A recursion theoretic analysis of the clopen Ramsey theorem

1984 ◽  
Vol 49 (2) ◽  
pp. 376-400 ◽  
Author(s):  
Peter Clote
Keyword(s):  
Order Type ◽  
Theoretic Analysis ◽  
Ramsey Theorem ◽  
Homogeneous Set

AbstractSolovay has shown that if F: [ω]ω → 2 is a clopen partition with recursive code, then there is an infinite homogeneous hyperarithmetic set for the partition (a basis result). Simpson has shown that for every 0α, where α is a recursive ordinal, there is a clopen partition F: [ω]ω → 2 such that every infinite homogeneous set is Turing above 0α (an anti-basis result). Here we refine these results, by associating the “order type” of a clopen set with the Turing complexity of the infinite homogeneous sets. We also consider the Nash-Williams barrier theorem and its relation to the clopen Ramsey theorem.

On partitioning the infinite subsets of large cardinals

1984 ◽  
Vol 49 (2) ◽  
pp. 539-541 ◽  
Author(s):  
R. J. Watro
Keyword(s):  
Order Type ◽  
Large Cardinal ◽  
Infinite Cardinal ◽  
Product Topology ◽  
Weakly Compact ◽  
Ramsey Theorem ◽  
Borel Sets ◽  
Analytic Sets ◽  
The Axiom Of Choice ◽  
Usual Product

Let λ be an ordinal less than or equal to an infinite cardinal κ. For S ⊂ κ, [S]λ denotes the collection of all order type λ subsets of S. A set X ⊂ [κ]λ will be called Ramsey iff there exists p ∈ [κ]κ such that either [p]λ ⊂ X or [p]λ ∩ X = ∅. The set p is called homogeneous for X.The infinite Ramsey theorem implies that all subsets of [ω]n are Ramsey for n < ω. Using the axiom of choice, one can define a non-Ramsey subset of [ω]ω. In [GP], Galvin and Prikry showed that all Borel subsets of [ω]ω are Ramsey, where one topologizes [ω]ω as a subspace of Baire space. Silver [S] proved that analytic sets are Ramsey, and observed that this is best possible in ZFC.When κ > ω, the assertion that all subsets of [κ]n are Ramsey is a large cardinal hypothesis equivalent to κ being weakly compact (and strongly inaccessible). Again, is not possible in ZFC to have all subsets of [κ]ω Ramsey. The analogy to the Galvin-Prikry theorem mentioned above was established by Kleinberg, extending work by Kleinberg and Shore in [KS]. The set [κ]ω is given a topology as a subspace of κω, which has the usual product topology, κ taken as discrete. It was shown that all open subsets of [κ]ω are Ramsey iff κ is a Ramsey cardinal (that is, κ → (κ)<ω).In this note we examine the spaces [κ]λ for κ ≥ λ ≥ ω. We show that κ Ramsey implies all open subsets of [κ]λ are Ramsey for λ < κ, and that if κ is measurable, then all open subsets of [κ]κ are Ramsey. Let us remark here that we can with the same methods prove these results with “κ-Borel” in the place of “open”, where the κ-Borel sets are the smallest collection containing the opens and closed under complementation and intersections of length less than κ. Also, although here we consider just subsets of [κ]λ, it is no more difficult to show that partitions of [κ]λ into less than κ many κ-Borel sets have, under the appropriate hypothesis, size κ homogeneous sets.

scholarly journals THE TOPOLOGICAL PIGEONHOLE PRINCIPLE FOR ORDINALS

2016 ◽  
Vol 81 (2) ◽  
pp. 662-686 ◽  
Author(s):  
JACOB HILTON
Keyword(s):  
Order Type ◽  
Pigeonhole Principle ◽  
Correct Order ◽  
Partition Relation ◽  
Independence Result ◽  
Homogeneous Set ◽  
Image Position

AbstractGiven a cardinal κ and a sequence ${\left( {{\alpha _i}} \right)_{i \in \kappa }}$ of ordinals, we determine the least ordinal β (when one exists) such that the topological partition relation$$\beta \to \left( {top\,{\alpha _i}} \right)_{i \in \kappa }^1$$holds, including an independence result for one class of cases. Here the prefix “top” means that the homogeneous set must be of the correct homeomorphism class rather than the correct order type. The answer is linked to the nontopological pigeonhole principle of Milner and Rado.

ANALYSIS OF THE INFLUENCE OF DIMENSIONS OF THE ROPE MIDWATER TRAWLS ON THEIR PERFORMANCE SUBJECT TO GEOMETRIC PROPERTIES OF CONE SHELLS

2017 ◽  
pp. 25-33
Author(s):  
Alexander Vasilievich Dvernik
Keyword(s):  
Surface Properties ◽  
Conical Shell ◽  
Side Surface ◽  
Theoretic Analysis ◽  
Good Convergence ◽  
Large Size ◽  
Quality Coefficient ◽  
Geometric Quality ◽  
The Relationship ◽  
Parameters Calculation

The article studies different shell constructions of mid-water trawls and their properties. The problem settled is suggested to be solved taking into account real geometric interrelations between spacious and surface properties of cone shells. The author suggests to accept a so-called geometric quality coefficient as a criterion of the properties of a conical shell, which represents the ratio of the shell to the area of its side surface and by analogy to use it to the shell of the trawl. The relationship between the trawl dimensions and geometric quality coefficient have been studied. Comparing these figures with the actual characteristics of trawls showed good convergence. According to the results of theoretic analysis and parameters calculation, trawl large-size shells will always have advantages in geometric characteristics over mid-size and, especially, small-size shells. The results of the analysis can be used for approximate calculations of the parameters of the trawl and justification of ways to improve the performance of existing mid-water trawls.

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