Abstract
Let TT1 be the combinatorial principle stating that every finite coloring of the infinite full binary tree has a homogeneous isomorphic subtree. Let RT 22 and WKL0 denote respectively the principles of Ramsey’s theorem for pairs and the weak König lemma. It is proved that TT1 + RT 22 + WKL0 is Π 03 -conservative over the base system RCA0. Thus over RCA0, TT1 and Ramsey’s theorem for pairs prove the same Π 03 -sentences.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. T. Chong, W. Li, W. Wang and Y. Yang, On the strength of Ramsey’s theorem for trees, Advances in Mathematics 369 (2020), Article no. 107180.
C. T. Chong, T. A. Slaman and Y. Yang, The metamathematics of stable Ramsey’s theorem for pairs, Journal of the American Mathematical Society 27 (2014), 863–892.
C. T. Chong, T. A. Slaman and Y. Yang, The inductive strength of Ramsey’s theorem for pairs, Advances in Mathematics 308 (2017), 121–141.
J. Corduan, M. J. Groszek and J. R. Mileti, Reverse mathematics and Ramsey’s property for trees, Journal of Symbolic Logic 75 (2010), 945–954.
P. Gács, Every sequence is reducible to a random one, Information and Control 70 (1986), 186–192.
J. L. Hirst, Combinatorics in Subsystems of Second Order Arithmetic, Ph.D. thesis, The Pennsylvania State University, University Park, PA, 1987.
J. Ketonen and R. Solovay, Rapidly growing Ramsey functions, Annals of Mathematics 113 (1981), 267–314.
L. A. Kołodziejczyk and K. Yokoyama, Some upper bounds on ordinal-valued Ramsey numbers for colourings of pairs, Selecta Mathematica 26 (2020), Article no. 56.
A. P. Kreuzer and K. Yokoyama, On principles between Σ1-and Σ2-induction, and monotone enumerations, Journal of Mathematical Logic 16 (2016), Article no. 1650004.
A. Kučera, Measure, Π10-classes and complete extensions of PA, in Recursion Theory Week (Oberwolfach, 1984), Lecture Notes in Mathematics, Vol. 1141, Springer, Berlin, 1985, pp. 245–259.
J. B. Paris and L. A. S. Kirby, Σn-collection schemas in arithmetic, in Logic Colloquium’ 77 (Proc. Conf., Wrocław, 1977), Studies in Logic and the Foundations Mathematics, Vol. 96, North-Holland, Amsterdam-New York, 1978, pp. 199–209.
L. Patey and K. Yokoyama, The proof-theoretic strength of Ramsey’s theorem for pairs and two colors, Advances in Mathematics 330 (2018), 1034–1070.
S. G. Simpson, Subsystems of Second Order Arithmetic, Perspectives in Logic, Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY, 2009.
Author information
Authors and Affiliations
Corresponding author
Additional information
All the authors acknowledge the support of the NUS Institute for Mathematical Sciences where the work began. The authors thank the referee for helpful suggestions.
Chong’s research was partially supported by NUS grants C-146-000-042-001 and WBS: R389-000-040-101.
Wang’s research was partially supported by China NSF Grant 11971501.
Yang’s research was partially supported by MOE2019-T2-2-121.
Rights and permissions
About this article
Cite this article
Chong, C.T., Wang, W. & Yang, Y. Conservation strength of the infinite pigeonhole principle for trees. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2567-8
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s11856-023-2567-8