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Archive for Mathematical Logic

, Volume 58, Issue 3–4, pp 413–425 | Cite as

Dickson’s lemma and weak Ramsey theory

  • Yasuhiko OmataEmail author
  • Florian Pelupessy
Article

Abstract

We explore the connections between Dickson’s lemma and weak Ramsey theory. We show that a weak version of the Paris–Harrington principle for pairs in c colors and miniaturized Dickson’s lemma for c-tuples are equivalent over \(\mathsf {RCA}_0^{*}\). Furthermore, we look at a cascade of consequences for several variants of weak Ramsey’s theorem.

Keywords

Reverse mathematics Ramsey theory Dickson’s lemma 

Mathematics Subject Classification

03B30 05C55 03F30 03H15 

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Notes

Acknowledgements

The first author thanks his supervisor Professor Kazuyuki Tanaka for helpful discussions and his support. We also thank the anonymous referee for valuable comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan

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