The strength of the rainbow Ramsey Theorem
Barbara F. Csima and Joseph R. Mileti
Source: J. Symbolic Logic Volume 74, Issue 4
(2009), 1310-1324.
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.
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1254748693
Digital Object Identifier: doi:10.2178/jsl/1254748693
Zentralblatt MATH identifier: 05654332
Mathematical Reviews number (MathSciNet): MR2583822
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