Ramsey's theorem for singletons and strong computable reducibility
Abstract
We answer a question posed by Hirschfeldt and Jockusch 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 $c : \omega \to k$ such that for every stable coloring $d : [\omega]^2 \to \ell$ (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 Dzhafarov, 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 $\mathsf{RCA}_0$.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 arXiv:
 arXiv:1602.04481
 Bibcode:
 2016arXiv160204481D
 Keywords:

 Mathematics  Logic
 EPrint:
 13 pages