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THE STRENGTH OF RAMSEY’S THEOREM FOR COLORING RELATIVELYLARGE SETS

Published online by Cambridge University Press:  17 April 2014

LORENZO CARLUCCI  and
KONRAD ZDANOWSKI
LORENZO CARLUCCI
Affiliation:
DIPARTIMENTO DI INFORMATICA, UNIVERSITY OF ROME I “LA SAPIENZA” ROME, ITALY E-mail: carlucci@di.uniroma1.it
KONRAD ZDANOWSKI
Affiliation:
UNIVERSITY OF CARDINAL STEFAN WYSZYŃSKI, WARSAW, POLAND E-mail: k.zdanowski@uksw.edu.pl
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Abstract

We characterize the effective content and the proof-theoretic strength of aRamsey-type theorem for bi-colorings of so-called exactly largesets. An exactly large set is a set $X \subset {\bf{N}}$ such that ${\rm{card}}\left( X \right) = {\rm{min}}\left( X \right) + 1$. The theorem we analyze is as follows. For every infinitesubset M of N, for every coloring C of theexactly large subsets of M in two colors, there exists andinfinite subset L of M such thatC is constant on all exactly large subsets ofL. This theorem is essentially due to Pudlák andRödl and independently to Farmaki. We prove that—overRCA0 —this theorem is equivalent to closure under theωth Turing jump (i.e., under arithmetical truth).Natural combinatorial theorems at this level of complexity are rare. In terms ofReverse Mathematics we give the first Ramsey-theoretic characterization of ${\rm{ACA}}_0^ +$. Our results give a complete characterization of the theoremfrom the point of view of Computability Theory and of the Proof Theory ofArithmetic. This nicely extends the current knowledge about the strength ofRamsey’s Theorem. We also show that analogous results hold for arelated principle based on the Regressive Ramsey’s Theorem. Weconjecture that analogous results hold for larger ordinals.

Keywords

Ramsey’s Theoremrelatively large setstrue arithmetic

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 89 - 102
Copyright
Copyright © Association for Symbolic Logic 2014 

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