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8 - Partition regular equations

Published online by Cambridge University Press:  05 May 2013

I. Leader
Affiliation:
Centre for Mathematical Sciences
C. D. Wensley
Affiliation:
University of Wales, Bangor
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Summary

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, such as van der Waerden's theorem or Schur's theorem, 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. In this survey paper we will review some finite results and then proceed to discuss some features of the infinite case.

Introduction

Let A be an m × n matrix with rational entries. We say that A is partition regular if for every finite colouring of the natural numbers ℕ = {1, 2,…} there is a monochromatic vector x ∈ ℕn with Ax = 0. In other words, A is partition regular if for every positive integer k, and every function c : ℕ → {1,…, k}, there is a vector x = (x1,…,xn) ∈ ℕn with c(x1) = … = c(xn) such that Ax = 0. We may also speak of the ‘system of equations Ax = 0’ being partition regular.

Many of the classical results of Ramsey theory may naturally be considered as statements about partition regularity. For example, Schur's theorem [11], that in any finite colouring of the natural numbers we may solve x+y = z in one colour class, is precisely the assertion that the 1×3 matrix (1,1, −1) is partition regular.

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Publisher: Cambridge University Press
Print publication year: 2003

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  • Partition regular equations
  • Edited by C. D. Wensley, University of Wales, Bangor
  • Book: Surveys in Combinatorics 2003
  • Online publication: 05 May 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107359970.010
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  • Partition regular equations
  • Edited by C. D. Wensley, University of Wales, Bangor
  • Book: Surveys in Combinatorics 2003
  • Online publication: 05 May 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107359970.010
Available formats
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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Partition regular equations
  • Edited by C. D. Wensley, University of Wales, Bangor
  • Book: Surveys in Combinatorics 2003
  • Online publication: 05 May 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107359970.010
Available formats
×