## 23 Citations

Partition Regularity of Equations

- MathematicsNonstandard Methods in Ramsey Theory and Combinatorial Number Theory
- 2019

Diophantine equations have been studied throughout the history of mathematics. Initially considered by Diophantus of Alexandria, a Hellenistic mathematician from the third century CE, they include…

Ultrafilters, monads and combinatorial properties

- Mathematics
- 2017

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…

ec 2 01 7 Ultrafilters , monads and combinatorial properties

- Mathematics
- 2017

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…

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

- MathematicsMath. Log. Q.
- 2019

Nonstandard methods, based on iterated hyperextensions, are used to develop applications to Ramsey theory of the theory of monads of ultrafilters, which extends to arbitrary sets and properties methods previously used to study partition regular Diophantine equations on N.

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

- Mathematics
- 2019

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…

Nonstandard characterizations of tensor products and monads in the theory of ultra lters

- Mathematics
- 2018

We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultralters. This is performed by studying in detail arbitrary tensor…

On Rado conditions for nonlinear Diophantine equations

- MathematicsEur. J. Comb.
- 2021

Partition regularity of affine algebraic plane curves

- Mathematics
- 2022

We characterise all systems of Diophantine equations in two variables admitting inﬁnitely many monochromatic solutions in N . Extending Rado’s Theorem, we prove that a system of polynomial equations…

Rado equations solved by linear combinations of idempotent ultrafilters

- MathematicsTopology and its Applications
- 2022

Rado's criterion over squares and higher powers

- MathematicsJournal of the European Mathematical Society
- 2021

We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado's characterisation of a partition regular equation remains valid…

## References

SHOWING 1-10 OF 60 REFERENCES

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

- Mathematics
- 2013

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…

Image Partition Regularity of Matrices

- MathematicsCombinatorics, Probability and Computing
- 1993

Many of the classical results of Ramsey Theory, including those of Hilbert, Schur, and van der Waerden, are naturally stated as instances of the following problem: given a u × ν matrix A with…

Monochromatic sums and products in N

- Mathematics
- 2017

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair {x+y, xy}. We answer this question affirmatively in a strong sense by exhibiting a…

IP-sets and polynomial recurrence

- MathematicsErgodic Theory and Dynamical Systems
- 1996

Abstract We combine recurrence properties of polynomials and IP-sets and show that polynomials evaluated along IP-sequences also give rise to Poincaré sets for measure-preserving systems, that is,…

The sumset phenomenon

- Mathematics
- 2001

Answering a problem posed by Keisler and Leth, we prove a theorem in non-standard analysis to reveal a phenomenon about sumsets, which says that if two sets A and B are large in terms of measure,…

New polynomial and multidimensional extensions of classical partition results

- MathematicsJ. Comb. Theory, Ser. A
- 2017

HYPERINTEGERS AND NONSTANDARD TECHNIQUES IN COMBINATORICS OF NUMBERS

- Mathematics
- 2012

In literature, many important combinatorial properties of subsets of N have been studied both with nonstandard techniques and from the point of view of N. In this thesis we mix these two different…