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

N1 - Funding Information:
The first author was partially supported by NSF grant DMS-1400267.

PY - 2017

Y1 - 2017

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 -