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On uniform relationships between combinatorial problems

Authors: François G. Dorais, Damir D. Dzhafarov, Jeffry L. Hirst, Joseph R. Mileti and Paul Shafer
Journal: Trans. Amer. Math. Soc. 368 (2016), 1321-1359
MSC (2010): Primary 03B30, 05D10; Secondary 05D40, 05D15, 03D32, 03D80, 03F35
Published electronically: April 9, 2015
MathSciNet review: 3430365
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Abstract: The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverse mathematics. In this setting, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take place in classical logic, they may in principle involve appeals to multiple applications of a particular theorem, or to non-uniform decisions about how to proceed in a given construction. In practice, however, if a theorem $ \mathsf {Q}$ implies a theorem $ \mathsf {P}$, it is usually because there is a direct uniform translation of the problems represented by $ \mathsf {P}$ into the problems represented by $ \mathsf {Q}$, in a precise sense formalized by Weihrauch reducibility.

We study this notion of uniform reducibility in the context of several natural combinatorial problems, and compare and contrast it with the traditional notion of implication in reverse mathematics. We show, for instance, that for all $ n,j,k \geq 1$, if $ j < k$, then Ramsey's theorem for $ n$-tuples and $ k$ many colors is not uniformly, or Weihrauch, reducible to Ramsey's theorem for $ n$-tuples and $ j$ many colors. The two theorems are classically equivalent, so our analysis gives a genuinely finer metric by which to gauge the relative strength of mathematical propositions.

We also study Weak König's Lemma, the Thin Set Theorem, and the Rainbow Ramsey's Theorem, along with a number of their variants investigated in the literature. Weihrauch reducibility turns out to be connected with sequential forms of mathematical principles, where one wishes to solve infinitely many instances of a particular problem simultaneously. We exploit this connection to uncover new points of difference between combinatorial problems previously thought to be more closely related.

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Additional Information

François G. Dorais
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551

Damir D. Dzhafarov
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Jeffry L. Hirst
Affiliation: Department of Mathematical Sciences, Appalachian State University, Boone, North Carolina 28608-2092

Joseph R. Mileti
Affiliation: Department of Mathematics and Statistics, Grinnell College, Grinnell, Iowa 50112

Paul Shafer
Affiliation: Department of Mathematics, Ghent University, Ghent, Belgium

Received by editor(s): October 30, 2012
Received by editor(s) in revised form: December 2, 2013, and March 16, 2014
Published electronically: April 9, 2015
Additional Notes: The second author was partially supported by an NSF Postdoctoral Fellowship
The third author was partially supported by grant ID#20800 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation
The fifth author was supported by the Fondation Sciences Mathématiques de Paris and is also an FWO Pegasus Long Postdoctoral researcher
Article copyright: © Copyright 2015 American Mathematical Society