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Ramsey’s theorem for singletons and strong computable reducibility. (English) Zbl 1423.03159
Summary: We answer a question posed by D. R. Hirschfeldt and C. G. Jockusch jun. [J. Math. Log. 16, No. 1, Article ID 1650002, 59 p. (2016; Zbl 1373.03068)] by showing that whenever $$k > \ell$$, Ramsey’s theorem for singletons and $$k$$-colorings, $$\mathsf {RT}^1_k$$, is not strongly computably reducible to the stable Ramsey’s theorem for $$\ell$$-colorings, $$\mathsf {SRT}^2_\ell$$. Our proof actually establishes the following considerably stronger fact: given $$k > \ell$$, there is a coloring $$c : \omega \rightarrow k$$ such that for every stable coloring $$d : [\omega]^2 \rightarrow \ell$$ (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 the first author [J. Symb. Log. 81, No. 4, 1405–1431 (2016; Zbl 1368.03044)], as it follows that the cohesive principle, $$\mathsf {COH}$$, is not strongly computably reducible to the stable Ramsey’s theorem for all colorings, $$\mathsf {SRT}^2_{<\infty}$$. The latter is the strongest partial result to date in the direction of giving a negative answer to the longstanding open question of whether $$\mathsf {COH}$$ is implied by the stable Ramsey’s theorem in $$\omega$$-models of $$\mathsf {RCA}_0$$.

##### MSC:
 03D80 Applications of computability and recursion theory 03F35 Second- and higher-order arithmetic and fragments 05D10 Ramsey theory 03B30 Foundations of classical theories (including reverse mathematics) 03D30 Other degrees and reducibilities in computability and recursion theory
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