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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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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.
PARTITIONING α–LARGE SETS:
"... Abstract. Let α → (β) r m denote the property: if A is an α–large set of natural numbers and [A] r is partitioned into m parts, then there exists a β– large subset of A which is homogeneous for this partition. Here the notion of largeness is in the sense of the so–called Hardy hierarchy. We give a l ..."
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Abstract. Let α → (β) r m denote the property: if A is an α–large set of natural numbers and [A] r is partitioned into m parts, then there exists a β– large subset of A which is homogeneous for this partition. Here the notion of largeness is in the sense of the so–called Hardy hierarchy. We give a lower bound for α in terms of β, m,r for some specific β. This paper is a continuation of our work [2] and [3] on partitions of finite sets, where the notion of largeness is in the sense of Hardy hierarchy. We give some lower bounds for partitions. All the definitions involving ordinals below ε0, fundamental sequences, the notion of α–largeness, etc. are defined in [2]. In order to avoid repetition we assume the reader to have a copy of [2] in hand. We define only the notions needed, which do not occur in [2]. We stress that the ideas below go back to J. Ketonen and R. Solovay [10]. Because of the nature of their problem, that is, describing the order of growth of the function shown by J. Paris and L. Harrington [17] to grow faster than any recursive function provably total in Peano arithmetic, they were interested merely in the existence of
Ramsey Theory Applications
"... There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science. Relations of Ramseytype theorems to various fields in mathematics are well documente ..."
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There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science. Relations of Ramseytype theorems to various fields in mathematics are well documented in published books and monographs. The main objective of this survey is to list applications mostly in theoretical computer science of the last two decades not contained in these. 1
www.math.ohiostate.edu/~friedman/ TABLE OF CONTENTS
"... this paper as reaxiomatizations of set theory. A vital feature of the standard set theories associated with the axiomatizations presented here is that they are missing the axiom of choice. This is an essential feature. For instance, ZF does not prove the existence of a standard model of each theorem ..."
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this paper as reaxiomatizations of set theory. A vital feature of the standard set theories associated with the axiomatizations presented here is that they are missing the axiom of choice. This is an essential feature. For instance, ZF does not prove the existence of a standard model of each theorem of ZFC; in fact, ZF does not prove the existence of a standard model of Zermelo set theory with the axiom of choice. Thus in this paper, we relate our axiomatizations to extensions of ZF by large cardinal axioms. In each case, we have chosen an appropriate version of the large cardinal axiom so that if ZF is replaced by ZFC then the resulting system is equivalent to a system which is familiar in the set theory literature. But one would like to know the relationship between the system with ZF and the system with ZFC. This relationship cannot be gauged by considering standard models. {PAGE } The normal way of gauging this relationship is through