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The strength of the rainbow Ramsey Theorem

Published online by Cambridge University Press:  12 March 2014

Barbara F. Csima  and
Joseph R. Mileti
Barbara F. Csima
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, On, N2L 3G1, Canada, URL: www.math.uwaterloo.ca/~csima, E-mail: csima@math.uwaterloo.ca
Joseph R. Mileti
Affiliation:
Department of Mathematics and Statistics, Grinnell College, Grinnel, Ia 50112-1690, USA, E-mail: miletijo@grinnell.edu
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009 , pp. 1310 - 1324
Copyright
Copyright © Association for Symbolic Logic 2009

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