# PARTITION REGULARITY WITHOUT THE COLUMNS PROPERTY

@inproceedings{Barber2015PARTITIONRW, title={PARTITION REGULARITY WITHOUT THE COLUMNS PROPERTY}, author={Ben Barber and Neil Hindman and Imre Leader and Dona Strauss}, year={2015} }

A finite or infinite matrix A with rational entries is called parti- tion regular if whenever the natural numbers are finitely coloured there is a monochromatic vector x with Ax = 0. Many of the classical theorems of Ram- sey Theory may naturally be interpreted as assertions that particular matrices are partition regular. In the finite case, Rado proved that a matrix is partition regular if and only it satisfies a computable condition known as the columns property. The first requirement of the…

## 12 Citations

### Maximality of Infinite Partition Regular Matrices

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- 2014

A finite or infinite matrix $A$ with rational entries (and only finitely many non-zero entries in each row) is called image partition regular if, whenever the natural numbers are finitely coloured,…

### Extensions of Infinite Partition Regular Systems

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The aim in this paper is to investigate maximality questions for image partition regular matrices, and some algebraic properties of $\beta {\mathbb N}$, the Stone- C ech compactification of the natural numbers.

### Distinguishing subgroups of the rationals by their Ramsey properties

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### Exponential Patterns in Arithmetic Ramsey Theory

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We show that for every finite colouring of the natural numbers there exists $a,b >1$ such that the triple $\{a,b,a^b\}$ is monochromatic. We go on to show the partition regularity of a much richer…

### On Cogrowth, Amenability, and the Spectral Radius of a Random Walk on a Semigroup

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We introduce two natural notions of cogrowth for finitely generated semigroups —one local and one global — and study their relationship with amenability and random walks. We establish the minimal…

### Partition regularity and multiplicatively syndetic sets

- MathematicsActa Arithmetica
- 2020

We show how multiplicatively syndetic sets can be used in the study of partition regularity of dilation invariant systems of polynomial equations. In particular, we prove that a dilation invariant…

### C O ] 7 J un 2 01 6 Ramsey properties of nonlinear Diophantine equations

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- 2018

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.…

### Fermat-Like Equations that are not Partition Regular

- MathematicsComb.
- 2018

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 xn + ym = zk with k ∉ {n,…

### Recent results on partition regularity of infinite matrices

- Mathematics
- 2015

We survey results obtained in the last ten years on image and kernel partition regularity of infinite matrices.

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