Ramsey's theorem for singletons and strong computable reducibility

Damir Dzhafarov 1 Ludovic Patey 2, 3 Reed Solomon 1 Linda Brown Westrick 1
3 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
Abstract : We answer a question posed by Hirschfeldt and Jockusch by showing that whenever k > , Ramsey's theorem for singletons and k-colorings, RT 1 k , is not strongly computably reducible to the stable Ramsey's theorem for-colorings, SRT 2. Our proof actually establishes the following considerably stronger fact: given k > , there is a coloring c : ω → k such that for every stable coloring d : [ω] 2 → (computable from c or not), there is an infinite homogeneous set H for d that computes no infinite homogeneous set for c. This also answers a separate question of Dzhafarov, as it follows that the cohesive principle, COH, is not strongly computably reducible to the stable Ramsey's theorem for all colorings, SRT 2 <∞. The latter is the strongest partial result to date in the direction of giving a negative answer to the longstanding open question of whether COH is implied by the stable Ramsey's theorem in ω-models of RCA 0 .
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Damir Dzhafarov, Ludovic Patey, Reed Solomon, Linda Brown Westrick. Ramsey's theorem for singletons and strong computable reducibility. Proceedings of the American Mathematical Society, American Mathematical Society, 2017, 145 (3), pp.1343 - 1355. ⟨10.1090/proc/13315⟩. ⟨hal-01888750⟩

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