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RAMSEY-LIKE THEOREMS AND MODULI OF COMPUTATION

Part of: General logic Computability and recursion theory

Published online by Cambridge University Press:  27 October 2020

LUDOVIC PATEY
LUDOVIC PATEY*
Affiliation:
CN RS, INSTITUT CAMILLE JORDAN UNIVERSITÉ CLAUDE BERNARD LYON 1 43 BOULEVARD DU 11 NOVEMBRE 1918 F-69622 VILLEURBANNE CEDEX, FRANCEE-mail: ludovic.patey@computability.frURL: http://ludovicpatey.com
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Abstract

Ramsey’s theorem asserts that every k-coloring of $[\omega ]^n$ admits an infinite monochromatic set. Whenever $n \geq 3$ , there exists a computable k-coloring of $[\omega ]^n$ whose solutions compute the halting set. On the other hand, for every computable k-coloring of $[\omega ]^2$ and every noncomputable set C, there is an infinite monochromatic set H such that $C \not \leq _T H$ . The latter property is known as cone avoidance.

In this article, we design a natural class of Ramsey-like theorems encompassing many statements studied in reverse mathematics. We prove that this class admits a maximal statement satisfying cone avoidance and use it as a criterion to re-obtain many existing proofs of cone avoidance. This maximal statement asserts the existence, for every k-coloring of $[\omega ]^n$ , of an infinite subdomain $H \subseteq \omega $ over which the coloring depends only on the sparsity of its elements. This confirms the intuition that Ramsey-like theorems compute Turing degrees only through the sparsity of its solutions.

Keywords

Ramsey’s theoryreverse mathematicscomputability theorymodulus

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 03D80: Applications of computability and recursion theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 72 - 108
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Bovykin, A. and Weiermann, A., The strength of infinitary Ramseyan principles can be accessed by their densities. Annals of Pure and Applied Logic , vol. 168 (2017), no. 9, pp. 1700–1709.10.1016/j.apal.2017.03.005CrossRefGoogle Scholar
Cholak, P. A., Giusto, M., Hirst, J. L., and Jockusch, C. G. Jr, Free sets and reverse mathematics . Reverse Mathematics , vol. 21 (2001), pp. 104–119.Google Scholar
Cholak, P. A., Jockusch, C. G., and Slaman, T. A., On the strength of Ramsey’s theorem for pairs , this Journal , vol. 66 (2001), no. 1, pp. 1–55.Google Scholar
Cholak, P. A. and Patey, L., Thin set theorems and cone avoidance , Transactions of the American Mathematical Society . (2019). to appear.10.1090/tran/7987CrossRefGoogle Scholar
Chubb, J., Hirst, J. L., and McNicholl, T. H., Reverse mathematics, computability, and partitions of trees , this Journal , vol. 74 (2009), no. 1, pp. 201–215.Google Scholar
Csima, B. F., Hirschfeldt, D. R., Knight, J. F., and Soare, R. I., Bounding prime models , this Journal , vol. 69 (2004), pp. 1117–1142.Google Scholar
Csima, B. F. and Mileti, J. R., The strength of the rainbow Ramsey theorem , this Journal , vol. 74 (2009), no. 4, pp. 1310–1324.Google Scholar
Dorais, F. G., Dzhafarov, D. D., Hirst, J. L., Mileti, J. R., and Shafer, P., On uniform relationships between combinatorial problems . Transactions of the American Mathematical Society , vol. 368 (2016), no. 2, pp. 1321–1359.CrossRefGoogle Scholar
Dzhafarov, D. D. and Jockusch, C. G., Ramsey’s theorem and cone avoidance , this Journal , vol. 74 (2009), no. 2, pp. 557–578.Google Scholar
Dzhafarov, D. D. and Patey, L., Coloring trees in reverse mathematics . Advances in Mathematics , vol. 318 (2017), pp. 497–514.CrossRefGoogle Scholar
Friedman, H. M., Fom:53:free sets and reverse math and fom:54:recursion theory and dynamics, Available at https://www.cs.nyu.edu/pipermail/fom/.Google Scholar
Groszek, M. J. and Slaman, T. A., Moduli of Computation (talk) , Buenos Aires, Argentina, 2007.Google Scholar
Hirschfeldt, D. R. and Jockusch, C. G., On notions of computability-theoretic reduction between ${\varPi}_2^1$ principles . Journal of Mathematical Logic , vol. 16 (2016), no. 1, p. 1650002, 59.CrossRefGoogle Scholar
Hirschfeldt, D. R. and Shore, R. A., Combinatorial principles weaker than Ramsey’s theorem for pairs , this Journal , vol. 72 (2007), no. 1, pp. 171–206.Google Scholar
Jockusch, C. G., Ramsey’s theorem and recursion theory , this Journal , vol. 37 (1972), no. 2, pp. 268–280.Google Scholar
Lerman, M., Degrees of Unsolvability: Local and Global Theory. Perspectives in Mathematical Logic , Springer-Verlag, Berlin, 1983.10.1007/978-3-662-21755-9CrossRefGoogle Scholar
Liu, L., RT2 2 does not imply WKL0 , this Journal , vol. 77 (2012), no. 2, pp. 609–620.Google Scholar
Mileti, J. R., Partition theorems and computability theory , (Ph.D. thesis), University of Illinois at Urbana-Champaign, ProQuest LLC, Ann Arbor, MI, 2004.Google Scholar
Monin, B. and Patey, L., Pigeons do not jump high, to appear. 2018. Available at https://arxiv.org/abs/1803.09771.Google Scholar
Patey, L., Combinatorial weaknesses of Ramseyan principles, In preparation. 2015. Available at http://ludovicpatey.com/media/research/combinatorial-weaknesses-draft.pdf.Google Scholar
Patey, L., Iterative forcing and hyperimmunity in reverse mathematics , CiE. Evolving Computability (A. Beckmann, V. Mitrana, and M. Soskova, editors), Lecture Notes in Computer Science, vol. 9136, Springer International Publishing, Berlin, Germany, 2015, pp. 291–301 (English).10.1007/978-3-319-20028-6_30CrossRefGoogle Scholar
Patey, L., Somewhere over the rainbow Ramsey theorem for pairs, Submitted. 2015. Available at http://arxiv.org/abs/1501.07424.Google Scholar
Patey, L., Open questions about Ramsey-type statements in reverse mathematics . The Bulletin of Symbolic Logic , vol. 22 (2016), no. 2, pp. 151–169.CrossRefGoogle Scholar
Patey, L., The reverse mathematics of Ramsey-type theorems , Ph.D. thesis, Université Paris Diderot, 2016.Google Scholar
Patey, L., Iterative forcing and hyperimmunity in reverse mathematics . Computability , vol. 6 (2017), no. 3, pp. 209–221.CrossRefGoogle Scholar
Seetapun, D. and Slaman, T. A., On the strength of Ramsey’s theorem . Notre Dame Journal of Formal Logic , vol. 36 (1995), no. 4, pp. 570–582.CrossRefGoogle Scholar
Simpson, S. G., Subsystems of Second Order Arithmetic , Cambridge University Press, Cambridge, MA, 2009.CrossRefGoogle Scholar
Solovay, R. M., Hyperarithmetically encodable sets . Transactions of the American Mathematica Society , vol. 239 (1978), pp. 99–122.CrossRefGoogle Scholar
Wang, W., Some logically weak Ramseyan theorems . Advances in Mathematics , vol. 261 (2014), pp. 1–25.CrossRefGoogle Scholar