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Iterated hyper-extensions and an idempotent ultrafilter proof of Rado's Theorem


Author: Mauro Di Nasso
Journal: Proc. Amer. Math. Soc. 143 (2015), 1749-1761
MSC (2010): Primary 03H05; Secondary 03E05, 05D10, 11D04
Published electronically: December 8, 2014
MathSciNet review: 3314087
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Abstract: By using nonstandard analysis, and in particular iterated hyper-extensions, we give foundations to a peculiar way of manipulating ultrafilters on the natural numbers and their pseudo-sums. The resulting formalism is suitable for applications in Ramsey theory of numbers. To illustrate the use of our technique, we give a (rather) short proof of Milliken-Taylor's Theorem and a ultrafilter version of Rado's Theorem about partition regularity of diophantine equations.


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Additional Information

Mauro Di Nasso
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: dinasso@dm.unipi.it

DOI: https://doi.org/10.1090/S0002-9939-2014-12342-2
Keywords: Nonstandard analysis, ultrafilters, Ramsey theory, diophantine equations
Received by editor(s): April 12, 2013
Received by editor(s) in revised form: July 30, 2013
Published electronically: December 8, 2014
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.