Fermat-Like Equations that are not Partition Regular

Nasso, Mauro Di; Riggio, Maria

doi:10.1007/s00493-016-3640-2

Cited by 11 publications

(10 citation statements)

References 14 publications

(13 reference statements)

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“…To our knowledge, the last progress done in this area about is found in [12], where M. Riggio and the first named author used nonstandard analysis to identify a large class of Fermat-like equations that are not partition regular, the simplest examples being x m + y n = z k where k / ∈ {n, m}. 3 At the moment this paper was completed, it was breaking news that M. J. H. Heule, O. Kullmann and V. W. Marek [18] solved a problem posed by P. Erdős and R. Graham in the 1970s, namely the Boolean Pythagorean triples problem, that asked whether the equation x 2 + y 2 = z 2 is partition regular for 2-colorings of N. By using a computer-assisted proof, they have been able to prove that any 2-coloring of {1, 2, .…”

Section: Multiplicative Rado's Theorema Nonlinear Diophantine Equatimentioning

confidence: 99%

“…In the last years, the interest on problems related to the partition regularity of nonlinear Diophantine equations has been rising constantly (see, e.g., [28,23,6,37,33,34,3,12,13,18]). We hope that this paper will contribute to a general Ramsey theory of nonlinear Diophantine equations.…”

Section: Final Remarks and Open Questionsmentioning

confidence: 99%

“…Example 3.18. In [12], it is proved that the following polynomials x n +y m = z k are not PR for k / ∈ {n, m}. This result is obtained as a particular case of Corollary 3.12.…”

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confidence: 92%

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Ramsey properties of nonlinear Diophantine equations

Nasso

Baglini

2018

Advances in Mathematics

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We prove general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado's Theorem by covering large classes of nonlinear equations. Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters βN, grounding on combinatorial properties of positive density sets and IP sets. Necessary 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.

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Section: Multiplicative Rado's Theorema Nonlinear Diophantine Equatimentioning

confidence: 99%

Section: Final Remarks and Open Questionsmentioning

confidence: 99%

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Ramsey properties of nonlinear Diophantine equations

Nasso

Baglini

2018

Advances in Mathematics

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“…From the previous theorem, we see that many "Fermat-like" equations are not partition regular: In [41], the previous corollary is extended to allow m and n to be equal, in which case the equations are shown to be not partition regular (as long as, in the case when m = n = k − 1, one excludes the trivial solution x = y = z = 2). The methods are similar to the previous proof.…”

Section: Non-partition Regularity Of Some Equationsmentioning

confidence: 99%

“…MANY STARS: ITERATED NONSTANDARD EXTENSIONSNotes and referencesIterated hyperextensions were introduced in[36], where they are used to give a new approach to the proof of Rado's theorem in the theory of partition regularity of equations (see Chapter 10 below). Further applications to the study of partition regularity of equations are obtained in[82,81,83,41,37]. A survey on the main properties of iterated hyperextensions, also in relation with hyperfinite generators of ultrafilters, is presented in[35].…”

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Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

Nasso

Goldbring

Lupini

2019

Lecture Notes in Mathematics

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AcknowledgementsThe collaboration between the authors first began when they participated in an American Institute of Mathematics (AIM) Structured Quartet Research Ensemble (or SQuaRE) program together with Renling Jin, Steven Leth, and Karl Mahlburg. We thus want to thank AIM for all of their support during our three year participation in the SQuaRE program as well as their encouragement to organize a larger workshop on the subject. A preliminary version of this manuscript was distributed during that workshop and we want to thank the participants for their valuable comments. In particular, Steven Leth and Terence Tao gave us a tremendous amount of feedback and for that we want to give them an extra expression of gratitude. i ii INTRODUCTION iii and measurable sets in a dynamical system), with the extra feature that the dynamical system obtained perfectly reflects all the combinatorial properties of the set that one started with. The achievements of the nonstandard approach in this area include the work of Leth on arithmetic progressions in sparse sets, Jin's theorem on sumsets, as well as Jin's Freiman-type results on inverse problems for sumsets. More recently, these methods have also been used by Jin, Leth, Mahlburg, and the present authors to tackle a conjecture of Erdős concerning sums of infinite sets (the so-called B +C conjecture), leading to its eventual solution by Moreira, Richter, and Robertson.Nonstandard methods are also tightly connected with ultrafilter methods. This has been made precise and successfully applied in recent work of Di Nasso, where he observed that there is a perfect correspondence between ultrafilters and elements of the nonstandard universe up to a natural notion of equivalence. On the one hand, this allows one to manipulate ultrafilters as nonstandard points, and to use ultrafilter methods to prove the existence of certain combinatorial configurations in the nonstandard universe. One the other hand, this gives an intuitive and direct way to infer, from the existence of certain ultrafilter configurations, the existence of corresponding standard combinatorial configurations via the fundamental principle of transfer in the nonstandard method. This perspective has successfully been applied by Di Nasso, Luperi Baglini, and co-authors to the study of partition regularity problems for Diophantine equations over the integers, providing in particular a far-reaching generalization of the classical theorem of Rado on partition regularity of systems of linear equations. Unlike Rado's theorem, this recent generalization also includes equations that are not linear.Finally, it is worth mentioning that many other results in combinatorics can be seen, directly or indirectly, as applications of the nonstandard method. For instance, the groundbreaking work of Hrushovski and Breuillard-Green-Tao on approximate groups, although not originally presented in this way, admit a natural nonstandard treatment. The same applies to the work of Bergelson and Tao on recurrence in quasirandom groups.The goal of t...

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Nonstandard characterisations of tensor products and monads in the theory of ultrafilters

Baglini

2019

Mathematical Logic Qtrly

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We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as by characterising their combinatorial properties by means of their monads. This extends to arbitrary sets and properties methods previously used to study partition regular Diophantine equations on . Several applications are described by means of multiple examples.

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Fermat-Like Equations that are not Partition Regular

Abstract: Abstract. By means of elementary conditions on coefficients, we isolate a large class of Fermat-like Diophantine equations that are not partition regular, the simplest examples being x n +y m = z k with k / ∈ {n, m}.

Cited by 11 publications

References 14 publications

Ramsey properties of nonlinear Diophantine equations

Ramsey properties of nonlinear Diophantine equations

Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

Nonstandard characterisations of tensor products and monads in the theory of ultrafilters

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