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12
A classification of rapidly growing Ramsey functions
 PROC. AMER. MATH. SOC
, 2003
"... Let f be a numbertheoretic function. A finite set X of natural numbers is called flarge if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding flargeness condition. We classify those functions f for which the statement PHf i ..."
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Cited by 16 (5 self)
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Let f be a numbertheoretic function. A finite set X of natural numbers is called flarge if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding flargeness condition. We classify those functions f for which the statement PHf is independent of first order (Peano) arithmetic PA.Iffis a fixed iteration of the binary length function, then PHf is independent. On the other hand PHlog ∗ is provable in PA. More precisely let fα(i):=i  H −1 α (i) where  i h denotes the htimes iterated binary length of i and H−1 α denotes the inverse function of the αth member Hα of the Hardy hierarchy. Then PHfα is independent of PA (for α ≤ ε0) iffα = ε0.
Partition Theorems and Computability Theory
 Bull. Symbolic Logic
, 2004
"... The computabilitytheoretic and reverse mathematical aspects of various combinatorial principles, such as König’s Lemma and Ramsey’s Theorem, have received a great deal of attention and are active areas of research. We carry on this study of effective combinatorics by analyzing various partition the ..."
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Cited by 8 (1 self)
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The computabilitytheoretic and reverse mathematical aspects of various combinatorial principles, such as König’s Lemma and Ramsey’s Theorem, have received a great deal of attention and are active areas of research. We carry on this study of effective combinatorics by analyzing various partition theorems (such as Ramsey’s Theorem) with the aim of understanding the complexity of solutions to computable instances in terms of the Turing degrees and the arithmetical hierarchy. Our main focus is the study of the effective content of two partition theorems allowing infinitely many colors: the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our results on the complexity of solutions rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study unearths some interesting relationships between these two partition theorems, Ramsey’s Theorem, and Konig’s Lemma, and these connections will be emphasized. We also study Ramsey degrees, i.e. those Turing degrees which are able to compute homogeneous sets for every computable 2coloring of pairs of natural numbers, in an attempt to further understand the effective content of Ramsey’s Theorem for exponent 2. We establish some new results about these degrees, and obtain as a corollary the nonexistence of a “universal ” computable 2coloring of pairs of natural numbers.
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 8 (2 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
Unprovability of sharp versions of Friedman’s sineprinciple
 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 135
, 2007
"... For every n ≥ 1 and every function F of one argument, we introduce the statement SPn F:“forall m, thereisNsuch that for any set A = {a1,a2,...,aN} of rational numbers, there is H ⊆ A of size m such that for any two nelement subsets ai1
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Cited by 5 (2 self)
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For every n ≥ 1 and every function F of one argument, we introduce the statement SPn F:“forall m, thereisNsuch that for any set A = {a1,a2,...,aN} of rational numbers, there is H ⊆ A of size m such that for any two nelement subsets ai1 <ai2 < ·· · <ain and ai1 <ak2 < ·· · <akn in H, wehave sin(ai1 · ai2 ···ain) − sin(ai1 · ak2 ···akn)  <F(i1)”. We prove that for n ≥ 2 and any function F(x) eventually dominated by ( 2 3)log(n−1) (x) n+1, the principle SPF is not provable in IΣn. In particular, the statement ∀nSPn ( 2 3)log(n−1) is not provable in Peano Arithmetic. In dimension 2, the result is: IΣ1 does not prove SP2 F,whereF(x)=(2 3) A−1 (x) √ x and A−1 is the inverse of the Ackermann function.
Regressive Ramsey numbers are Ackermannian
 J. Combin. Theory Ser. A
, 1999
"... Abstract. We give an elementary proof of the fact that regressive Ramsey numbers are Ackermannian. This fact was first proved by Kanamori and McAloon with mathematical logic techniques. Nous vivons encore sous le règne de la logique, voilà, bien entendu, à quoi je voulais en venir. Mais les procédés ..."
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Cited by 5 (2 self)
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Abstract. We give an elementary proof of the fact that regressive Ramsey numbers are Ackermannian. This fact was first proved by Kanamori and McAloon with mathematical logic techniques. Nous vivons encore sous le règne de la logique, voilà, bien entendu, à quoi je voulais en venir. Mais les procédés logiques, de nos jours, ne s’appliquent plus qu’à la résolution de problèmes d’intérêt secondaire. [1, 1924, p. 13] 649 revision:19980508 modified:20020227 1.
Brief introduction to unprovability
"... Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. ..."
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Cited by 1 (0 self)
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Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the ParisHarrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its modeltheoretic techniques and, finally, a modeltheoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logicaware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.
Regressive Ramsey Numbers Are Ackermannian
 J. Combin. Theory Ser. A
, 1999
"... We give an elementary proof of the fact that regressive Ramsey numbers are Ackermannian. This fact was first proved by Kanamori and McAloon with mathematical logic techniques. Nous vivons encore sous le r`egne de la logique, voil`a, bien entendu, `a quoi je voulais en venir. Mais les proc'ed'es logi ..."
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We give an elementary proof of the fact that regressive Ramsey numbers are Ackermannian. This fact was first proved by Kanamori and McAloon with mathematical logic techniques. Nous vivons encore sous le r`egne de la logique, voil`a, bien entendu, `a quoi je voulais en venir. Mais les proc'ed'es logiques, de nos jours, ne s'appliquent plus qu'`a la r'esolution de probl`emes d'int'eret secondaire. [1, 1924, p. 13] 1.