Skip to main content
SpringerLink
Log in

Infinite partition regular matrices

  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

We consider infinite matrices with entries fromℤ (and only finitely many nonzero entries on any row). A matrixA is partition regular overℕ provided that, whenever the setℕ of positive integers is partitioned into finitely many classes there is a vector\(\overrightarrow x\) with entries inℤ such that all entries ofA \(A\overrightarrow x\) lie in the same cell of the partition. We show that, in marked contrast with the situation for finite matrices, there exists a finite partition ofℕ no cell of which contains solutions for all partition regular matrices and determine which of our pairs of matrices must always have solutions in the same cell of a partition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. Bergelson, andN. Hindman: A combinatorially large cell of a partition ofN, J. Comb. Theory (Series A).48 (1988), 39–52.

    Google Scholar 

  2. W. Deuber: Partitionen und lineare Gleichungssysteme,Math. Z. 133 (1973), 109–123.

    Google Scholar 

  3. W. Deuber, andN. Hindman: Partitions and sums of (m,p,c)-sets,J. Comb. Theory (Series A)45 (1987), 300–302.

    Google Scholar 

  4. R. Ellis: Lectures on topological dynamics, Benjamin, New York, 1969.

    Google Scholar 

  5. N. Hindman: Ultrafilters and Combinatorial number theory, inNumber Theory Carbondale 1979, M. Nathansen ed., Lecture Notes in Math.151 (1979), 119–184.

  6. N. Hindman, andI. Leader: Image partition regularity of matrices,Comb. Prob. and Comp. 2 (1993), 437–463.

    Google Scholar 

  7. N. Hindman, andH. Lefmann: Partition Regularity of (M,P,C)-systems,J. Comb. Theory (Series A),64 (1993), 1–9.

    Google Scholar 

  8. K. Milliken: Ramsey's Theorem with sums or unions,J. Comb. Theory (Series A)18 (1975), 276–290.

    Google Scholar 

  9. A. Taylor: A canonical partition relation for finite subsets of ω,J. Comb. Theory (Series A),21 (1976), 137–146.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501, Bielefeld, Germany

    Walter A. Deuber

  2. Department of Mathematics, Howard University, 20059, Washington, D.C., USA

    Neil Hindman

  3. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, England

    Imre Leader

  4. Fachbereich Informatik, LS II, Universität Dortmund, D-44221, Dortmund, Germany

    Hanno Lefmann

Authors
  1. Walter A. Deuber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Neil Hindman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Imre Leader
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hanno Lefmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deuber, W.A., Hindman, N., Leader, I. et al. Infinite partition regular matrices. Combinatorica 15, 333–355 (1995). https://doi.org/10.1007/BF01299740

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01299740

Mathematics Subject Classification (1991)

Search

Navigation