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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Studia Logicaarrow_drop_down
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Article . 1988 . Peer-reviewed
License: Springer TDM
Data sources: Crossref
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
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Data sources: zbMATH Open
DBLP
Article . 1988
Data sources: DBLP
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Nonstandard combinatorics

Authors: Joram Hirshfeld;

Nonstandard combinatorics

Abstract

This interesting paper is mostly concerned with Ramsey type theorems, where a theorem is of this type if it takes the form ``if certain sets are partitioned, then at least one of these parts has some particular property.'' Since a standard partition is usually considered to be a finite collection, then nonstandard methods should be of considerable interest relative to intuitive comprehension. The author shows this to be the case with his nonstandard proofs of the (infinite) Ramsey Theorem and Hindman's Theorem. His method of establishing the Ramsey Theorem by using a nonstandard natural number to direct our choice of a standard set of natural number is a useful contribution. Further, for abstract Ramsey properties, the author gives a significant nonstandard characterization for a divisible property. Nonstandard analysis has allowed the author to more closely analyze the Finite Ramsey Theorem and he is thereby able to give an interesting generalization. The paper has many examples, a few typographical errors, that should prove to be of no difficulty to the reader, and concludes with the following Compactness type theorem. Let S be a collection of finite subsets of X and let r be a fixed number. Call a subset of X bad if it contains no set of S. Then: If every finite subset of X has a partition into r bad sets, then X has a partition into r bad sets.

Related Organizations
Keywords

Ramsey type theorems, Combinatorial aspects of partitions of integers, Nonstandard models in mathematics, Hindman's Theorem

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
6
Average
Top 10%
Average
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