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Mat. Zametki, 1998, Volume 64, Issue 3, Pages 341–350 (Mi mz1404)  

This article is cited in 4 scientific papers (total in 4 papers)

On a cardinal group invariant related to decompositions of Abelian groups

T. O. Banakh

Ivan Franko National University of L'viv

Abstract: For each Abelian group $G$, a cardinal invariant $\chi(G)$ is introduced and its properties are studied. In the special case $G=\mathbb Z^n$, the cardinal $\chi\mathbb Z^n)$ is equal to the minimal cardinality of an essential subset of $\mathbb Z^n$, i.e., a of a subset $A\subset\mathbb Z^n$ such that, for any coloring of the group $\mathbb Z^n$ in $n$ colors, there exists an infinite one-color subset that is symmetric with respect to some point $\alpha$ of $A$. The estimate $n(n+1)/2\le\chi(\mathbb Z^n)<2^n$ is proved for all $n$ and the relation $\chi(\mathbb Z^n)=n(n+1)/2$ for $n\le3$. The structure of essential subsets of cardinality $\chi(\mathbb Z^n)$ in $\mathbb Z^n$ is completely described for $n\le3$.

DOI: https://doi.org/10.4213/mzm1404

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English version:
Mathematical Notes, 1998, 64:3, 295–302

Bibliographic databases:

UDC: 519.4
Received: 01.08.1997

Citation: T. O. Banakh, “On a cardinal group invariant related to decompositions of Abelian groups”, Mat. Zametki, 64:3 (1998), 341–350; Math. Notes, 64:3 (1998), 295–302

Citation in format AMSBIB
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\paper On a cardinal group invariant related to decompositions of Abelian groups
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\yr 1998
\vol 64
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\pages 341--350
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\crossref{https://doi.org/10.4213/mzm1404}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1680158}
\zmath{https://zbmath.org/?q=an:0933.20043}
\transl
\jour Math. Notes
\yr 1998
\vol 64
\issue 3
\pages 295--302
\crossref{https://doi.org/10.1007/BF02314837}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000079258700002}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. T. O. Banakh, Ya. B. Vorobets, O. V. Verbitskii, “Ramsay problems for spaces with symmetries”, Izv. Math., 64:6 (2000), 1091–1127  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Banakh T.O., Kmit I.Y., Verbitsky O.V., “On asymmetric colorings of integer grids”, Ars Combinatoria, 62 (2002), 257–271  mathscinet  zmath  isi
    3. Banakh T., Dudko A., Repovs D., “Symmetric monochromatic subsets in colorings of the Lobachevsky plane”, Discrete Mathematics and Theoretical Computer Science, 12:1 (2010), 12–U2  mathscinet  isi
    4. Banakh T., Chervak O., “Centerpole Sets for Colorings of Abelian Groups”, J. Algebr. Comb., 34:2 (2011), 267–300  crossref  mathscinet  zmath  isi  scopus  scopus
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