
doi: 10.1007/bf00370553
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.
Ramsey type theorems, Combinatorial aspects of partitions of integers, Nonstandard models in mathematics, Hindman's Theorem
Ramsey type theorems, Combinatorial aspects of partitions of integers, Nonstandard models in mathematics, Hindman's Theorem
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