The Strength of the Rainbow Ramsey Theorem

Journal of Symbolic Logic 74 (4):1310 - 1324 (2009)
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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

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Citations of this work

Cohesive sets and rainbows.Wei Wang - 2014 - Annals of Pure and Applied Logic 165 (2):389-408.
Degrees bounding principles and universal instances in reverse mathematics.Ludovic Patey - 2015 - Annals of Pure and Applied Logic 166 (11):1165-1185.

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References found in this work

On the Strength of Ramsey's Theorem.David Seetapun & Theodore A. Slaman - 1995 - Notre Dame Journal of Formal Logic 36 (4):570-582.
Calibrating randomness.Rod Downey, Denis R. Hirschfeldt, André Nies & Sebastiaan A. Terwijn - 2006 - Bulletin of Symbolic Logic 12 (3):411-491.
Computability Theory.Barry Cooper - 2010 - Journal of the Indian Council of Philosophical Research 27 (1).

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