Ramsey’s theorem for singletons and strong computable reducibility

Damir D. Dzhafarov, Ludovic Patey, Reed Solomon, Linda Brown Westrick

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6 Scopus citations

Abstract

We answer a question posed by Hirschfeldt and Jockusch by showing that whenever k > ℓ Ramsey’s theorem for singletons and k-colorings, RT1/k, is not strongly computably reducible to the stable Ramsey’s theorem for _-colorings, SRT2/ ℓ Our proof actually establishes the following considerably stronger fact: given k > ℓ there is a coloring c: ω → ℓ such that for every stable coloring d: [ω]2 → ℓ (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, SRT2<. 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 RCA0.

Original languageEnglish (US)
Pages (from-to)1343-1355
Number of pages13
JournalProceedings of the American Mathematical Society
Volume145
Issue number3
DOIs
StatePublished - 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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