Some variations of the Hardy hierarchy

Kotlarski, Henryk; Piekart, Bożena

doi:10.1002/malq.200410043

Cited by 1 publication

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“…When this paper was being refereed, we wrote another paper [11], where we define several variations of the notion of largeness in the sense of Hardy hierarchy and prove Ratajczyk's main lemma for each of them. The problem of working out some variants of Hardy hierarchy is also considered in [4].…”

Section: Proof By Induction Onmentioning

confidence: 99%

“…The problem of working out some variants of Hardy hierarchy is also considered in [4]. In our opinion the notion of largeness suggested by the estimation lemma (called estimative largeness in [11]), in particular the work with pseudonorm rather than with fundamental sequences will be most convenient in this sort of combinatorics. When Andreas Weiermann read this paper, he used ideas in it to prove Ackermannian lower bounds for diagonally iterated dense(2,2) sets.…”

Section: Proof By Induction Onmentioning

confidence: 99%

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Partitioning 𝛼–large sets: Some lower bounds

Bigorajska

Kotlarski

2006

Trans. Amer. Math. Soc.

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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 ε 0 , 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 ω-large homogeneous sets. We generalize this to higher β. We use some serious technical simplifications of the ideas of [10] from section 6.3 in [9]. On a more personal level we are highly influenced by the work of Z. Ratajczyk; see [18], [13], [14] and his final [19]. In particular, the idea of the notion of arbitrary set of natural numbers (not only interval) being α-large is due to Ratajczyk. It should be noted that the idea of Hardy hierarchy was developed by several schools; see, e.g., [7], [8] and [6]. The idea of working with α-large sets may be used to eliminate the so-called cut elimination from the proofs of consistencies of theories like Peano arithmetic; see Ratajczyk's papers mentioned above and [20,21] and [1]. See also [10,2] and [3] for some upper bounds for this sort of Ramsey theorem.The new (when compared with [10] and other papers in this area of combinatorics) idea is to use expansions of ordinals to bases other than ω. The estimation lemma is the basic tool which allows us to estimate how large a homogeneous set is, despite the fact that partitions were constructed with the use of expansions to bases of the form ω m .We have organized the paper as follows. In section 1 we give the basic construction of partitions of finite sets of natural numbers to be generalized in later sections. The results are well known; we included this section for motivational purposes. In section 2 we give a proof of some version of results due to J. Ketonen and

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Section: Proof By Induction Onmentioning

confidence: 99%

Section: Proof By Induction Onmentioning

confidence: 99%