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ON THE UNIFORM COMPUTATIONAL CONTENT OF RAMSEY’S THEOREM

Published online by Cambridge University Press:  09 January 2018

VASCO BRATTKA  and
TAHINA RAKOTONIAINA
VASCO BRATTKA
Affiliation:
FACULTY OF COMPUTER SCIENCE UNIVERSITÄT DER BUNDESWEHR MÜNCHEN NEUBIBERG, GERMANY and DEPARTMENT OF MATHEMATICS & APPLIED MATHEMATICS UNIVERSITY OF CAPE TOWN RONDEBOSCH, SOUTH AFRICAE-mail: Vasco.Brattka@cca-net.de
TAHINA RAKOTONIAINA
Affiliation:
DEPARTMENT OF MATHEMATICS & APPLIED MATHEMATICS UNIVERSITY OF CAPE TOWN RONDEBOSCH, SOUTH AFRICAE-mail: fenoira@gmail.com
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Abstract

We study the uniform computational content of Ramsey’s theorem in the Weihrauch lattice. Our central results provide information on how Ramsey’s theorem behaves under product, parallelization, and jumps. From these results we can derive a number of important properties of Ramsey’s theorem. For one, the parallelization of Ramsey’s theorem for cardinality n ≥ 1 and an arbitrary finite number of colors k ≥ 2 is equivalent to the n-th jump of weak Kőnig’s lemma. In particular, Ramsey’s theorem for cardinality n ≥ 1 is ${\bf{\Sigma }}_{n + 2}^0$-measurable in the effective Borel hierarchy, but not ${\bf{\Sigma }}_{n + 1}^0$-measurable. Secondly, we obtain interesting lower bounds, for instance the n-th jump of weak Kőnig’s lemma is Weihrauch reducible to (the stable version of) Ramsey’s theorem of cardinality n + 2 for n ≥ 2. We prove that with strictly increasing numbers of colors Ramsey’s theorem forms a strictly increasing chain in the Weihrauch lattice. Our study of jumps also shows that certain uniform variants of Ramsey’s theorem that are indistinguishable from a nonuniform perspective play an important role. For instance, the colored version of Ramsey’s theorem explicitly includes the color of the homogeneous set as output information, and the jump of this problem (but not the uncolored variant) is equivalent to the stable version of Ramsey’s theorem of the next greater cardinality. Finally, we briefly discuss the particular case of Ramsey’s theorem for pairs, and we provide some new separation techniques for problems that involve jumps in this context. In particular, we study uniform results regarding the relation of boundedness and induction problems to Ramsey’s theorem, and we show that there are some significant differences with the nonuniform situation in reverse mathematics.

Keywords

03D3003D7803F6005D10computable analysisWeihrauch latticeRamsey’s theorem
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1278 - 1316
Copyright
Copyright © The Association for Symbolic Logic 2017 

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