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Partition regularity without the columns property


Authors: Ben Barber, Neil Hindman, Imre Leader and Dona Strauss
Journal: Proc. Amer. Math. Soc. 143 (2015), 3387-3399
MSC (2010): Primary 05D10
DOI: https://doi.org/10.1090/S0002-9939-2015-12519-1
Published electronically: February 20, 2015
MathSciNet review: 3348781
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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.


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Additional Information

Ben Barber
Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Email: b.a.barber@bham.ac.uk

Neil Hindman
Affiliation: Department of Mathematics, Howard University, Washington, DC 20059
Email: nhindman@aol.com

Imre Leader
Affiliation: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom
Email: i.leader@dpmms.cam.ac.uk

Dona Strauss
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9J2, United Kingdom
Email: d.strauss@hull.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2015-12519-1
Received by editor(s): January 6, 2014
Received by editor(s) in revised form: April 4, 2014
Published electronically: February 20, 2015
Additional Notes: The second author acknowledges support received from the National Science Foundation (USA) via Grant DMS-1160566.
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2015 American Mathematical Society