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On uniform relationships between combinatorial problems
| Citations: | 8 - 2 self |
BibTeX
@MISC{Dorais_onuniform,
author = {François G. Dorais and Damir D. Dzhafarov and Jeffry L. Hirst and Joseph R. Mileti and Paul Shafer},
title = {On uniform relationships between combinatorial problems },
year = {}
}
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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 nonuniform decisions about how to proceed in a given construction. In practice, however, if a theorem Q implies a theorem P, it is usually because there is a direct uniform translation of the problems represented by P into the problems represented by 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 ≥ 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.
Keyphrases
uniform relationship combinatorial problem reverse mathematics many color weihrauch reducibility relative strength many instance particular problem mathematical logic weak nig lemma weak formal theory mathematical theorem rainbow ramsey theorem computable mathematics thin set theorem several natural combinatorial problem logical strength new point traditional notion sequential form combinatorial problem active area mathematical proposition classical logic genuinely finer mathematical principle direct uniform translation ramsey theorem particular theorem uniform reducibility precise sense common framework
