The weakness of being cohesive, thin or free in reverse mathematics (Q503277)

A combinatorial principle \(P\) is computably reducible to a principle \(Q\), denoted \( P \leq_c Q\), if given any instance \(p\) of \(P\) one can compute from \(p\) an instance \(q\) of \(Q\) such that for any solution \(s\) of \(q\), a solution of \(p\) can be computed from \(s \oplus p\). The author proves the nonexistence of myriad computable reductions and associated results in reverse mathematics. For example, for stable Ramsey's theorem, he proves that for \(k>l\) and \(n \geq 2\), \(\text{SRT}^n_k \not\leq_c\text{SRT}^n_l\), extending recent work of \textit{D. D. Dzhafarov} [J. Symb. Log. 81, No. 4, 1405--1431 (2016; Zbl 1368.03044)]. For the thin set theorem, he proves that for \(n\geq2\) and \(l>k\geq 2\), \(\text{TS}^n_k \not\leq_c \text{TS}^n_l\). This result is used to construct an infinite chain of statements which are strictly descending in reverse mathematical strength. With regard to thin sets and free sets, the author answers many questions found in [\textit{P. A. Cholak} et al., in: Reverse mathematics 2001. Wellesley, MA: A K Peters; Urbana, IL: Association for Symbolic Logic (ASL). 104--119 (2005; Zbl 1092.03031); \textit{A. Montalbán}, Bull. Symb. Log. 17, No. 3, 431--454 (2011; Zbl 1233.03023); \textit{D. R. Hirschfeldt}, Slicing the truth. On the computable and reverse mathematics of combinatorial principles. Hackensack, NJ: World Scientific (2014; Zbl 1304.03001)]. The article also answers questions related to cohesive sets and the principle COH posed by \textit{W. Wang} [J. Nanjing Univ., Math. Biq. 30, No. 1, 40--47 (2013; Zbl 1313.03019)]. Proof techniques include forcing on Mathias conditions and the preservation of non-c.e.~definitions of \textit{W. Wang} [J. Symb. Log. 81, No. 4, 1531--1554 (2016; Zbl 1436.03099)].