an:06668907
Zbl 1423.03159
Dzhafarov, Damir D.; Patey, Ludovic; Solomon, Reed; Westrick, Linda Brown
Ramsey's theorem for singletons and strong computable reducibility
EN
Proc. Am. Math. Soc. 145, No. 3, 1343-1355 (2017).
00361743
2017
j
03D80 03F35 05D10 03B30 03D30
Summary: We answer a question posed by \textit{D. R. Hirschfeldt} and \textit{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 \( \mathsf {RCA}_0\).
Zbl 1373.03068; Zbl 1368.03044