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26
On the strength of Ramsey’s Theorem for pairs
- Journal of Symbolic Logic
, 2001
"... Abstract. We study the proof–theoretic strength and effective content denote Ram-of the infinite form of Ramsey’s theorem for pairs. Let RT n k sey’s theorem for k–colorings of n–element sets, and let RT n < ∞ denote (∀k)RTn k. Our main result on computability is: For any n ≥ 2 and any computable (r ..."
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Cited by 26 (5 self)
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Abstract. We study the proof–theoretic strength and effective content denote Ram-of the infinite form of Ramsey’s theorem for pairs. Let RT n k sey’s theorem for k–colorings of n–element sets, and let RT n < ∞ denote (∀k)RTn k. Our main result on computability is: For any n ≥ 2 and any computable (recursive) k–coloring of the n–element sets of natural numbers, there is an infinite homogeneous set X with X ′ ′ ≤T 0 (n). Let I�n and B�n denote the �n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low2) to models is conservative of arithmetic enables us to show that RCA0 + I �2 + RT2 2 over RCA0 + I �2 for �1 1 statements and that RCA0 + I �3 + RT2 < ∞ is �1 1-conservative over RCA0 + I �3. It follows that RCA0 + RT2 2 does not imply B �3. In contrast, J. Hirst showed that RCA0 + RT2 < ∞ does imply B �3, and we include a proof of a slightly strengthened version of this result. It follows that RT2 < ∞ is strictly stronger than RT2 2 over RC A0. 1.
Ramsey's theorem and the pigeonhole principle in intuitionistic mathematics
- University of Utrecht, Dept of Philosophy
, 1992
"... At first sight, the argument which F. P. Ramsey gave for (the infinite case of) his famous theorem from 1927, is hopelessly unconstructive. If suitably reformulated, the theorem is true intuitionistically as well as classically: we offer a proof which should convince both the classical and the intui ..."
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Cited by 15 (2 self)
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At first sight, the argument which F. P. Ramsey gave for (the infinite case of) his famous theorem from 1927, is hopelessly unconstructive. If suitably reformulated, the theorem is true intuitionistically as well as classically: we offer a proof which should convince both the classical and the intuitionistic reader. 1.
Combinatorial Principles Weaker than Ramsey's Theorem for Pairs
"... We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order ha ..."
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Cited by 14 (5 self)
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We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (ChainAntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is well-known that Ramsey's Theorem for pairs (RT
On Presentations of Algebraic Structures
- in Complexity, Logic and Recursion Theory
, 1995
"... This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC-509. This paper is dedicat ..."
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Cited by 13 (6 self)
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This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC-509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995
On the Strength of Ramsey's Theorem
- Notre Dame J. Formal Logic
, 1995
"... this paper we study the logical strength of Ramsey's Theorem (1930), especially of Ramsey's Theorem for partitions of pairs into two pieces. ..."
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Cited by 11 (0 self)
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this paper we study the logical strength of Ramsey's Theorem (1930), especially of Ramsey's Theorem for partitions of pairs into two pieces.
Computability-Theoretic and Proof-Theoretic Aspects of Partial and Linear Orderings
- Israel Journal of mathematics
"... Szpilrajn's Theorem states that any partial order P = hS;
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Cited by 7 (0 self)
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Szpilrajn's Theorem states that any partial order P = hS; <P i has a linear extension L = hS; <L i. This is a central result in the theory of partial orderings, allowing one to de ne, for instance, the dimension of a partial ordering. It is now natural to ask questions like \Does a well-partial ordering always have a well-ordered linear extension?" Variations of Szpilrajn's Theorem state, for various (but not for all) linear order types , that if P does not contain a subchain of order type , then we can choose L so that L also does not contain a subchain of order type . In particular, a well-partial ordering always has a well-ordered extension.
Questions in Computable Algebra and Combinatorics
, 1999
"... this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these ..."
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Cited by 5 (0 self)
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this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25
Generalized Cohesiveness
- The Journal of Symbolic Logic
, 1997
"... . We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey 's Theorem. An infinite set A of natural numbers is n--cohesive (respectively, n--r--cohesive) if A is almost homogeneous for every computably enumerable (respectively, computab ..."
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Cited by 5 (3 self)
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. We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey 's Theorem. An infinite set A of natural numbers is n--cohesive (respectively, n--r--cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2--coloring of the n--element sets of natural numbers. (Thus the 1--cohesive and 1--r--cohesive sets coincide with the cohesive and r--cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n--cohesive and n--r--cohesive sets. For example, we show that for all n 2, there exists a \Delta 0 n+1 n--cohesive set. We improve this result for n = 2 by showing that there is a \Pi 0 2 2--cohesive set. We show that the n--cohesive and n--r--cohesive degrees together form a linear, non--collapsing hierarchy of degrees for n 2. In addition, for n 2 we characterize the jumps of n--cohesive degrees as exactly the degrees 0 (n+1) and s...
The canonical Ramsey theorem and computability theory
"... Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable in ..."
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Cited by 4 (1 self)
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Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable instances of these problems in terms of the Turing degrees and the arithmetical hierarchy. We succeed in giving a sharp characterization for the Canonical Ramsey Theorem for exponent 2 and for the Regressive Function Theorem for all exponents. These results rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study also unearths some interesting relationships between these two partition theorems, Ramsey’s Theorem, and Konig’s Lemma. 1
