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

DOI:
https://doi.org/10.1090/S0002-9947-02-03191-4

Published electronically:
November 7, 2002

MathSciNet review:
1938754

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A finite or infinite matrix is *image partition regular* provided that whenever is finitely colored, there must be some with entries from such that all entries of 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 must have images in every central subset of . 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 is finitely colored, there must exist injective sequences and in with all sums of the forms and with 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.

**1.**W. Deuber,*Partitionen und lineare Gleichungssysteme*, Math. Zeit.**133**(1973), 109-123. MR**48:3753****2.**W. Deuber, N. Hindman, I. Leader, and H. Lefmann,*Infinite partition regular matrices*, Combinatorica**15**(1995), 333-355. MR**96i:05173****3.**H. Furstenberg,*Recurrence in ergodic theory and combinatorial number theory*, Princeton University Press, Princeton, 1981. MR**82j:28010****4.**R. Graham, B. Rothschild, and J. Spencer,*Ramsey Theory*, Wiley, New York, 1990. MR**90m:05003****5.**N. Hindman and I. Leader,*Image partition regularity of matrices*, Comb. Prob. and Comp.**2**(1993), 437-463. MR**95j:05167****6.**N. Hindman, I. Leader, and D. Strauss,*Image partition regular matrices - bounded solutions and preservation of largeness*, Discrete Math.**242**(2002), 115-144. MR**2002j:05146****7.**N. Hindman and D. Strauss,*Algebra in the Stone-Cech compactification - theory and applications*, W. de Gruyter & Co., Berlin, 1998. MR**99j:54001****8.**K. Milliken,*Ramsey's Theorem with sums or unions*, J. Combinatorial Theory (Series A)**18**(1975), 276-290. MR**51:10106****9.**R. Rado,*Studien zur Kombinatorik*, Math. Zeit.**36**(1933), 242-280.**10.**I. Schur,*Über die Kongruenz*, Jahresbericht der Deutschen Math.-Verein.**25**(1916), 114-117.**11.**A. Taylor,*A canonical partition relation for finite subsets of*, J. Combinatorial Theory (Series A)**21**(1976), 137-146. MR**54:12530****12.**B. van der Waerden,*Beweis einer Baudetschen Vermutung*, Nieuw Arch. Wiskunde**19**(1927), 212-216.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
05D10,
22A15,
54H13

Retrieve articles in all journals with MSC (2000): 05D10, 22A15, 54H13

Additional Information

**Neil Hindman**

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

Email:
nhindman@aol.com

**Imre Leader**

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

Email:
I.Leader@dpmms.cam.ac.uk

**Dona Strauss**

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

Email:
d.strauss@maths.hull.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-02-03191-4

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