Partition regularity without the columns property

Abstract

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 propertyin 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.

Ramsey properties of nonlinear Diophantine equations

“…Here is the formulation for a single equation. 1 Rado's Theorem. A linear Diophantine equation with no constant term c 1 x 1 + · · · + c n x n = 0 is partition regular on N if and only if the following condition is satisfied:…”

Exponential patterns in arithmetic Ramsey theory

“…is equal to L (log (b log a)) + 2 = L log b + log(2) a + 2 ≥ L(log b) + 2 = L(b) + 1, as desired. To bound L a b from above, we again writeL(a b ) = L log b + log (2) a + 2 ≤ max{L(b) − 1, L(log (r) a) + (r − 2)} + 3.Now since log (r) a ≤ b the above is at most L (b) + (r + 1).…”

“…such that all the finite sums of distinct x i receive the same colour. Other examples of infinite linear systems that admit monochromatic solutions in an arbitrary finite colouring are known [2], [17], [27], however it appears that we are far from a classification of such infinite systems of equations [16].…”

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

“…The focus so far has been chiefly on the finite case (most notably on face monoids of hyperplane arrangements), but the extension to finitely generated semigroups is arguably the next natural step. Finally, amenability for semigroups has been an active area of research from the 1950s [12] to the present day; see [2,3,4,10,23] for some examples of recent developments and applications in other areas and [18,38] for advances relating specifically to finitely generated semigroups. It therefore seems apposite to ask whether whether the concept of cogrowth makes sense in a semigroup setting, and if so whether it is capable of forming a similar bridge between these areas.…”