Open Problems in Partition Regularity

Authors:
Neil Hindman;Imre Leader;Dona Strauss
Affiliations:
Department of Mathematics, Howard University, Washington, DC 20059, USA (e-mail: nhindman@aol.com http://members.aol.com/nhindman/);Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: i.leader@dpmms.cam.ac.uk);Department of Pure Mathematics, University of Hull, Hull HU6 7RX, UK (e-mail: d.strauss@hull.ac.uk)
Venue:
Combinatorics, Probability and Computing
Year:
2003

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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.While in the finite case partition regularity is well understood, very little is known in the infinite case. Our aim in this paper is to present some of the natural and appealing open problems in the area.