# Computable Reductions and Reverse Mathematics

Conference paper

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## Abstract

Recent work in reverse mathematics on combinatorial principles below Ramsey’s theorem for pairs has made use of a variety of computable reductions to give a finer analysis of the relationships between these principles. We use three concrete examples to illustrate this work, survey the known results and give new negative results concerning \(\mathsf {RT}^1_k\), \(\mathsf {SRT}^2_\ell \) and \(\mathsf {COH}\). Motivated by these examples, we introduce several variations of \(\mathsf {ADS}\) and describe the relationships between these principles under Weihrauch and strong Weihrauch reductions.

## References

- 1.Astor, E.P., Dzhafarov, D.D., Solomon, R., Suggs, J.: The uniform content of partial and linear orders (in preparation)Google Scholar
- 2.Cholak, P.A., Jockusch Jr., C.G., Slaman, T.A.: On the strength of Ramsey’s theorem for pairs. J. Symb. Log.
**66**, 1–55 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Chong, C.T., Lempp, S., Yang, Y.: On the role of collection principles for \(\Sigma ^0_2\) formulas in second-order reverse mathematics. Proc. Am. Math. Soc.
**138**, 1093–1100 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Chong, C.T., Slaman, T.A., Yang, Y.: The metamathematics of stable Ramsey’s theorem for pairs. J. Am. Math Soc.
**27**, 863–892 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Dorais, F.G., Dzhafarov, D.D., Hirst, J.L., Mileti, J.R., Shafer, P.: On uniform relationships between combinatorial problems. Trans. Am. Math. Soc.
**368**(2), 1321–1359 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Downey, R., Hirschfeldt, D.R., Lempp, S., Solomon, R.: A \(\Delta ^0_2\) set with no infinite low set in either it or its complement. J. Symb. Log.
**66**, 1371–1381 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Dzhafarov, D.D.: Strong reducibilities between combinatorial principles. J. Symb. Log. (to appear)Google Scholar
- 8.Dzhafarov, D.D., Patey, L., Solomon, R., Westrick, L.B.: Ramsey’s theorem for singletons and strong computable reducibility (submitted)Google Scholar
- 9.Hirschfeldt, D.R., Jockusch Jr., C.G.: On notions of computability theoretic reduction between \(\Pi ^1_2\) principles. J. Math. Logic (to appear)Google Scholar
- 10.Hirschfeldt, D.R., Shore, R.A.: Combinatorial principles weaker than Ramsey’s theorem for pairs. J. Symb. Log.
**72**, 171–206 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Patey, L.: The weakness of being cohesive, thin or free in reverse mathematics. Isr. J. Math. (to appear)Google Scholar
- 12.Rakotoniaina, T.: The computational strength of Ramsey’s theorem. Ph.D. thesis, University of Cape Town (2015)Google Scholar

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© Springer International Publishing Switzerland 2016