Any FIP real computes a 1-generic

Cholak, Peter; Downey, Rodney G.; Igusa, Greg

doi:10.48550/arxiv.1502.03785

Cited by 2 publications

(11 citation statements)

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“…It can be seen as a sequential version of Ramsey's theorem for singletons and admits characterizations in terms of degrees whose jump computes a path through a Π 0,∅ ′ 1 class [16]. The decomposition of RT 2 2 in terms of COH and stable Ramsey's theorem for pairs (SRT 2 2 ) has been reused in the analysis of many consequences of Ramsey's theorem [14]. The link between cohesiveness and SRT 2 2 is still an active research subject [4,8,12,26].…”

Section: Cohesivenessmentioning

confidence: 99%

“…The decomposition of RT 2 2 in terms of COH and stable Ramsey's theorem for pairs (SRT 2 2 ) has been reused in the analysis of many consequences of Ramsey's theorem [14]. The link between cohesiveness and SRT 2 2 is still an active research subject [4,8,12,26].…”

Section: Cohesivenessmentioning

confidence: 99%

“…However, Ramsey theory is known to provide a large class of theorems escaping this phenomenon. Indeed, consequences of Ramsey's theorem for pairs (RT 2 2 ) usually belong to their own subsystem. Therefore, they received a lot of attention from the reverse mathematics community [3,13,14,27].…”

Section: Introductionmentioning

confidence: 99%

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Dominating the Erdős–Moser theorem in reverse mathematics

Patey

2017

Annals of Pure and Applied Logic

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The Erdős-Moser theorem (EM) states that every infinite tournament has an infinite transitive subtournament. This principle plays an important role in the understanding of the computational strength of Ramsey's theorem for pairs (RT 2 2 ) by providing an alternate proof of RT 2 2 in terms of EM and the ascending descending sequence principle (ADS). In this paper, we study the computational weakness of EM and construct a standard model (ω-model) of simultaneously EM, weak König's lemma and the cohesiveness principle, which is not a model of the atomic model theorem. This separation answers a question of Hirschfeldt, Shore and Slaman, and shows that the weakness of the Erdös-Moser theorem goes beyond the separation of EM from ADS proven by Lerman, Solomon and Towsner. LUDOVIC PATEYJockusch, Lempp and Slaman [13] proved that COH contains a model with no diagonally non-computable function, thus COH does not imply SRT 2 2 over RCA 0 . Cooper [6] proved that every degree above 0 ′ is the jump of a minimal degree. Therefore there exists a p-cohesive set of minimal degree. The Erdős-Moser theoremThe Erdős-Moser theorem is a principle coming from graph theory. It provides together with the ascending descending principle (ADS) an alternative proof of Ramsey's theorem for pairs (RT 2 2 ). Indeed, every coloring f : [ω] 2 → 2 can be seen as a tournament R such that R(x, y) holds if x < y and f (x, y) = 1, or x > y and f (y, x) = 0. Every infinite transitive subtournament induces a linear order whose infinite ascending or descending sequences are homogeneous for f .EM is the statement "Every infinite tournament T has an infinite transitive subtournament." SEM is the restriction of EM to stable tournaments.Bovykin and Weiermann [1] introduced the Erdős-Moser theorem in reverse mathematics and proved that EM together with the chain-antichain principle (CAC) is equivalent to RT 2 2 over RCA 0 . This was refined into an equivalence between EM + ADS and RT 2 2 by Montalbán (see [1]), and the equivalence still holds between the stable versions of the statements. Lerman, Solomon and Towsner [19] proved that EM is strictly weaker than RT 2 2 by constructing an ωmodel of EM which is not a model of the stable ascending descending sequence (SADS). SADS is the restriction of ADS to linear orders of order type ω + ω * [14]. The author noticed in [22] that their construction can be adapted to obtain a separation of EM from the stable thin set theorem for pairs (STS (2)). Wang strengthened this separation by constructing in [31] a standard model of many theorems, including EM, COH and weak König's lemma (WKL) which is neither a model of STS(2) nor a model of SADS. The author later refined in [24,26] the forcing technique of Lerman, Solomon and Towsner and showed that it is strong enough to obtain the same separations as Wang.On the lower bounds side, Lerman, Solomon and Towsner [19] showed that EM implies the omitting partial types principle (OPT) over RCA 0 + BΣ 0 2 and Kreuzer proved in [18] that SEM implies BΣ 0 2 over RCA 0 . The stat...

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Section: Cohesivenessmentioning

confidence: 99%

Section: Cohesivenessmentioning

confidence: 99%

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Dominating the Erdős–Moser theorem in reverse mathematics

Patey

2017

Annals of Pure and Applied Logic

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“…Downey et al [14] first established a link between 1-genericity and FIP by proving that 1-GEN implies FIP over RCA 0 . Later, Cholak et al [5] proved the other direction.…”

Section: Question 42 Does Em Admit a Universal Instance?mentioning

confidence: 95%

“…It happens that most consequences of RT 2 2 in reverse mathematics do not admit a universal instance. The most notable exceptions are the rainbow Ramsey theorem for pairs [12,55,62,63], the finite intersection property [5,14,20] and DNR. It is natural to wonder, given the fact that RWKL is a consequence of both SRT 2 2 and of WKL 0 , whether RWKL admits a universal instance.…”

Section: A Ramsey-type Weak König's Lemmamentioning

confidence: 99%

Open Questions About Ramsey-Type Statements in Reverse Mathematics

Patey¹

2016

Bull. symb. log

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Abstract. Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey's theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.

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Any FIP real computes a 1-generic

Cited by 2 publications

References 5 publications

Dominating the Erdős–Moser theorem in reverse mathematics

Dominating the Erdős–Moser theorem in reverse mathematics

Open Questions About Ramsey-Type Statements in Reverse Mathematics

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