Infinite partition regular matrices: solutions in central sets
Authors:
Neil Hindman, Imre Leader and Dona Strauss
Journal:
Trans. Amer. Math. Soc. 355 (2003), 12131235
MSC (2000):
Primary 05D10; Secondary 22A15, 54H13
Published electronically:
November 7, 2002
MathSciNet review:
1938754
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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 MillikenTaylor Theorem, or variants thereof.
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W. Deuber, N. Hindman, I. Leader, and H. Lefmann, Infinite partition regular matrices, Combinatorica 15 (1995), 333355. MR 96i:05173
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N. Hindman and I. Leader, Image partition regularity of matrices, Comb. Prob. and Comp. 2 (1993), 437463. MR 95j:05167
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N. Hindman, I. Leader, and D. Strauss, Image partition regular matrices  bounded solutions and preservation of largeness, Discrete Math. 242 (2002), 115144. MR 2002j:05146
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N. Hindman and D. Strauss, Algebra in the StoneCech compactification  theory and applications, W. de Gruyter & Co., Berlin, 1998. MR 99j:54001
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K. Milliken, Ramsey's Theorem with sums or unions, J. Combinatorial Theory (Series A) 18 (1975), 276290. MR 51:10106
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 1.
 W. Deuber, Partitionen und lineare Gleichungssysteme, Math. Zeit. 133 (1973), 109123. MR 48:3753
 2.
 W. Deuber, N. Hindman, I. Leader, and H. Lefmann, Infinite partition regular matrices, Combinatorica 15 (1995), 333355. 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), 437463. 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), 115144. MR 2002j:05146
 7.
 N. Hindman and D. Strauss, Algebra in the StoneCech 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), 276290. MR 51:10106
 9.
 R. Rado, Studien zur Kombinatorik, Math. Zeit. 36 (1933), 242280.
 10.
 I. Schur, Über die Kongruenz , Jahresbericht der Deutschen Math.Verein. 25 (1916), 114117.
 11.
 A. Taylor, A canonical partition relation for finite subsets of , J. Combinatorial Theory (Series A) 21 (1976), 137146. MR 54:12530
 12.
 B. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskunde 19 (1927), 212216.
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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:
http://dx.doi.org/10.1090/S0002994702031914
PII:
S 00029947(02)031914
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 DMS0070593.
Article copyright:
© Copyright 2002
American Mathematical Society
