We present general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado’s Theorem by covering large classes of nonlin-ear equations. The goal is to contribute to an overall theory of Ramsey properties of (nonlinear) Diophantine equations that encompasses the known results in this area under a unified framework. Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters βN, grounding on combi-natorial properties of positive density sets and IP sets. Neces-sary conditions are proved by a new technique in nonstandard analysis, based on the use of the relation of u-equivalence for the hypernatural numbers ∗N.

Ramsey properties of nonlinear Diophantine equations / M. Di Nasso, L. Luperi Baglini. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 324(2018), pp. 84-117. [10.1016/j.aim.2017.11.003]

Ramsey properties of nonlinear Diophantine equations

L. Luperi Baglini
2018

Abstract

We present general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado’s Theorem by covering large classes of nonlin-ear equations. The goal is to contribute to an overall theory of Ramsey properties of (nonlinear) Diophantine equations that encompasses the known results in this area under a unified framework. Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters βN, grounding on combi-natorial properties of positive density sets and IP sets. Neces-sary conditions are proved by a new technique in nonstandard analysis, based on the use of the relation of u-equivalence for the hypernatural numbers ∗N.
Ramsey Theory; Diophantine equations; Ultrafilters; Nonstandard analysis
Settore MAT/01 - Logica Matematica
2018
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/651033
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