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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Infinite partition regular matrices: solutions in central sets

Authors: Neil Hindman, Imre Leader and Dona Strauss
Journal: Trans. Amer. Math. Soc. 355 (2003), 1213-1235
MSC (2000): Primary 05D10; Secondary 22A15, 54H13
Published electronically: November 7, 2002
MathSciNet review: 1938754
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Abstract: A finite or infinite matrix $A$ is image partition regular provided that whenever ${\mathbb N}$ is finitely colored, there must be some $\vec {x}$with entries from ${\mathbb N}$ such that all entries of $A\vec {x}$ are in the same color class. In contrast to the finite case, infinite image partition regular matrices seem very hard to analyze: they do not enjoy the closure and consistency properties of the finite case, and it is difficult to construct new ones from old. In this paper we introduce the stronger notion of central image partition regularity, meaning that $A$ must have images in every central subset of ${\mathbb N}$. We describe some classes of centrally image partition regular matrices and investigate the extent to which they are better behaved than ordinary image partition regular matrices. It turns out that the centrally image partition regular matrices are closed under some natural operations, and this allows us to give new examples of image partition regular matrices. In particular, we are able to solve a vexing open problem by showing that whenever ${\mathbb N}$ is finitely colored, there must exist injective sequences $\langle x_n\rangle_{n=0}^\infty$ and $\langle z_n\rangle_{n=0}^\infty$ in ${\mathbb N}$ with all sums of the forms $x_n+x_m$ and $z_n+2z_m$ with $n<m$ in the same color class. This is the first example of an image partition regular system whose regularity is not guaranteed by the Milliken-Taylor Theorem, or variants thereof.

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Additional Information

Neil Hindman
Affiliation: Department of Mathematics, Howard University, Washington, DC 20059

Imre Leader
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB2 1SB, United Kingdom

Dona Strauss
Affiliation: Department of Pure Mathematics, University of Hull, Hull HU6 7RX, United Kingdom

Received by editor(s): May 10, 2001
Published electronically: November 7, 2002
Additional Notes: The first author acknowledges support received from the National Science Foundation (USA) via grant DMS-0070593.
Article copyright: © Copyright 2002 American Mathematical Society

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