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OPEN QUESTIONS IN REVERSE MATHEMATICS
, 2010
"... The objective of this paper is to provide a source of open questions in reverse mathematics and to point to areas where there could be interesting developments. The questions I discuss are mostly known and come from somewhere in the literature. My objective was to compile them in one place and discu ..."
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The objective of this paper is to provide a source of open questions in reverse mathematics and to point to areas where there could be interesting developments. The questions I discuss are mostly known and come from somewhere in the literature. My objective was to compile them in one place and discuss them in the context of related work. The list is definitely not comprehensive, and my
On uniform relationships between combinatorial problems
"... 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 ..."
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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 ntuples and k many colors is not uniformly, or Weihrauch, reducible to Ramsey’s theorem for ntuples 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.
Some logically weak Ramseyan theorems
 Institute of Logic and Cognition and Department of Philosophy, Sun Yatsen University, 135 Xingang Xi Road, Guangzhou 510275, P.R. China Email address: wwang.cn@gmail.com
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On notions of computability theoretic reduction between Π12 principles
, 2015
"... Several notions of computability theoretic reducibility between Π12 principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each n> 3, there is ..."
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Several notions of computability theoretic reducibility between Π12 principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each n> 3, there is an instance of RTn2 all of whose solutions have PA degree over ∅(n−2), and use this to show that König’s Lemma lies strictly between RT22 and RT 3 2 under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer [ta] on comparing versions of Ramsey’s Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey theoretic properties, we show that for each m> 2, there is an (m + 1)coloring c of N such
, Rainbow Ramsey Theorem for colorings of triples, and prove that RCA0 +RRT32 implies neither WKL0 nor
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A). The Thin Set Theorem. (Harvey
"... the set of all increasing ordered ntuples of elements of A (or nelement subsets of ..."
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the set of all increasing ordered ntuples of elements of A (or nelement subsets of