Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Partition regularity without the columns property

Authors: Ben Barber, Neil Hindman, Imre Leader and Dona Strauss
Journal: Proc. Amer. Math. Soc. 143 (2015), 3387-3399
MSC (2010): Primary 05D10
Published electronically: February 20, 2015
MathSciNet review: 3348781
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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. In the finite case, Rado proved that a matrix is partition regular if and only it satisfies a computable condition known as the columns property. The first requirement of the columns property is that some set of columns sums to zero.

In the infinite case, much less is known. There are many examples of matrices with the columns property that are not partition regular, but until now all known examples of partition regular matrices did have the columns property. Our main aim in this paper is to show that, perhaps surprisingly, there are infinite partition regular matrices without the columns property -- in fact, having no set of columns summing to zero.

We also make a conjecture that if a partition regular matrix (say with integer coefficients) has bounded row sums then it must have the columns property, and prove a first step towards this.

References [Enhancements On Off] (What's this?)

  • [1] L. N. Argabright and C. O. Wilde, Semigroups satisfying a strong Følner condition, Proc. Amer. Math. Soc. 18 (1967), 587-591. MR 0210797 (35 #1683)
  • [2] Ben Barber, Neil Hindman, and Imre Leader, Partition regularity in the rationals, J. Combin. Theory Ser. A 120 (2013), no. 7, 1590-1599. MR 3092686,
  • [3] H. G. Dales, A. T.-M. Lau, and D. Strauss, Banach algebras on semigroups and on their compactifications, Mem. Amer. Math. Soc. 205 (2010), no. 966, vi+165. MR 2650729 (2011d:43001),
  • [4] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625 (82j:28010)
  • [5] Neil Hindman, Finite sums from sequences within cells of a partition of $ N$, J. Combinatorial Theory Ser. A 17 (1974), 1-11. MR 0349574 (50 #2067)
  • [6] Neil Hindman, Imre Leader, and Dona Strauss, Infinite partition regular matrices: solutions in central sets, Trans. Amer. Math. Soc. 355 (2003), no. 3, 1213-1235. MR 1938754 (2003h:05187),
  • [7] Neil Hindman, Imre Leader, and Dona Strauss, Open problems in partition regularity, Combin. Probab. Comput. 12 (2003), no. 5-6, 571-583. Special issue on Ramsey theory. MR 2037071 (2005e:05147),
  • [8] Neil Hindman and Dona Strauss, Density in arbitrary semigroups, Semigroup Forum 73 (2006), no. 2, 273-300. MR 2280825 (2009b:11019),
  • [9] Neil Hindman and Dona Strauss, Sets satisfying the central sets theorem, Semigroup Forum 79 (2009), no. 3, 480-506. MR 2564059 (2011m:22005),
  • [10] Neil Hindman and Dona Strauss, Density and invariant means in left amenable semigroups, Topology Appl. 156 (2009), no. 16, 2614-2628. MR 2561213 (2011b:43003),
  • [11] Neil Hindman and Dona Strauss, Algebra in the Stone-Čech compactification, de Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012. Theory and applications; Second revised and extended edition [of MR1642231]. MR 2893605
  • [12] Keith R. Milliken, Ramsey's theorem with sums or unions, J. Combinatorial Theory Ser. A 18 (1975), 276-290. MR 0373906 (51 #10106)
  • [13] Richard Rado, Studien zur Kombinatorik, Math. Z. 36 (1933), no. 1, 424-470 (German). MR 1545354,
  • [14] I. Schur, Über die Kongruenz $ x^m+y^m=z^m\ \pmod p$, Jahresbericht der Deutschen Math.-Verein. 25 (1916), 114-117.
  • [15] Alan D. Taylor, A canonical partition relation for finite subsets of $ \omega $, J. Combinatorial Theory Ser. A 21 (1976), no. 2, 137-146. MR 0424571 (54 #12530)
  • [16] B. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskunde 19 (1927), 212-216.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 05D10

Retrieve articles in all journals with MSC (2010): 05D10

Additional Information

Ben Barber
Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom

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

Imre Leader
Affiliation: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom

Dona Strauss
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9J2, United Kingdom

Received by editor(s): January 6, 2014
Received by editor(s) in revised form: April 4, 2014
Published electronically: February 20, 2015
Additional Notes: The second author acknowledges support received from the National Science Foundation (USA) via Grant DMS-1160566.
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society