≤ T P .Cholak & al. used in [ ] the existence of a low subuniform degree to deduce theexistence, for every set P ≫ ; ′ , of an r-cohesive set C such that C ′ ≤ T P . We can applya similar reasoning for ; ′ -computable sets, using the fact that degrees bounding COH are somehow subuniform degrees for ∆ approximations. LUDOVIC PATEY
Theorem 2.4
For every set P ≫ ; ′′ , there exists an ~ R -cohesive set C such that C ′′ ≤ T C ⊕; ′′ ≤ T P , where ~ R is the (non-uniformly computable) sequence of all ; ′ -computablesets. Proof.
Let ~ U be the uniformly computable sequence of sets defined by U e , x = { s : Φ ; ′ s e , s ( x ) = } Fix a low ~ U -cohesive set C and its C -computable bijection f : ω → C . Every set P ≫ ; ′′ , P ≫ C ′′ . Consider the uniformly C ′ -computable sequence of sets V e = { x : lim s Φ ; ′ f ( s ) e , s ( x ) = } The sequence ~ V contains every ; ′ -computable set. In particular, every ~ V -cohesive set is ~ R -cohesive. By a relativization of Wang’s result, there exists an ~ V -cohesive set C suchthat ( C ⊕ C ) ′′ ≤ T C ⊕ C ′′ = T C ⊕ ; ′′ ≤ T P . (cid:3) The proof of previous theorem shows that an application of
COH followed by anapplication of
COH [ ; ′ ] are enough to obtain a set of degree bounding COH [ ; ′ ] . Thefollowing question remains open: Question 2.5
Does
COH [ ; ′ ] admit a universal instance ?3. D EGREES BOUNDING THE ATOMIC MODEL THEOREM
The atomic model theorem is a statement of model theory admitting a simple, purelycomputability theoretic characterization over ω -models. This statement happens tohave a weak computational content and is therefore a consequence of many otherprinciples in reverse mathematics. For those reasons, the atomic model theorem is agood candidate for factorizing proofs of properties which are closed upward by theconsequence relation. Definition 3.1 (Atomic model theorem) A formula ϕ ( x , . . . , x n ) of T is an atom of atheory T if for each formula ψ ( x , . . . , x n ) , one of T ⊢ ϕ → ψ and T ⊢ ϕ → ¬ ψ holds,but not both. A theory T is atomic if, for every formula ψ ( x , . . . , x n ) consistent with T , there exists an atom ϕ ( x , . . . , x n ) of T extending it, i.e. one such that T ⊢ ϕ → ψ .A model A of T is atomic if every n -tuple from A satisfies an atom of T . AMT is thestatement “Every complete atomic theory has an atomic model”.
AMT has been introduced as a principle by Hirschfeldt & al. in [ ] . They provedthat WKL and AMT are incomparable on ω -models, proved over RCA that AMT is strictly weaker than
SADS . The author proved in [ ] that STS ( ) implies AMT over
RCA . In this section we use the fact that AMT is not bounded by any ∆ low degree to deduce that none of AMT , SADS and
SCAC admits a universal instance.The principle
AMT has been proven in [
15, 6 ] to be computably equivalent to thefollowing principle: Definition 3.2 (Escape property) For every ∆ function f , there exists a function g such that f ( x ) ≤ g ( x ) for infinitely many x . EGREES BOUNDING PRINCIPLES AND UNIVERSAL INSTANCES 5
This equivalence does not hold over
RCA as, unlike AMT , the escape propertyimplies IΣ over BΣ [ ] . Using this characterization, we can easily deduce the twofollowing theorems: Theorem 3.3 (Hirschfeldt & al. [ ] ) There is no low ∆ degree bounding AMT . Theorem 3.4
No principle P having an ω -model with only low sets and such that AMT ≤ c P admits a universal instance.Theorem 3.3 and Theorem 3.4 can be easily proven using the following characteri-zation of ∆ low sets in terms of domination: Lemma 3.5 (Martin, [ ] ) A set A ≤ T ; ′ is low iff there exists an f ≤ T ; ′ dominatingevery A -computable function. Proof.
A set A is low iff ; ′ is high relative to A . As a set X is high relative to a set A ≤ T ; ′ iff it computes a function dominating every A -computable function, we conclude. (cid:3) Remark.
As explained Conidis in [ ] , Theorem 3.3 cannot be extended to every low sets: Soare [ ] constructed a low set bounding the escape property using a forcingargument. So there exists a low degree bounding AMT . Proof of Theorem 3.4.
Suppose for the sake of contradiction that P has a universal in-stance U and an ω -model M with only low sets. As U is computable, U ∈ M . Let X ∈ M be a (low) solution to U . In particular, X is low and ∆ , so by Lemma 3.5and the computable equivalence of AMT and the escape property, there exists a com-putable instance Y of AMT such that X does not compute a solution to Y . As AMT ≤ c P , there exists a Y -computable (hence computable) instance Z of P such that everysolution to Z computes a solution to Y . Thus X does not compute a solution to Z ,contradicting universality of U . (cid:3) Hirschfeldt & al. proved in [ ] the existence of an ω -model of SADS and
SCAC with only low sets. Therefore we obtain another proof that neither
SADS nor
SCAC admits a universal instance. The result was first proven in [ ] using an ad-hoc notionof reducibility. Corollary 3.6
None of
AMT , SADS and
SCAC admit a universal instance.Previous argument can not directly be applied to
SRT , SEM or STS ( ) as noneof those principles admit an ω -model with only low sets [
10, 17, 23 ] . HoweverLemma 3.4 can be extended to principles such that every computable instance hasa ∆ low solution. It is currently unknown whether every ∆ set admits a ∆ low infinite subset in either it or its complement. A positive answer would lead to a proofthat SRT , SEM and
STS ( ) have no universal instance, and more importantly, wouldprovide an ω -model of SRT not model of DNR [ ; ′ ] as explained in [ ] . We shall seelater that none of SRT , SEM and
STS ( ) admits a universal instance.4. D EGREES BOUNDING
STS ( ) AND
SADS
Mileti originally proved in [ ] that no principle P having an ω -model with onlylow sets and satisfying SRT ≤ c P admits a universal instance, and deduced that none LUDOVIC PATEY of SRT and RT admit one. In this section, we reapply his argument to much weakerstatements and derive non-universality results to a large range of principles in reversemathematics. Thin set theorem and ascending descending sequence are example ofstatements weak enough to be a consequence of many others, and surprisingly strongenough to diagonalize against low sets. Definition 4.1 (Thin set) Let k ∈ ω and f : [ ω ] k → ω . A set A is thin for f if f ([ A ] n ) = ω , that is, if the set A “avoids” at least one color. TS ( k ) is the statement“every function f : [ ω ] k → ω has an infinite set thin for f ”. STS ( k ) is the restrictionof TS ( k ) to stable functions.Cholak & al. studied extensively thin set principle in [ ] . Some of the results wherealready stated by Friedman without giving a proof, notably there exists an ω -modelof WKL which is not a model of TS ( ) , and the arithmetical comprehension axiom( ACA ) does not imply ( ∀ k ) TS ( k ) over RCA . Wang showed in [ ] that ( ∀ k ) TS ( k ) does not imply ACA on ω -models. Rice [ ] proved that STS ( ) implies DNR over
RCA . The author proved in [ ] that RCA ⊢ TS ( ) → RRT . Definition 4.2 (Ascending descending sequence)
ADS is the statement “Every linearorder admits an infinite ascending or descending sequence”.
SADS is the restriction of
ADS to order types ω + ω ∗ .Tennenbaum [ ] constructed a computable linear order of order type ω + ω ∗ withno computable ascending or descending sequence. Therefore SADS does not holdover
RCA . Hirschfeldt & Shore [ ] studied ADS within the framework of reversemathematics, proving that
ADS imply both
COH and BΣ over RCA and that SADS implies
AMT over
RCA . They constructed an ω -model of ADS not model of
DNR ,and an ω -model of COH + WKL not model of SADS .The study of degrees bounding a statement and the existence of a universal instanceare closely related. As does Mileti in [ ] , we deduce two kind of theorems by theapplication of his proof technique. Theorem 4.3
There exists no low degree bounding any of STS ( ) or SADS . Theorem 4.4
No principle P having an ω -model with only low sets and such that anyof STS ( ) , SADS is computably reducible to P admits a universal instance.The proof of the two theorems is split into three lemmas. Lemma 4.7 provides ageneral way of obtaining bounding and universality results, assuming the ability of aprinciple to diagonalize against a particular set. Lemma 4.8 and Lemma 4.9 state thedesired diagonalization for respectively STS ( ) and SADS . Corollary 4.5
None of the following principles admits a universal instance: RT , RWKL [ ; ′ ] , FS ( ) , TS ( ) , CAC , ADS and their stable versions.
Proof.
Each of the above mentioned principles is a consequence of RT over RCA andcomputably implies either SADS or STS ( ) . See [ ] for RWKL [ ; ′ ] , [ ] for FS ( ) and TS ( ) , and [ ] for CAC and
ADS . By Theorem 3.1 of [ ] , there exists an ω -model of RT having only low sets. We conclude by Theorem 4.4. (cid:3) EGREES BOUNDING PRINCIPLES AND UNIVERSAL INSTANCES 7
In order to prove Theorem 4.3 and Theorem 4.4, we need the following theoremproven by Mileti. It simply consists of applying a relativized version of the low basistheorem to a Π class of completions of the enumeration of all partial computable sets. Theorem 4.6 (Mileti, Corollary 5.4.5 of [ ] ) For every set X , there exists f : ω →{
0, 1 } low over X such that for every X -computable set Z , there exists an e ∈ ω with Z = (cid:8) a ∈ ω : f ( e , a ) = (cid:9) . Lemma 4.7
Fix an n ∈ ω and two principles P and Q such that P ≤ c Q . Suppose thatfor any f : ω → {
0, 1 } satisfying f ′′ ≤ T ; ( n + ) , there exists a computable instance I of P such that for each e ∈ ω , if { a ∈ ω : f ( e , a ) = } is infinite then it is not a solutionto I . Then the following holds:(i) For any degree d low over ; ( n ) there is a computable instance U of P suchthat d does not bound a solution to U .(ii) There is no degree low over ; ( n ) bounding P .(iii) If every computable instance I of Q has a solution low over ; ( n ) , then Q hasno universal instance. Proof. (i) Consider any set X of degree low over ; ( n ) . By Theorem 4.6, there exists afunction f : ω → {
0, 1 } low over X , hence low over ; ( n ) , such that any X -computable set Z is of the form { a ∈ ω : f ( e , a ) = } for some e ∈ ω . Take acomputable instance I of P having no solution of the form { a ∈ ω : f ( e , a ) = } for any e ∈ ω . Then X does not compute a solution to I .(ii) Immediate from (i).(iii) Take any computable instance U of Q . By assumption, U has a solution X low over ; ( n ) . By (i), there exists an instance I of P such that X does not computea solution to I . As P ≤ c Q , there exists an I -computable (hence computable)instance J of Q such that any solution to J computes a solution to I . Then X does not compute a solution to J , hence U is not a universal instance. (cid:3) We will prove the following lemmas which, together with Lemma 4.7, are sufficientto deduce Theorem 4.3 and Theorem 4.4.
Lemma 4.8
Fix a set X . Suppose f : ω → {
0, 1 } satisfies f ′′ ≤ T X ′′ . There exists an X -computable stable coloring g : [ ω ] → ω such that for all e ∈ ω , if { a ∈ ω : f ( e , a ) = } is infinite then it is not thin for g . Lemma 4.9
Fix a set X . Suppose f : ω → {
0, 1 } satisfies f ′′ ≤ T X ′′ . There exists astable X -computable linear order L such that for all e ∈ ω , if { a ∈ ω : f ( e , a ) = } isinfinite then it is neither an ascending nor a descending sequence in L .Before proving the two remaining lemmas, we relativize the results to colorings overarbitrary tuples. Theorem 4.10
For any n , there exists no degree low over ; ( n ) bounding STS ( n + ) . LUDOVIC PATEY
Proof.
Apply Lemma 4.8 relativized to X = ; ( n ) together with Lemma 4.7. Simplynotice that if f : [ ω ] n → ω is a ; ′ -computable coloring, the computable coloring g : [ ω ] n + → ω obtained by an application of Schoenfield’s limit lemma is such thatevery infinite set thin for g is thin for f . (cid:3) Theorem 4.11
For any n , no principle P having an ω -model with only low over ; ( n ) sets and such that STS ( n + ) ≤ c P admits a universal instance. Proof.
Same reasoning as Theorem 4.4 using the notice in the proof of Theorem 4.10. (cid:3)
Theorem 4.12
For any n , none of RT n + , RWKL [ ; ( n + ) ] , FS ( n + ) , TS ( n + ) andtheir stable versions admits a universal instance. Proof.
Fix an n ∈ ω . Each of the above cited principles P satisfies STS ( n + ) ≤ c P andis a consequence of RT n + over ω -models. Cholak & al. [ ] proved the existence ofan ω -model of RT n + having only low over ; ( n ) sets. Apply Theorem 4.11. (cid:3) We now turn to the proofs of Lemma 4.8, and Lemma 4.9.
Proof of Lemma 4.8.
For each e ∈ ω , let Z e = (cid:8) a ∈ ω : f ( e , a ) = (cid:9) . The proof is verysimilar to [
20, Theorem 5.4.2. ] . We build a ; ′ -computable function c : ω → ω suchthat for all e ∈ ω , if Z e is infinite then it is not thin for c . Given such a function c ,we can then apply Schoenfield’s limit lemma to obtain a stable computable function h : [ ω ] → ω such that for each x ∈ ω , lim s h ( x , s ) = c ( x ) . Every set thin for h is thinfor c , and therefore for all e ∈ ω , if Z e is infinite then it is not thin for h .Suppose by Kleene’s fixpoint theorem that we are given a Turing index d of thefunction c . The construction is done by a finite injury priority argument satisfying thefollowing requirements for each e , i ∈ ω : R e , i : Z e is finite or ( ∃ a )[ f ( e , a ) = Φ ; ′ d ( a ) = i ] The requirements are ordered in a standard way, that is, following the pairing of theindexes. Notice that each of these requirement is Σ f , and furthermore we can effec-tively find an index for each as such. Therefore, for each e and i ∈ ω , we can effectivelyfind an integer m e , i such that R e , i is satisfied if and only if m e , i ∈ f ′′ . By Schoen-field’s limit Lemma relativized to ; ′ and low -nes of f , there exists a ; ′ -computablefunction g : ω → m , we have m ∈ f ′′ ↔ lim s g ( m , s ) = m f ′′ ↔ lim s g ( m , s ) =
0. Notice that for all e and i ∈ ω , R e , i is satisfied if and onlyif lim s g ( m e , i , s ) = s , assume we have defined c ( u ) for every u < s . If there exists a leaststrategy R e , i (in priority order) with 〈 e , i 〉 < s such that g ( m e , i , s ) =
0, set c ( s ) = i .Otherwise set c ( s ) =
0. This ends the construction. We now turn to the verification.
Claim.
Every requirement R e , i is satisfied. Proof.
By induction over ordered pairs 〈 e , i 〉 in lexicographic order. Suppose that R e ′ , i ′ is satisfied for all (cid:10) e ′ , i ′ (cid:11) < 〈 e , i 〉 , but R e , i is not satisfied. Then there exists a threshold t ≥ 〈 e , i 〉 such that g ( m e ′ , i ′ , s ) = (cid:10) e ′ , i ′ (cid:11) < 〈 e , i 〉 and g ( m e , i , s ) = s ≥ t . By construction, c ( s ) = i for every s ≥ t . As Z e is infinite, there exists an element s ∈ Z e such that c ( s ) = i , so Z e is not thin for c with witness i and therefore R e , i issatisfied. Contradiction. (cid:3) EGREES BOUNDING PRINCIPLES AND UNIVERSAL INSTANCES 9 (cid:3)
Proof of Lemma 4.9.
For each e ∈ ω , let Z e = (cid:8) a ∈ ω : f ( e , a ) = (cid:9) . The proof is verysimilar to [
20, Theorem 5.4.2. ] . We build a ∆ set U together with a stable computablelinear order L such that U is the ω part of L , that is, U is the collection of elements L -below cofinitely many other elements. We furthermore ensure that for each e ∈ ω , if Z e is infinite, then it intersects both U and U . Therefore, if Z e is infinite, it is neither anascending, nor a descending sequence in L as otherwise it would be included in either U or U .Assume by Kleene’s fixpoint theorem that we are given the Turing index d of U . Theset U is built by a finite injury priority construction with the following requirementsfor each e ∈ ω : • R e : Z e is finite or ( ∃ a )[ f ( e , a ) = Φ ; ′ d ( a ) = ] • R e + : Z e is finite or ( ∃ a )[ f ( e , a ) = Φ ; ′ d ( a ) = ] Notice again that each of these requirement is Σ f , and furthermore we can effec-tively find an index for each as such. Therefore, for each i ∈ ω , we can effectivelyfind an m i such that R i is satisfied if and only if m i ∈ f ′′ . By two applications ofSchoenfield’s limit Lemma and low -ness of f , there exists a computable function g : ω → m ∈ ω , we have m ∈ f ′′ ↔ lim t lim s g ( m , s , t ) = m f ′′ ↔ lim t lim s g ( m , s , t ) =
0. Notice that for all i ∈ ω , R i is satisfied ↔ lim t lim s g ( m i , s , t ) = U = ; and every integer is a decision-maker and follows itself. We say that R i requires attention for u at stage s if i ≤ u ≤ s , u is decision-maker and g ( m i , s , u ) = s +
1, assume we have decided u < L v or u > L v for every u , v < s . Set u < L s if u ∈ U s and u > L s if u U s . Initially set U s + = U s . For each decision-maker u ≤ s which has not been claimed at stage s + R i , i < u requires attention, say that the least such R i claims u and act as follows.(a) If i = e and u U s , then add [ u , s ] to U s + . Elements of [ u + s ] follow u andare no more considered as decision-makers from now on and at any furtherstage.(b) If i = e + u ∈ U s , then remove [ u , s ] from U s + . As well, elements of [ u + s ] are no more decision-makers and follow u .The go to next decision-maker u ≤ s . This ends the construction. An immediateverification shows that at every stage, • if u stops being a decision-maker it never becomes again a decision-maker • if u follows v then v ≤ u , v is a decision-maker, every w between v and u follows v and thus u will never follow any w > v .So the decision-maker that u follows eventually stabilizes. As well, because g is limit-computable, each decision-maker eventually stops increasing the number of followersand therefore there are infinitely many decision-makers. Claim. L is a linear order.
Proof. As L is a tournament, it suffices to check there is no 3-cycle. By symmetry,we check only the case where u < L s < L v < L u forms a 3-cycle with s the maximalelement in < ω order. By construction, this means that u ∈ U s , v U s . If u < ω v , then u U v and so there exists a decision-maker w ≤ ω u and an even number i ≤ w such that R i requires attention for w at a stage t ≥ v . Case (a) of the construction appliesand the interval [ w + t ] is included U at least until stage s . As v ∈ [ w + t ] , v ∈ U s contradicting our hypothesis. Case u > ω v is symmetric. (cid:3) Claim. U is ∆ . Proof.
Suppose for the sake of absurd that there exists a least element u entering U and leaving it infinitely many times. Such u must be a decision-maker, otherwise itwould not be the least one. Let R i be the least requirement claiming u infinitely manytimes. As lim s g ( m i , s , u ) exists, it will claim u cofinitely many times and therefore u will be in U or in U cofinitely many times. Contradiction. (cid:3) It immediately follows that L is stable. Claim.
Every requirement R i is satisfied. Proof.
By induction over R i in priority order. Suppose that R j is satisfied for all j < i ,but R i is not satisfied. Then there exists a threshold t ≥ i such that lim s g ( m j , s , t ) = j < i and lim s g ( m i , s , t ) = t ≥ t .Then for every decision-maker u ≥ t , R i will claim u cofinitely many times, andtherefore u will be in U if i is even and in U if i is odd. As every element follows theleast decision-maker below itself, every v above the least decision-maker greater than t will be in U if i is even and in U if i is odd. So if Z e is infinite, there will be such a v ∈ Z e satisfying R i . Contradiction. (cid:3)(cid:3)
5. D
EGREES BOUNDING THE E RDÖS M OSER THEOREM
Another approach to the strength analysis of Ramsey’s theorem for pairs consists inseeing a coloring f : [ ω ] → T such that T ( x , y ) holds for x < y if and only if f ( x , y ) =
1. The Erd˝os Moser theorem states the existence of aninfinite transitive subtournament, that is, an infinite subset on which the tournamentbehaves like a linear order. Therefore the Erd˝os Moser theorem can be seen as aprinciple reducing instances of RT into instances of ADS . Definition 5.1 (Erd˝os Moser theorem) A tournament T on a domain D ⊆ N is an ir-reflexive binary relation on D such that for all x , y ∈ D with x = y , exactly one of T ( x , y ) or T ( y , x ) holds. A tournament T is transitive if the corresponding relation T is transitive in the usual sense. A tournament T is stable if ( ∀ x ∈ D )[( ∀ ∞ s ) T ( x , s ) ∨ ( ∀ ∞ s ) T ( s , x )] . EM is the statement “Every infinite tournament T has an infinite tran-sitive subtournament.” SEM is the restriction of EM to stable tournaments.Bovykin and Weiermann proved in [ ] that EM + ADS is equivalent to RT over RCA , equivalence still holding between their stable versions. Lerman & al. [ ] proved over RCA + BΣ that EM implies OPT and constructed an ω -model of EM not model of SRT . Kreuzter & al. proved in [ ] that SEM implies BΣ over RCA .Bienvenu & al. proved in [ ] that RCA ⊢ SEM → RWKL , hence there exists an ω -model of RRT not model of SEM . Wang constructed in [ ] an ω -model of EM + COH not model of
STS ( ) . Finally, the author proved in [ ] that RCA ⊢ EM → [ STS ( ) ∨ COH ] .The following notion of minimal interval plays a fundamental role in the analysis of EM . See [ ] for a background analysis of EM . EGREES BOUNDING PRINCIPLES AND UNIVERSAL INSTANCES 11
Definition 5.2 (Minimal interval) Let T be an infinite tournament and a , b ∈ T besuch that T ( a , b ) holds. The interval ( a , b ) is the set of all x ∈ T such that T ( a , x ) and T ( x , b ) hold. Let F ⊆ T be a finite transitive subtournament of T . For a , b ∈ F such that T ( a , b ) holds, we say that ( a , b ) is a minimal interval of F if there is no c ∈ F ∩ ( a , b ) , i.e. no c ∈ F such that T ( a , c ) and T ( c , b ) both hold.We provide in the next subsections two different proofs of the existence of a low degree bounding EM . More precisely, we construct a low set G which is, up to finitechanges, transitive for every infinite computable tournament.The author proved in [ ] that [ STS ( ) ∨ COH ] ≤ c EM . Therefore every low degree bounding EM bounds also COH . The proof does not seem adaptable to provethat
COH is a consequence of EM even in ω -models. However we can prove a weakerstatement: Lemma 5.3
For every set X , there exists an infinite X -computable tournament T suchthat for every infinite T -transitive subtournament U , U ⊆ ∗ X or U ⊆ ∗ X . Proof.
Fix a set X . We define a tournament T as follows: For each a < b , set T ( a , b ) to hold iff a ∈ X and b ∈ X or a X and b X . Suppose for the sake of absurd that U is an infinite transitive subtournament of T which intersects infinitely often X and X .Take any a , c ∈ U ∩ X and b , d ∈ U ∩ X such that a < b < c < d . Then T ( a , c ) , T ( c , b ) , T ( b , d ) and T ( d , a ) hold contradicting transitivity of U . (cid:3) Using previous lemma, the constructed set G must be cohesive and therefore pro-vides another proof of the existence of a low cohesive set. Finally, we can deduce astatement slightly weaker than Theorem 4.10 simply by the existence of a low degreebounding EM . Lemma 5.4
There exists a set C such that there is no low over C degree d ≫ SADS C . Proof.
Fix a low set C ≫ EM ; and a set X low over C . By low -ness of C , X is low .Consider the stable coloring f : [ ω ] → [ ] , such that X computes no infinite f -homogeneous set. We can see f as a stable tournament T such that for each x < y , T ( x , y ) holds iff f ( x , y ) =
1. As C ≫ EM ; , there exists aninfinite C -computable transitive subtournament U of T . U is a stable linear order suchthat every infinite ascending or descending sequence is f -homogeneous. Therefore X computes no infinite ascending or descending sequence in U . (cid:3) The following question remains open:
Question 5.5
Does EM admit a universal instance ?5.1. A low degree bounding EM using first jump control. The following theoremuses the proof techniques introduced in [ ] for producing low sets by controlling thefirst jump. It is done in the same spirit as Theorem 3.6 in [ ] . Theorem 5.6
For every set P ≫ ; ′ , there exists a set G ≫ EM ; such that G ′ ≤ T P .Before proving Theorem 5.6, we introduce the notion of Erd˝os Moser condition . Definition 5.7 An Erd˝os Moser condition (EM condition) for an infinite tournament T is a Mathias condition ( F , X ) where(a) F ∪ { x } is T -transitive for each x ∈ X (b) X is included in a minimal T -interval of F .Extension is usual Mathias extension. EM conditions have good properties for tour-naments as state following lemmas. Given a tournament T and two sets E and F , wedenote by E → T F the formula ( ∀ x ∈ E )( ∀ y ∈ F ) T ( x , y ) holds. Lemma 5.8
Fix an EM condition ( F , X ) for a tournament T . For every x ∈ F , { x } → T X or X → T { x } . Proof.
Fix an x ∈ F . Let ( u , v ) be the minimal T -interval containing X , where u , v maybe respectively −∞ and + ∞ . By definition of interval, { u } → T X → T { v } . By definitionof minimal interval, T ( x , u ) or T ( v , x ) holds. Suppose the former holds. By transitivityof F ∪ { y } for every y ∈ X , T ( x , y ) holds, therefore { x } → T Y . In the latter case, bysymmetry, Y → T { x } . (cid:3) Lemma 5.9
Fix an EM condition c = ( F , X ) for a tournament T , an infinite subset Y ⊆ X and a finite T -transitive set F ⊂ X such that F < Y and [ F → T Y ∨ Y → T F ] .Then d = ( F ∪ F , Y ) is a valid extension of c . Proof.
Properties of a Mathias condition for d are immediate. We prove property (a).Fix an x ∈ Y . To prove that F ∪ F ∪ { x } is T -transitive, it suffices to check that thereexists no 3-cycle in F ∪ F ∪ { x } . Fix three elements u < v < w ∈ F ∪ F ∪ { x } . • Case 1: { u , v , w } ∩ F = ; . Then u ∈ F as F < F < { x } and u < v < w . If v ∈ F then using the fact that F ∪ { x } ⊂ X and property (a) of condition c , { u , v , w } is T -transitive. If v F , then by Lemma 5.8, { u } → T X ( ⊇ F ∪ { x } ) or X → T { u } hence { u } → T { v , w } or { v , w } → T { u } so { u , v , w } is T -transitive. • Case 2: { u , v , w } ∩ F = ; . Then at least u , v ∈ F because F < { x } . If w ∈ F ,then { u , v , w } is T -transitive by T -transitivity of F . Otherwise, as F → T Y or Y → T F , { u , v } → T { w } or { w } → T { u , v } and { u , v , w } is T -transitive.We now prove property (b). Let ( u , v ) be the minimal T -interval of F in which X (hence Y ) is included by property (b) of condition c . u and v may be respectively −∞ and + ∞ . By assumption, either F → T Y or Y → T F . As F is a finite T -transitiveset, it has a minimal and a maximal element, say x and y . If F → T Y then Y isincluded in the T -interval ( y , v ) . Symmetrically, if Y → T F then Y is included in the T -interval ( u , x ) . To prove minimality for the first case, assume that some w is in theinterval ( y , v ) . Then w F by minimality of the interval ( u , v ) w.r.t. F , and w F bymaximality of y . Minimality for the second case holds by symmetry. (cid:3) Proof of Theorem 5.6.
Let C be a low set such that there exists a uniformly C -computableenumeration ~ T of infinite tournaments containing every computable tournament. Notethat P ≫ C ′ . Our forcing conditions are tuples ( σ , F , X ) where σ ∈ ω <ω and the fol-lowing holds:(a) ( F , X ) forms a Mathias condition and X is a set low over C .(b) ( F r [ σ ( ν )] , X ) is an EM condition for T ν for each ν < | σ | .A condition ( ˜ σ , ˜ F , ˜ X ) extends a condition ( σ , F , X ) if σ (cid:22) ˜ σ and ( ˜ F , ˜ X ) Mathias extends ( F , X ) . A set G satisfies the condition ( σ , F , X ) if G r [ σ ( ν )] is T ν -transitive for each EGREES BOUNDING PRINCIPLES AND UNIVERSAL INSTANCES 13 ν < | σ | and G satisfies the Mathias condition ( F , X ) . An index of a condition ( σ , F , X ) is a code of the tuple 〈 σ , F , e 〉 where e is a lowness index of X .The first lemma simply states that we can ensure that G will be infinite and eventu-ally transitive for each tournament in ~ T . Lemma 5.10
For every condition c = ( σ , F , X ) and every i , j ∈ ω , one can P -computean extension ( ˜ σ , ˜ F , ˜ X ) such that | ˜ σ | ≥ i and | ˜ F | ≥ j uniformly from i , j and an indexof c . Proof.
Let x be the first element of X . As X is low over C , x can be found C ′ -computablyfrom a lowness index of X . The condition ( ˜ σ , F , X ) is a valid extension of c where˜ σ = σ ⌢ x . . . x so that | ˜ σ | ≥ i . It suffices to prove that we can C ′ -compute an extension ( ˜ σ , ˜ F , ˜ X ) with | ˜ F | > | F | and iterate the process. Define the computable coloring g : X → | ˜ σ | by g ( s ) = ρ where ρ ∈ | ˜ σ | such that ρ ( ν ) = T ν ( x , s ) holds. One canfind uniformly in P a ρ ∈ | ˜ σ | such that the following C -computable set is infinite: Y = { s ∈ X r { x } : g ( s ) = ρ } By Lemma 5.9, (( F ∪ { x } ) r [
0, ˜ σ ( ν )] , Y ) is a valid EM extension for T ν . As Y is lowover C , ( ˜ σ , F ∪ { x } , Y ) is a valid extension for c . (cid:3) It remains to be able to decide e ∈ ( G ⊕ C ) ′ uniformly in e . We first need to definea forcing relation. Definition 5.11
Fix a condition c = ( σ , F , X ) and two integers e and x .1. c (cid:141) Φ G ⊕ Ce ( x ) ↑ if Φ ( F ∪ F ) ⊕ Ce ( x ) ↑ for all finite subsets F ⊆ X such that F is T ν -transitive simultaneously for each ν < | σ | .2. c (cid:141) Φ G ⊕ Ce ( x ) ↓ if Φ F ⊕ Ce ( x ) ↓ .Note that the way we defined our forcing relation c (cid:141) Ψ G ⊕ Ce ( x ) ↑ differs slightlyfrom the “true” forcing notion (cid:141) ∗ inherited by the notion of satisfaction of G . The trueforcing definition of this statement is the following: c (cid:141) ∗ Φ G ⊕ Ce ( x ) ↑ if Φ ( F ∪ F ) ⊕ Ce ( x ) ↑ for all finite extensible subsets F ⊆ X such that F is T ν -transitive simultaneously for each ν < | σ | , i.e. for all finite subsets F ⊆ X suchthat there exists an extension d = ( ˜ σ , F ∪ F , ˜ X ) .However c (cid:141) ∗ Φ G ⊕ Ce ( x ) ↑ is not a Π statement whereas c (cid:141) Φ G ⊕ Ce ( x ) ↑ is. Inparticular the fact that c (cid:141) Φ G ⊕ Ce ( x ) ↑ does not mean that c has an extension forcing itsnegation. This subtlety is particularly important in Lemma 5.13. The following lemmagives a sufficient constraint, namely being included in a part of a particular partition,on finite transitive sets to ensure that they are extensible . Lemma 5.12
Let c = ( σ , F , X ) be a condition and E ⊆ X be a finite set. There exists a2 | σ | partition ( E ρ : ρ ∈ | σ | ) of E and an infinite set Y ⊆ X low over C such that E < Y and for all ρ ∈ | σ | and ν < | σ | , if ρ ( ν ) = E ρ → T ν Y and if ρ ( ν ) = Y → T ν E ρ .Moreover this partition and a lowness index of Y can be uniformly P -computedfrom an index of c and the set E . Proof.
Given a set E , define P E to be the finite set of ordered 2 | σ | -partitions of E , thatis, P E = { ¬ E ρ : ρ ∈ | σ | ¶ : [ ρ ∈ | σ | E ρ = E and ρ = ξ → E ρ ∩ E ξ = ;} Define the C -computable coloring g : X → P E by g ( x ) = ( E x ρ : ρ ∈ | σ | ) where E x ρ = { a ∈ E : ( ∀ ν < | σ | )[ T ν ( a , x ) holds iff ρ ( ν ) = ] } . On can find uniformly in P apartition ( E ρ : ρ ∈ | σ | ) such that the following C -computable set is infinite: Y = { x ∈ X r E : g ( x ) = ( E ρ : ρ ∈ | σ | ) } By definition of g , for all ρ ∈ | σ | and ν < | σ | , if ρ ( ν ) = E ρ → T ν Y and if ρ ( ν ) = Y → T ν E ρ . (cid:3) We are now ready to prove the key lemma of this forcing, stating that we can P -decide whether or not e ∈ G ′ for any e ∈ ω . Lemma 5.13
For every condition ( σ , F , X ) and every e ∈ ω , there exists an extension d = ( ˜ σ , ˜ F , ˜ X ) such that one of the following holds:1. d (cid:141) Φ G ⊕ Ce ( e ) ↓ d (cid:141) Φ G ⊕ Ce ( e ) ↑ This extension can be P -computed uniformly from an index of c and e . Moreover thereis a C ′ -computable procedure to decide which case holds from an index of d . Proof.
Let k = | σ | . Using a C ′ -computable procedure, we can decide from an index of c and e whether there exists a finite set E ⊂ X such that for every 2 k -partition ( E i : i < k ) of E , there exists an i < k and a subset F ⊆ E i T ν -transitive simultaneously for each ν < k and satisfying Φ ( F ∪ F ) ⊕ Ce ( e ) ↓ .1. If such a set E exists, it can be C ′ -computably found. By Lemma 5.12, onecan P -computably find a 2 k -partition ( E ρ : ρ ∈ k ) of E and a set Y ⊆ X lowover C such that for all ρ ∈ k and ν < k , if ρ ( ν ) = E ρ → T ν Y andif ρ ( ν ) = Y → T ν E ρ . We can C ′ -computably find a ρ ∈ k and a set F ⊆ E ρ which is T ν -transitive simultaneously for each ν < k and satisfying Φ ( F ∪ F ) ⊕ Ce ( e ) ↓ . By Lemma 5.9, ( F r [ σ ( ν )]) ∪ F , Y ) is a valid EM extensionof ( F r [ σ ( ν )] , X ) for T ν for each ν < k . As Y is low over C , ( σ , F ∪ F , Y ) is a valid extension of c forcing Φ G ⊕ Ce ( e ) ↓ .2. If no such set exists, then by compactness, the Π C class of all 2 k -partitions ( X i : i < k ) of X such that for every i < k and every finite set F ⊆ X i whichis T ν -transitive simultaneously for each ν < k , Φ ( F ∪ F ) ⊕ Ce ( e ) ↑ is non-empty.In other words, the Π C class of all 2 k -partitions ( X i : i < k ) of X such thatfor every i < k , ( σ , F , X i ) (cid:141) Φ G ⊕ Ce ( e ) ↑ is non-empty. By the relativized lowbasis theorem, there exists a 2 k -partition ( X i : i < k ) of X low over C . Fur-thermore, a lowness index for this partition can be uniformly C ′ -computablyfound. Using P , one can find an i < k such that X i is infinite. ( σ , F , X i ) is avalid extension of c forcing Φ G ⊕ Ce ( e ) ↑ . (cid:3) Using Lemma 5.10 and Lemma 5.13, one can P -compute an infinite decreasingsequence of conditions c = ( ε , ; , ω ) ≥ c ≥ . . . such that for each s > EGREES BOUNDING PRINCIPLES AND UNIVERSAL INSTANCES 15 | σ s | ≥ s , | F s | ≥ s c s (cid:141) Φ G ⊕ Cs ( s ) ↓ or c s (cid:141) Φ G ⊕ Cs ( s ) ↑ where c s = ( σ s , F s , X s ) . The resulting set G = S s F s is T ν -transitive up to finite changesfor each ν ∈ ω and G ′ ≤ T P . (cid:3) A low degree bounding EM using second jump control. We now use the sec-ond proof technique used in [ ] for producing a low set. It consists of directly con-trolling the second jump of the produced set. Theorem 5.14
There exists a low degree bounding EM . Proof.
Similar to Theorem 5.6, we fix a low set C such that there exists a uniformly C -computable enumeration ~ T of infinite tournaments containing every computabletournament. In particular P ≫ C ′ .Our forcing conditions are the same as in Theorem 5.6. We can release the con-straints of infinity and lowness over C for X in a condition ( σ , F , X ) . This gives thenotion of precondition . The forcing relations extend naturally to preconditions. Definition 5.15
Fix a finite set of Turing indexes ~ e . A condition ( σ , F , X ) is ~ e-small ifthere exists a number x and a sequence ( σ i , F i , X i : i < n ) such that for each i < n (i) ( σ i , F i , X i ) is a precondition extending c (ii) ( X i : i < n ) is a partition of X ∩ ( x , + ∞ ) (iii) max ( X i ) < x or ( σ i , F ∪ F i , X i ) (cid:141) ( ∃ e ∈ ~ e )( ∃ y < x )Φ G ⊕ Ce ( y ) ↑ A condition is ~ e-large if it is not ~ e -small.A condition ( ˜ σ , ˜ F , ˜ X ) is a finite extension of ( σ , F , X ) if ˜ X = ∗ X . Finite extensions donot play the same fundamental role as in the original forcing in [ ] as adding elementsto the set F may require to remove infinitely many elements of the promise set X toobtain a valid extension. We nevertheless prove the following traditional lemma. Lemma 5.16
Fix an ~ e -large condition c = ( σ , F , X ) .1. If ~ e ′ ⊆ ~ e then c is ~ e ′ -large.2. If d is a finite extension of c then d is ~ e -large. Proof.
Clause 1 is trivial as ~ e appears only in a universal quantification in the definitionof ~ e -largeness. We prove clause 2. Let d = ( ˜ σ , ˜ F , ˜ X ) be an ~ e -small finite extension of c .We will prove that c is ~ e -small. Let x ∈ ω and ( σ i , F i , X i : i < n ) witness ~ e -smallness of d . Let y = max ( x , X r ˜ X ) . For each i < n , set ˜ F i = ( ˜ F r F ) ∪ F i and ˜ X i = X i ∩ ( y , + ∞ ) .Then y and ( σ i , ˜ F i , ˜ X i : i < n ) witness ~ e -smallness of c . (cid:3) Lemma 5.17
There exists a C ′′ -effective procedure to decide, given an index of a con-dition c and a finite set of Turing indexes ~ e , whether c is ~ e -large. Furthermore, if c is ~ e -small, there exists sets ( X i : i < n ) low over C witnessing this, and one may C ′ -compute a value of n , x , lowness indexes for ( X i : i < n ) and the correspondingsequences ( σ i , F i , X i : i < n ) which witness that c is ~ e -small. Proof.
Fix a condition c = ( σ , F , X ) The predicate “ ( σ , F , X ) is ~ e -small” can be expressedas a Σ statement ( ∃ z )( ∃ Z ) P ( z , Z , F , X , ~ν , ~ e ) where P is a Π C predicate. Here z codes n and x , and Z codes ( X i : i < n ) . ( ∃ Z ) P ( z , Z , F , X , σ , ~ e ) is a Π C ⊕ X predicate by compactness. As X is low over C and F and σ are finite, one can compute a ∆ C index for the same predicate P with parame-ter z , an index of c and ~ e , from a lowness index for X , F and σ . Therefore there existsa Σ C statement with parameters an index of c and ~ e which holds iff c is ~ e -small.If c is ~ e -small, there exists sets ( X i : i < n ) low over X (hence low over C ) witnessingit by the low basis theorem relativized to C . By the uniformity of the proof of the lowbasis theorem, one can compute lowness indexes of ( X i : i < n ) uniformly from alowness index of X . (cid:3) As the extension produced in Lemma 5.10 is not a finite extension, we need to refineit to ensure largeness preservation.
Lemma 5.18
For every ~ e -large condition c = ( σ , F , X ) and every i , j ∈ ω , one can P -compute an ~ e -large extension ( ˜ σ , ˜ F , ˜ X ) such that ˜ σ ≥ i and | ˜ F | ≥ j uniformly from anindex of c , i , j and ~ e . Proof.
Let x be the first element of X . As X is low over C , x can be found C ′ -computablyfrom a lowness index of X . The condition d = ( ˜ σ , F , X ) is a valid extension of c where ˜ σ = σ ⌢ x . . . x so that | ˜ σ | ≥ i . As d is a finite extension of c , it is ~ e -large byLemma 5.16. It suffices to prove that we can C ′ -compute an ~ e -large extension ( ˜ σ , ˜ F , ˜ X ) with | ˜ F | > | F | and iterate the process. Define the C -computable coloring g : X → | ˜ σ | as in Lemma 5.10. For each ρ ∈ | ˜ σ | , define the following set: Y ρ = { s ∈ X r { x } : g ( s ) = ρ } There must be a ρ ∈ | ˜ σ | such that Y ρ is infinite and ( ˜ σ , F ∪{ x } , Y ρ ) is ~ e -large, otherwisethe witnesses of ~ e -smallness for each ρ ∈ | ˜ σ | would witness ~ e -smallness of c . ByLemma 5.17, one can C ′′ -find a ρ ∈ | ˜ σ | such that ( ˜ σ , F ∪ { x } , Y ρ ) is ~ e -large. As seenin Lemma 5.18, ( ˜ σ , F , { x } , Y ρ ) is a valid extension. (cid:3) The following lemma is a refinement of Lemma 5.12 controlling largeness preserva-tion.
Lemma 5.19
Let c = ( σ , F , X ) be an ~ e -large condition and E ⊆ X be a finite set. Thereis a 2 | σ | partition ( E ρ : ρ ∈ | σ | ) of E and an infinite set Y ⊆ X low over C such that E < Y and1. for all ρ ∈ | σ | and ν < | σ | , if ρ ( ν ) = E ρ → T ν Y and if ρ ( ν ) = Y → T ν E ρ .2. ( σ , F ∪ F , Y ) is an ~ e -large condition extending d for every ρ ∈ | σ | and everyfinite set F ⊆ E ρ which is T ν -transitive for each ν < | σ | Moreover this partition and a lowness index of Y can be uniformly C ′′ -computed froman index of c and the set E . Proof.
Given a set E , recall from Lemma 5.12 that P E is the finite set or ordered2 k -partitions of E . Define again the computable coloring g : X → P E by g ( x ) = D E x ρ : ρ ∈ | σ | E where E x ρ = { a ∈ E : ( ∀ ν < | σ | )[ T ν ( a , x ) holds iff ρ ( ν ) = ] } . Iffor each partition ( E ρ : ρ ∈ | σ | ) , there exists a ρ ∈ | σ | and a F ⊆ E ρ which is T ν -transitive simultaneously for each ν < | σ | and such that ( σ , F ∪ F , Y ) is ~ e -small EGREES BOUNDING PRINCIPLES AND UNIVERSAL INSTANCES 17 where Y = { x ∈ X r E : g ( x ) = ( E ρ : ρ ∈ | σ | ) } Then we could construct a witness of ~ e -smallness of c using smallness witnesses of ( σ , F ∪ F , Y ) for each partition ( E ρ : ρ ∈ | σ | ) . Therefore there must exist a partition ( E ρ : ρ ∈ | σ | ) such that Y is infinite and d = ( σ , F ∪ F , Y ) is ~ e -large for every ρ ∈ | σ | and every F ⊆ E ρ which is T ν -transitive for each ν < | σ | .By Lemma 5.17, such partition can be found C ′′ -computably. By definition of g ,for all ρ ∈ | σ | and ν < k , if ρ ( ν ) = E ρ → T ν Y and if ρ ( ν ) = Y → T ν E ρ . Therefore, by Lemma 5.9, (( F r [ σ ( ν )]) ∪ F , Y ) is a valid EM extension of ( F r [ σ ( ν )] , X ) for T ν for each ν < | σ | , so d is a valid condition. (cid:3) Lemma 5.20
Suppose that c = ( σ , F , X ) is ~ e -large. For every y ∈ ω and e ∈ ~ e , thereexists an ~ e -large extension d such that d (cid:141) Φ G ⊕ Ce ( y ) ↓ . Furthermore, an index for d can be computed from an oracle for C ′ from an index of c , e and y . Proof.
Let k = | σ | . As c is ~ e -large, then by a compactness argument, there exists afinite set E ⊂ X such that for every 2 k -partition ( E i : i < k ) of E , there exists an i < k and a finite subset F ⊆ E i which is T ν -transitive simultaneously for each ν < k , and Φ ( F ∪ F ) ⊕ Ce ( y ) ↓ . Moreover this set E can be C ′ -computably found. By Lemma 5.19, oncan uniformly C ′′ -find a partition ( E ρ : ρ ∈ k ) of E and a lowness index for an infiniteset Y ⊆ X low over C such that1. for all ρ ∈ k and ν < k , if ρ ( ν ) = E ρ → T ν Y and if ρ ( ν ) = Y → T ν E ρ .2. ( σ , F ∪ F , Y ) is an ~ e -large condition extending c for every ρ ∈ k and everyfinite set finite set F ⊆ E ρ which is T ν -transitive for each ν < k We can then produce by a C ′ -computable search a ρ ∈ k and a finite set F ⊆ E ρ which is T ν -transitive for each ν < k and such that Φ ( F ∪ F ) ⊕ Ce ( y ) ↓ . By Lemma 5.9, (( F r [ σ ( ν )]) ∪ F , Y ) is a valid EM extension of ( F r [ σ ( ν )] , X ) for T ν for each ν < k . As Y is low over C , ( σ , F ∪ F , Y ) is a valid ~ e -large extension. (cid:3) Lemma 5.21
Suppose that c = ( σ , F , X ) is ~ e -large and ( ~ e ∪ { u } ) -small. There exists a ~ e -large extension d such that d (cid:141) Φ G ⊕ Cu ( y ) ↑ for some y ∈ ω . Furthermore one canfind an index for d by applying a C ′′ -computable function to an index of c , ~ e and u . Proof.
By Lemma 5.17, we may choose the sets ( X i : i < n ) witnessing that c is ( ~ e ∪{ u } ) -small to be low over C . Fix the corresponding x and ( σ i , F i : i < n ) . Consider the i ’ssuch that ( σ i , F i , X i ) (cid:141) Φ G ⊕ Cu ( y ) ↑ for some y < x . As c is ~ e -large, there must be onesuch i < n such that ( σ i , F i , X i ) is an ~ e -large condition. By Lemma 5.17 we can find C ′′ -computably such i < n . ( σ i , F i , X i ) is the desired extension. (cid:3) Using previous lemmas, we can C ′′ -compute an infinite descending sequence ofconditions c = ( ε , ; , ω ) ≥ c ≥ . . . together with an infinite increasing sequence ofTuring indexes ~ e = ; ⊆ ~ e ⊆ . . . such that for each s > | σ s | ≥ s , | F s | ≥ s , c s is ~ e s -large2. Either s ∈ ~ e s or c s (cid:141) Φ G ⊕ Cs ( y ) ↑ for some y ∈ ω c s (cid:141) Φ G ⊕ Ce ( x ) ↓ if s = 〈 e , x 〉 and e ∈ ~ e s where c s = ( σ s , F s , X s ) . The resulting set G = S s F s is T ν -transitive up to finite changessimultaneously for each ν ∈ ω and G ′′ ≤ T C ′′ ≤ T ; ′′ . (cid:3)
6. D
EGREE BOUNDING THE RAINBOW R AMSEY THEOREM
The rainbow Ramsey theorem intuitively states that when a coloring over tuplesuses each color a bounded number of times then it has an infinite subset on which eachcolor is used at most once. This statement has been extensively studied over the pastfew years [
8, 7, 26, 23 ] . Remarkably, the restriction of the rainbow Ramsey theoremto coloring over pairs of integers coincides with a well-known notion of algorithmicrandomness. Definition 6.1 (Rainbow Ramsey theorem) Let n , k ∈ ω . A coloring function f : [ ω ] n → ω is k-bounded if for every y ∈ ω , (cid:12)(cid:12) f − ( y ) (cid:12)(cid:12) ≤ k . A set R is a rainbow for f if f ↾ [ R ] n is injective. RRT nk is the statement “Every k -bounded function f : [ ω ] n → ω has an infinite rainbow”.A proof of the rainbow Ramsey theorem is due to Galvin who noticed that it followseasily from Ramsey’s theorem. Hence every computable 2-bounded coloring function f over n -tuples has an infinite Π n rainbow. Csima and Mileti proved in [ ] that every2-random is RRT -bounding and deduced that RRT implies neither SADS nor
WKL over ω -models. Conidis & Slaman adapted in [ ] the argument from Cisma and Miletito obtain RCA ⊢ - RAN → RRT . Definition 6.2
A function f : ω → ω is diagonally non-computable (DNC) relative to X if f ( e ) = Φ Xe ( e ) for each e ∈ ω . DNR [ ; ′ ] is the statement “For every set X , there existsa function DNC relative to the jump of X ”. Theorem 6.3 (J.S. Miller [ ] ) RRT and DNR [ ; ′ ] are computably equivalent. Corollary 6.4
RRT admits a universal instance. Proof. If P and Q are two principles computably equivalent and Q admits a universalinstance, then so does P . As DNR [ ; ′ ] admits a universal instance (any function DNCrelative to ; ′ ), so does RRT . (cid:3) Corollary 6.5
For every X ≫ ; ′ , there exists a Y ≫ RRT ; such that Y ′ ≤ T X . Proof.
Let f : [ ω ] → ω be a universal instance of RRT . By Csima & Mileti [ ] , RRT ≤ c RT , so there exists a computable coloring g : [ ω ] → g -homogeneous set computes an infinite f -rainbow, hence bounds RRT . ByCholak & al. [ ] , for every X ≫ ; ′ there exists an infinite f -homogeneous set H suchthat H ′ ≤ T X . In particular H ≫ RRT ; . (cid:3) Corollary 6.6
There exists a low degree bounding RRT . Proof.
By the relativized low basis theorem, there exists a set X ≫ ; ′ low over ; ′ . ByCorollary 6.5, there exists a set Y ≫ RRT ; such that Y ′ ≤ T X , hence Y ′′ ≤ T X ′ ≤ T ; ′′ .So Y is low . (cid:3) We can generalize Corollary 6.6 to colorings over arbitrary tuples. For this, we needto restrict ourselves to the study of a particular class of colorings.
EGREES BOUNDING PRINCIPLES AND UNIVERSAL INSTANCES 19
Definition 6.7
A coloring f : [ ω ] n + → ω is normal if f ( σ , a ) = f ( τ , b ) for each σ , τ ∈ [ ω ] n , whenever a = b .Wang proved in [ ] that for every 2-bounded coloring f : [ ω ] n → ω , every f -random computes an infinite set X on which f is normal. The author refined in [ ] this result by proving that every function d.n.c. relative to f computes such a set. Theorem 6.8
For each n ≥
0, there exists a set X ≫ RRT n + ; low over ; ( n ) . Proof.
We prove by induction over n that for every set A there exists a set X low over A ( n ) such that X ≫ RRT n + A . Case n = A , there exists a set X low over A ( n ) such that X ≫ RRT n + A . Fix a set A ,an A -random set R low over A and a set C low over A ⊕ R such that C ′ ≫ ( A ⊕ R ) ′ . Inparticular R ⊕ C is low over A . By induction hypothesis, there exists a set X low over ( A ⊕ R ⊕ C ) ( n + ) such that X ≫ RRT n + ( A ⊕ R ⊕ C ) ′ . In particular X is low over A ( n + ) .We claim that X ≫ RRT n + A .Fix an A -computable 2-bounded coloring f : [ ω ] n + → ω . By relativizing Lemma 4.3in [ ] , every A -random computes an infinite set Y such that f restricted to Y is nor-mal. So X ⊕ R computes such a set Y . For each σ , τ ∈ [ Y ] n + , let U σ , τ = { s ∈ Y : f ( σ , s ) = f ( τ , s ) } By Jockusch & Frank [ ] , as C ′ ≫ ( A ⊕ R ) ′ , A ⊕ R ⊕ C computes an infinite ~ U -cohesiveset Z ⊆ Y . In particular the following limit exists˜ f ( σ ) = lim s ∈ Z min { τ ≤ lex σ : f ( σ , s ) = f ( τ , s ) } ˜ f is a 2-bounded ( A ⊕ R ⊕ C ) ′ -computable coloring of ( n + ) -tuples, so X bounds aninfinite ˜ f -rainbow H . A ⊕ H computes an infinite f -rainbow, so X bounds an infinite f -rainbow. (cid:3) A stable rainbow Ramsey theorem.
A common process in the strength analysisof a principle consists of splitting the statement into a stable and a cohesive version.The standard notion of stability does not apply for the rainbow Ramsey theorem asno stable coloring is k -bounded for some k ∈ ω . Nevertheless one can define certainnotions of stability for the rainbow Ramsey theorem [ ] . Mileti proved in [ ] thatthe only ∆ degree bounding SRT is ′ . In fact, his priority argument can be adaptedto prove the same result on a much weaker principle coinciding with a stable versionof the rainbow Ramsey theorem for pairs. Definition 6.9
A coloring f : [ ω ] → ω is rainbow-stable if for every x ∈ ω , one of thefollowing holds:(a) There exists a y = x such that ( ∀ ∞ s ) f ( x , s ) = f ( y , s ) (b) ( ∀ ∞ s ) (cid:12)(cid:12)(cid:8) y = x : f ( x , s ) = f ( y , s ) (cid:9)(cid:12)(cid:12) = SRRT is the statement “every rainbow-stable 2-bounded coloring f : [ ω ] → ω hasa rainbow.”Introduced by the author in [ ] , he proved that SRRT is computably reducibleto SEM and
STS ( ) . This principle admits many computably equivalent formulations. We are particularly interested in a characterization which can be seen as a stable notionof
DNR [ ; ′ ] . Definition 6.10
Given a function f : ω → ω , a function g is f -diagonalizing if ( ∀ x )[ f ( x ) = g ( x )] . SDNR [ ; ′ ] is the statement “Every ∆ function f : ω → ω has an f -diagonalizingfunction”. Theorem 6.11 (Patey [ ] ) SRRT and SDNR [ ; ′ ] are computably equivalent.The following theorem extends Mileti’s result to SDNR [ ; ′ ] . As SDNR [ ; ′ ] is com-putably below many stable principles, we shall deduce a few more non-universalityresults. Theorem 6.12
For every ∆ incomplete set X , there exists a ∆ function f : ω → ω such that X computes no f -diagonalizing function. Corollary 6.13 A ∆ degree d bounds SRRT iff d = ′ . Proof. As SRRT ≤ c SRT , any computable instance of SRRT has a ∆ solution.So ′ bounds SRRT . If d is incomplete, then by Theorem 6.12 and by SRRT = c SDNR [ ; ′ ] , there is a computable instance of SRRT such that d bounds no solution. (cid:3) Corollary 6.14
No statement P such that SRRT ≤ c P ≤ c SRT admits a universalinstance. Proof. By [
13, Corollary 4.6 ] every ∆ set or its complement has an incomplete ∆ in-finite subset. As P ≤ c SRT ≤ c D , every computable instance U of P has a ∆ incom-plete solution X . By Theorem 6.12, there exists a computable coloring f : [ ω ] → ω such that X computes no infinite f -rainbow. As SRRT ≤ c P , there exists a computableinstance of P such that X does not compute a solution to it. Hence U is not a universalinstance of P . (cid:3) Corollary 6.15
None of
SRRT , SEM , STS ( ) and SFS ( ) admits a universal instance. Proof of Theorem 6.12.
The proof is an adaptation of [
20, Theorem 5.3.7 ] . Supposethat D is a ∆ incomplete set. We will construct a ∆ coloring f : ω → ω such that D does not compute any f -diagonalizing function. We want to satisfy the followingrequirements for each e ∈ ω : R e : If Φ De is total, then there is an a such that Φ De ( a ) = f ( a ) .For each e ∈ ω , define the partial function u e by letting u e ( a ) be the use of Φ De oninput a if Φ De ( a ) ↓ and letting u e ( a ) ↑ otherwise. We can assume w.l.o.g. that whenever u e ( a ) ↓ then u e ( a ) ≥ a . Also define a computable partial function θ by letting θ ( a ) =( µ t )[ a ∈ ; ′ t ] if a ∈ ; ′ and θ ( a ) ↑ otherwise.The local strategy for satisfying a single requirement R e works as follows. If R e receives attention at stage s , this strategy does the following. First it identifies a num-ber a ≥ e that is not restrained by strategies of higher priority such that the followingconditions holds: EGREES BOUNDING PRINCIPLES AND UNIVERSAL INSTANCES 21 (i) Φ D s e , s ( a ) ↓ (ii) u e , s ( a ) < max ( θ s ( a )) If no such number a exists, the strategy does nothing. Otherwise it puts a restrainton a and commits to assigning f s ( a ) = Φ D s e , s ( a ) . For any such a , this commitment willremain active as long as the strategy has a restraint on this element. Having done allthis, the local strategy is declared to be satisfied and will not act again unless either ahigher priority puts restraints on a , or the value of u e , s ( a ) or θ s ( a ) changes. In bothcases the strategy gets injured and has to reset, releasing all its restraints.To finish stage s , the global strategy assigns values f s ( y ) for all y ≤ s as follows: if y is commited to some value assignment of f s ( y ) due to a local strategy, then define f s ( y ) to be this value. If not, let f s ( y ) =
0. This finishes the construction and we nowturn to the verification.For each e , a ∈ ω , let Z e , a = (cid:8) s ∈ ω : R e restrains a at stage s (cid:9) . Claim.
For each e , a ∈ ω ,(a) if Φ De ( a ) ↑ then Z e , a is finite;(b) if Φ De ( a ) ↓ = Z e , a is either finite or cofinite. Proof.
By induction on the priority order. We consider Z e , a , assuming that for all R e ′ ofhigher priority, the set Z e ′ , a is either finite or cofinite. First notice that Z e , a = ; if a < e or a
6∈ ; ′ , so we may assume that a ≥ e and a ∈ ; ′ . If there exists e ′ < e such that Z e ′ , a is cofinite, then Z e , a is finite because at most one requirement may claim a at a givenstage. Suppose that Z e ′ , a is finite for all e ′ < e . Fix t such that for all e ′ < e and s ≥ t R e ′ does not restrain a at stage s . and θ s ( a ) = θ ( a ) .Suppose that Φ De ( a ) ↑ . Fix t ≥ t such that D ( b ) = D s ( b ) for all b ≤ θ ( a ) andall s ≥ t . Then for all s ≥ t , if Φ D s e , s ( a ) ↓ then we must have u e , s ( a ) > θ ( a ) becauseotherwise Φ De ( a ) ↓ . It follows that for all s ≥ t , requirement R e does not restrain a atstage s . Hence Z e , a is finite.Suppose now that Φ De ( a ) ↓ . Fix t ≥ t such that for all s ≥ t we have Φ D s e , s ( a ) ↓ and D s ( c ) = D ( c ) for every c ≤ u e ( a ) . For every s ≥ t , u e , s ( a ) = u e , t ( a ) and θ s ( a ) = θ t ( a ) for each i ≤ a . So properties (i) and (ii) will either hold at each stage s ≥ t , or nothold at each stage s ≥ t . Therefore Z e , a is either finite or cofinite. (cid:3) Claim.
Each requirement R e is satisfied. Proof.
Suppose that Φ De is total for some e ∈ ω . We will prove that Φ De is not an f -diagonalizing function. Let A = ¦ a ≥ e : ( ∀ e ′ < e ) Z e ′ , a is finite © . Notice that A iscofinite since for each e ′ < e , there is at most one a such that Z e ′ , a is cofinite. Define h : ω → ω as follows.If for all but finitely many k ∈ ω , we have k ∈ ; ′ → k ∈ ; ′ u e ( k ) , then ; ′ ≤ T u e ≤ T D ,contrary to hypothesis. Thus we may let a be the least element of { k ∈ A : k ∈ ; ′ r ; ′ u e ( k ) } greater than e . We then have(1) a ≥ e , Φ De ( a ) ↓ , θ ( a ) > u e ( a ) (2) For all e ′ < e , there exists t such that R e ′ does not claim a at any stage s ≥ t .Therefore we may fix t ≥ a such that for all s ≥ t , we have Φ D s e , s ( a ) ↓ , θ s ( a ) = θ ( a ) , u e , s ( a ) = u e ( a ) , and for each e ′ < e , R e ′ does not claim a at stage s . Thus, for every s ≥ t , requirement R e claims a ′ ≤ a at stage s . Since Z e , i is either finite or cofinitefor each i ≤ a , it follows that Z e , a is cofinite. By the above argument, we must have Φ De ( a ) ↓ , and by construction, f ( a ) = Φ De ( a ) . Therefore R e is satisfied. (cid:3) Claim.
The resulting function f s is ∆ . Proof.
Consider a particular element a . Because of Claim 1, if e > a then Z e , a = ; .We have then two cases: Either Z e , a is finite for all e ≤ a , in which case for all butfinitely many s , f s ( a ) =
0, or Z e , a is cofinite for some e . Then there is a stage s atwhich requirement R e has committed f s ( a ) = Φ De ( a ) for assignment and has neverbeen injured. Thus f is ∆ . (cid:3)(cid:3) Acknowledgements . The author is thankful to his PhD advisor Laurent Bienvenufor his availability during the different steps giving birth to this paper, and his usefulsuggestions increasing its readability. R
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