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… 

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References

SHOWING 1-10 OF 24 REFERENCES

Open Problems in Partition Regularity

TLDR
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

Sets satisfying the Central Sets Theorem

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

Density in Arbitrary Semigroups

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

A Canonical Partition Relation for Finite Subsets of omega

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