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Image Partition Regularity of Matrices

Published online by Cambridge University Press:  12 September 2008

Neil Hindman  and
Imre Leader
Neil Hindman
Affiliation:
Department of Mathematics, Howard University, Washington, D.C. 20059, U.S.A.
Imre Leader
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Cambridge University, England
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Abstract

Many of the classical results of Ramsey Theory, including those of Hilbert, Schur, and van der Waerden, are naturally stated as instances of the following problem: given a u × ν matrix A with rational entries, is it true, that whenever the set ℕ of positive integers is finitely coloured, there must exist some x∈ℕν such that all entries of Ax are the same colour? While the theorems cited are all consequences of Rado's theorem, the general problem had remained open. We provide here several solutions for the alternate problem, which asks that x∈ℕν. Based on this, we solve the general problem, giving various equivalent characterizations.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 4 , December 1993 , pp. 437 - 463
Copyright
Copyright © Cambridge University Press 1993

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References

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