Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T11:54:32.485Z Has data issue: false hasContentIssue false
  • English
  • Français

Rainbow Ramsey Theorem for Triples is Strictly Weaker than the Arithmetical Comprehension Axiom

Published online by Cambridge University Press:  12 March 2014

Wei Wang
Wei Wang*
Affiliation:
Institute of Logic and Cognition and Department of Philosophy, Sun Yat-Sen University, 135 Xingang XI Road, Guangzhou 510275, P.R. China, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that RCA0 + RRT ⊬ ACA0 where RRT is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow, regardless of the complexity of the given coloring. We also apply the proof of the cone avoidance theorem to the question whether RCA0 + RRT ⊦ ACA0 and obtain some partial answer.

Keywords

03B3003F3503D3203D80
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 824 - 836
Copyright
Copyright © Association for Symbolic Logic 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Cholak, Peter A., Jockusch, Carl G., and Slaman, Theodore A., On the strength of Ramsey's theorem for pairs, this Journal, vol. 66 (2001), no. 1, pp. 1–55.Google Scholar
[2] Csima, Barbara and Mileti, Joseph, The strength of the rainbow Ramsey theorem, this Journal, vol. 74 (2009), no. 4, pp. 1310–1324.Google Scholar
[3] Dzhafarov, Damir D. and Jockusch, Carl G. JR., Ramsey's theorem and cone avoidance, this Journal, vol. 74 (2009), no. 2, pp. 557–578.Google Scholar
[4] Hirschfeldt, Denis R. and Shore, Richard A., Combinatorial principles weaker than Ramsey's theorem for pairs, this Journal, vol. 72 (2007), no. 1, pp. 171–206.Google Scholar
[5] Jockusch, Carl G. JR., Ramsey's theorem and recursion theory, this Journal, vol. 37 (1972), no. 2, pp. 268–280.Google Scholar
[6] Jockusch, Carl G. JR. and Soare, Robert I., -classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[7] Kučera, Antonín, Measure, -classes and complete extensions of PA, Recursion theory week (Oberwolfach, 1984), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245–259.Google Scholar
[8] Mileti, Joseph Roy, Partition theorems and computability theory, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2004.Google Scholar
[9] Seetapun, David and Slaman, Theodore A., On the strength of Ramsey's theorem, Notre Dame Journal of Formal Logic, vol. 36 (1995), no. 4, pp. 570–582, Special Issue: Models of arithmetic.Google Scholar
[10] Simpson, Stephen G., Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999.CrossRefGoogle Scholar
[11] Soare, Robert I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle Scholar
[12] Specker, E., Ramsey's Theorem does not hold in recursive set theory, Logic Colloquium (Manchester, 1969), 1971, pp. 439–442.Google Scholar