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Transactions of the American Mathematical Society

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An infinitary extension of the Graham-Rothschild Parameter Sets Theorem


Authors: Timothy J. Carlson, Neil Hindman and Dona Strauss
Journal: Trans. Amer. Math. Soc. 358 (2006), 3239-3262
MSC (2000): Primary 05D10
DOI: https://doi.org/10.1090/S0002-9947-06-03899-2
Published electronically: February 20, 2006
MathSciNet review: 2216266
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Abstract: The Graham-Rothschild Parameter Sets Theorem is one of the most powerful results of Ramsey Theory. (The Hales-Jewett Theorem is its most trivial instance.) Using the algebra of $ \beta S$, the Stone-Cech compactification of a discrete semigroup, we derive an infinitary extension of the Graham-Rothschild Parameter Sets Theorem. Even the simplest finite instance of this extension is a significant extension of the original. The original theorem says that whenever $ k<m$ in $ \mathbb{N}$ and the $ k$-parameter words are colored with finitely many colors, there exist a color and an $ m$-parameter word $ w$ with the property that whenever a $ k$-parameter word of length $ m$ is substituted in $ w$, the result is in the specified color. The ``simplest finite instance'' referred to above is that, given finite colorings of the $ k$-parameter words for each $ k<m$, there is one $ m$-parameter word which works for each $ k$. Some additional Ramsey Theoretic consequences are derived.

We also observe that, unlike any other Ramsey Theoretic result of which we are aware, central sets are not necessarily good enough for even the $ k=1$ and $ m=2$ version of the Graham-Rothschild Parameter Sets Theorem.


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Additional Information

Timothy J. Carlson
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Email: carlson@math.ohio-state.edu

Neil Hindman
Affiliation: Department of Mathematics, Howard University, Washington, DC 20059
Email: nhindman@aol.com

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/S0002-9947-06-03899-2
Received by editor(s): February 20, 2004
Received by editor(s) in revised form: September 14, 2004
Published electronically: February 20, 2006
Additional Notes: The second author acknowledges support received from the National Science Foundation (USA) via grant DMS-0070593.
Article copyright: © Copyright 2006 American Mathematical Society

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