A combinatorial analysis of functions provably recursive in $ΙΣ_n$

MLQ - Math. Log. Quart.

Piekart²

2005

Section: Introductionmentioning

confidence: 99%

Some variations of the Hardy hierarchy

MLQ - Math. Log. Quart.

Piekart²

2005

“…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.…”

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confidence: 99%

Partitioning 𝛼–large sets: Some lower bounds

Bigorajska

Trans. Amer. Math. Soc.

2006

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

“…We believe that our approach, using the Hardy hierarchy, is much simpler than that of [3]. On a more personal level our work was influenced by Z. Ratajczyk's work (see [7,4,5,8]). …”

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confidence: 99%

Some combinatorics involving ξ-large sets

Bigorajska

2002

Fund. Math.

Abstract. We prove a version of the Ramsey theorem for partitions of (increasing) n-tuples. We derive this result from a version of König's infinity lemma for ξ-large trees. Here ξ < ε 0 and the notion of largeness is in the sense of Hardy hierarchy.In this paper we prove some Ramsey style results for partitions of ntuples of finite sets. This paper is a continuation of [1], and, in fact, in order to avoid repetitions we assume that the reader has a copy of [1] at hand.The ideas are taken from Ketonen-Solovay [3] (they were interested only in the existence of ω-large homogeneous sets, see [3, Theorem 5.6]). We believe that our approach, using the Hardy hierarchy, is much simpler than that of [3]. On a more personal level our work was influenced by Z. Ratajczyk's work (see [7,4,5,8]).The main result of this paper is a version of the Ramsey theorem involving partitions of increasing n-tuples of elements of large (in the Hardy sense) sets of natural numbers, where n is an arbitrary fixed positive integer. We begin with the case n = 2 for clarity.In both cases (n = 2 and arbitrary n) we have no idea how to get lower bounds. Ketonen and Solovay work merely with ω-large homogeneous sets and their ideas do not seem to generalize. The only results in this direction we know about are: the lower bound from [1, p. 36], which concerns the case n = 1, and the result due to Erdős and Mills (see [2, Theorem 2, p. 171]).