author={Ben Barber and Neil Hindman and Imre Leader and Dona Strauss},
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… 

Maximality of Infinite Partition Regular Matrices

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

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.

Exponential Patterns in Arithmetic Ramsey Theory

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

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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

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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

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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.

Fermat-Like Equations that are not Partition Regular

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

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



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The aim in this paper is to present some of the natural and appealing open problems in the area of partition regularity in the infinite case.

Partition regularity in the rationals

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Central subsets of a discrete semigroup S have very strong combinatorial properties which are a consequence of the Central Sets Theorem . We investigate here the class of semigroups that have a

Density in Arbitrary Semigroups

We introduce some notions of density in an arbitrary semigroup S which extend the usual notions in countable left amenable semigroups in which density is based on Folner sequences. The new notions

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AbstractWe introduce some notions of density in an arbitrary semigroup S which extend the usual notions in countable left amenable semigroups in which density is based on Folner sequences. The new

Finite Sums from Sequences Within Cells of a Partition of N

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Banach Algebras on Semigroups and on Their Compactifications

Let $S$ be a (discrete) semigroup, and let $\ell^{\,1}(S)$ be the Banach algebra which is the semigroup algebra of $S$. The authors study the structure of this Banach algebra and of its second dual.

Algebra in the Stone-Cech Compactification: Theory and Applications

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large