Partition regularity without the columns property

Ben Barber, Neil Hindman, Imre Leader, Dona Strauss

Research output: Contribution to journalArticle (Academic Journal)peer-review

8 Citations (Scopus)
270 Downloads (Pure)

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 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.
Original languageEnglish
Pages (from-to)3387-3399
Number of pages13
JournalProceedings of the American Mathematical Society
Volume143
DOIs
Publication statusPublished - 20 Feb 2015

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