Journal of Symbolic Logic

The strength of the rainbow Ramsey Theorem

Barbara F. Csima and Joseph R. Mileti

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Abstract

The Rainbow Ramsey Theorem is essentially an “anti-Ramsey" theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey's Theorem, even in the weak system RCA0 of reverse mathematics. We answer the question of the converse implication for pairs, showing that the Rainbow Ramsey Theorem for pairs is in fact strictly weaker than Ramsey's Theorem for pairs over RCA0. The separation involves techniques from the theory of randomness by showing that every 2-random bounds an ω-model of the Rainbow Ramsey Theorem for pairs. These results also provide as a corollary a new proof of Martin's theorem that the hyperimmune degrees have measure one.

Article information

Source
J. Symbolic Logic, Volume 74, Issue 4 (2009), 1310-1324.

Dates
First available in Project Euclid: 5 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1254748693

Digital Object Identifier
doi:10.2178/jsl/1254748693

Mathematical Reviews number (MathSciNet)
MR2583822

Zentralblatt MATH identifier
1188.03044

Citation

Csima, Barbara F.; Mileti, Joseph R. The strength of the rainbow Ramsey Theorem. J. Symbolic Logic 74 (2009), no. 4, 1310--1324. doi:10.2178/jsl/1254748693. https://projecteuclid.org/euclid.jsl/1254748693


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