Journal of Symbolic Logic

Rainbow Ramsey Theorem for triples is strictly weaker than the Arithmetical Comprehension Axiom

Wei Wang

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Abstract

We prove that $\operatorname{RCA}_0 + \operatorname{RRT}^3_2 \nvdash \operatorname{ACA}_0$ where $\operatorname{RRT}^3_2$ 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 $\operatorname{RCA}_0 + \operatorname{RRT}^4_2 \vdash \operatorname{ACA}_0$ and obtain some partial answer.

Article information

Source
J. Symbolic Logic, Volume 78, Issue 3 (2013), 824-836.

Dates
First available in Project Euclid: 6 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1389032277

Digital Object Identifier
doi:10.2178/jsl.7803070

Mathematical Reviews number (MathSciNet)
MR3135500

Zentralblatt MATH identifier
1300.03013

Subjects
Primary: 03B30, 03F35, 03D32, 03D80

Citation

Wang, Wei. Rainbow Ramsey Theorem for triples is strictly weaker than the Arithmetical Comprehension Axiom. J. Symbolic Logic 78 (2013), no. 3, 824--836. doi:10.2178/jsl.7803070. https://projecteuclid.org/euclid.jsl/1389032277


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