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Partition regularity without the columns property
A finite or infinite matrix A with rational entries is called partition
regular if whenever the natural numbers are finitely coloured there is a
monochromatic vector x with Ax=0. Many of the classical theorems of Ramsey
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 columns property is that some set of
columns sums to zero.
In the infinite case, much less is known. There are many examples of matrices
with the columns property that are not partition regular, but until now all
known examples of partition regular matrices did have the columns property. Our
main aim in this paper is to show that, perhaps surprisingly, there are
infinite partition regular matrices without the columns property --- in fact,
having no set of columns summing to zero.
We also make a conjecture that if a partition regular matrix (say with
integer coefficients) has bounded row sums then it must have the columns
property, and prove a first step towards this.Comment: 13 page
Centrally Image partition Regularity near 0
The notion of Image partition regularity near zero was first introduced by De
and Hindman. It was shown there that like image partition regularity over
the main source of infinite image partition regular matrices near
zero are Milliken- Taylor matrices. But Milliken- Taylor matrices are far apart
to have images in central sets. In this regard the notion of centrally image
partition regularity was introduced. In the present paper we propose the notion
centrally partition regular matrices near zero for dense sub semigroup of
(\ber^+,+) which are different from centrally partition regular matrices
unlike finite cases
Rado's theorem for rings and modules
We extend classical results of Rado on partition regularity of systems of
linear equations with integer coefficients to the case when the coefficient
ring is either an arbitrary domain or a noetherian ring. The crucial idea is to
study partition regularity for general modules rather than only for rings.
Contrary to previous techniques, our approach is independent of the
characteristic of the coefficient ring.Comment: 19 page
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