Ramsey-product subsets of a group

T. Banakh; I. Protasov; K. Protasova

doi:10.15330/ms.47.2.145-149

Given an infinite group

$G$ and a number vector

$\overrightarrow{m}=(m_{1},\ldots , m_{k} )\in\mathbb{Z}^{k}$ of finite length

$k$ , we say that a subset

$A$ of

$G$ is a Ramsey

$\overrightarrow{m}$ -product set if every infinite subset

$X\subset G$ contains distinct elements

$x_{1},\ldots, x_{k} \in X$ such that

$x^{m_{1}} _{\sigma(1)}\ldots x^{m_{k}} _{\sigma(k)} \in A$ for any permutation

$\sigma\in S_{k}$ . We use these subsets to characterize combinatorially some algebraically defined subsets of the Stone-Cech compactification

$\beta G$ of

$G$ .

1. V. Bergelson, N. Hindman, R. McCutcheon, Notions of size and combinatorial properties of quotient sets in semigroups, Topology Proc., 23 (1998), 23-60.

2. P. Erdos, R. Rado, Intersection theorems for systems of sets, J. London Math. Soc., 35 (1960), 85-90.

3. N. Hindman, A. Maleki, D. Strauss, Linear equations in the Stone-Cech compactification of N, Integers: Electronic Journal of Combinatorial Number Theory, 0 (2000), A02, 1-20.

4. N. Hindman, D. Strauss, Algebra in the Stone-Cech compactification, de Gruyter, Berlin, 1998.

5. I. Protasov, Equations in

$\beta G$ and resolvability of Abelian groups, Math. Notes, 6 (1999), 787-789.

6. I. Protasov, K. Protasova, On reccurence in G-spaces, Algebra Discrete Math., 23 (2017), no.2, 279-284.

7. O. Sipacheva, Large sets in Boolean and non-Boolean groups and topology, preprint (https://arxiv.org/abs/1709.02027).

	Ramsey-product subsets of a group
Author	T. Banakh, I. Protasov, K. Protasova t.o.banakh@gmail.com; i.v.protasov@gmail.com, ksuha@freenet.com.ua Ivan Franko National University of L'viv, Lviv, Ukraine; Faculty of Computer Science and Cybernetics, Kyiv University, Kyiv, Ukraine
Abstract	Given an infinite group $G$ and a number vector $\overrightarrow{m}=(m_{1},\ldots , m_{k} )\in\mathbb{Z}^{k}$ of finite length $k$ , we say that a subset $A$ of $G$ is a Ramsey $\overrightarrow{m}$ -product set if every infinite subset $X\subset G$ contains distinct elements $x_{1},\ldots, x_{k} \in X$ such that $x^{m_{1}} _{\sigma(1)}\ldots x^{m_{k}} _{\sigma(k)} \in A$ for any permutation $\sigma\in S_{k}$ . We use these subsets to characterize combinatorially some algebraically defined subsets of the Stone-Cech compactification $\beta G$ of $G$ .
Keywords	Stone-Cech compactification; product of ultrafilters; Ramsey product subset
DOI	doi:10.15330/ms.47.2.145-149
Reference	1. V. Bergelson, N. Hindman, R. McCutcheon, Notions of size and combinatorial properties of quotient sets in semigroups, Topology Proc., 23 (1998), 23-60. 2. P. Erdos, R. Rado, Intersection theorems for systems of sets, J. London Math. Soc., 35 (1960), 85-90. 3. N. Hindman, A. Maleki, D. Strauss, Linear equations in the Stone-Cech compactification of N, Integers: Electronic Journal of Combinatorial Number Theory, 0 (2000), A02, 1-20. 4. N. Hindman, D. Strauss, Algebra in the Stone-Cech compactification, de Gruyter, Berlin, 1998. 5. I. Protasov, Equations in $\beta G$ and resolvability of Abelian groups, Math. Notes, 6 (1999), 787-789. 6. I. Protasov, K. Protasova, On reccurence in G-spaces, Algebra Discrete Math., 23 (2017), no.2, 279-284. 7. O. Sipacheva, Large sets in Boolean and non-Boolean groups and topology, preprint (https://arxiv.org/abs/1709.02027).
Pages	145-149
Volume	47
Issue	2
Year	2017
Journal	Matematychni Studii
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