### Abstract

We answer a question posed by Hirschfeldt and Jockusch by showing that whenever k > ℓ 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}/ 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, 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}.

Original language | English (US) |
---|---|

Pages (from-to) | 1343-1355 |

Number of pages | 13 |

Journal | Proceedings of the American Mathematical Society |

Volume | 145 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2017 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*145*(3), 1343-1355. https://doi.org/10.1090/proc/13315

}

*Proceedings of the American Mathematical Society*, vol. 145, no. 3, pp. 1343-1355. https://doi.org/10.1090/proc/13315

**Ramsey’s theorem for singletons and strong computable reducibility.** / Dzhafarov, Damir D.; Patey, Ludovic; Solomon, Reed; Westrick, Linda Brown.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Ramsey’s theorem for singletons and strong computable reducibility

AU - Dzhafarov, Damir D.

AU - Patey, Ludovic

AU - Solomon, Reed

AU - Westrick, Linda Brown

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85007282887&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85007282887&partnerID=8YFLogxK

U2 - 10.1090/proc/13315

DO - 10.1090/proc/13315

M3 - Article

AN - SCOPUS:85007282887

VL - 145

SP - 1343

EP - 1355

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 3

ER -