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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some new results in multiplicative and additive Ramsey theory

Authors: Mathias Beiglböck, Vitaly Bergelson, Neil Hindman and Dona Strauss
Journal: Trans. Amer. Math. Soc. 360 (2008), 819-847
MSC (2000): Primary 05D10; Secondary 22A15
Published electronically: May 16, 2007
MathSciNet review: 2346473
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Abstract: There are several notions of largeness that make sense in any semigroup, and others such as the various kinds of density that make sense in sufficiently well-behaved semigroups including $ (\mathbb{N},+)$ and $ (\mathbb{N},\cdot)$. It was recently shown that sets in $ \mathbb{N}$ which are multiplicatively large must contain arbitrarily large geoarithmetic progressions, that is, sets of the form $ \big\{r^j(a+id):i,j\in\{0,1,\dotsc,k\}\big\}$, as well as sets of the form $ \big\{b(a+id)^j:i,j\in\{0,1,\dotsc,k\}\big\}$. Consequently, given a finite partition of $ \mathbb{N}$, one cell must contain such configurations. In the partition case we show that we can get substantially stronger conclusions. We establish some combined additive and multiplicative Ramsey theoretic consequences of known algebraic results in the semigroups $ (\beta\mathbb{N},+)$ and $ (\beta\mathbb{N},\cdot)$, derive some new algebraic results, and derive consequences of them involving geoarithmetic progressions. For example, we show that given any finite partition of $ \mathbb{N}$ there must be, for each $ k$, sets of the form $ \big\{b(a+id)^j:i,j\in\{0,1,\dotsc,k\}\big\}$ together with $ d$, the arithmetic progression $ \big\{a+id:i\in\{0,1,\dotsc,k\}\big\}$, and the geometric progression $ \big\{bd^j:j\in\{0,1,\dotsc,k\}\big\}$ in one cell of the partition. More generally, we show that, if $ S$ is a commutative semigroup and $ {\mathcal F}$ a partition regular family of finite subsets of $ S$, then for any finite partition of $ S$ and any $ k\in \mathbb{N}$, there exist $ b,r\in S$ and $ F\in {\mathcal F}$ such that $ rF\cup\{b(rx)^j:x \in F,j\in\{0,1,2,\ldots ,k\}\}$ is contained in a cell of the partition. Also, we show that for certain partition regular families $ {\mathcal F}$ and $ {\mathcal G}$ of subsets of $ \mathbb{N}$, given any finite partition of $ \mathbb{N}$ some cell contains structures of the form $ B \cup C \cup B\cdot C$ for some $ B\in {\mathcal F}, C\in {\mathcal G}$.

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Additional Information

Mathias Beiglböck
Affiliation: Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Wien, Austria

Vitaly Bergelson
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210

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

Dona Strauss
Affiliation: Mathematics Centre, University of Hull, Hull HU6 7RX, United Kingdom

Keywords: Ramsey theory, central sets, piecewise syndetic
Received by editor(s): October 21, 2005
Published electronically: May 16, 2007
Additional Notes: The first author thanks the Austrian Science Foundation FWF for its support through Projects nos. S8312 and P17627-N12. He also thanks Ohio State University for its hospitality in the spring of 2004 while much of this research was being conducted.
The second author acknowledges support received from the National Science Foundation via grant DMS-0345350.
The third author acknowledges support received from the National Science Foundation via grants DMS-0243586 and DMS-0554803.
Article copyright: © Copyright 2007 American Mathematical Society