RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (202 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 519.4

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
\Bibitem{Ban98} \by T.~O.~Banakh \paper On a cardinal group invariant related to decompositions of Abelian groups \jour Mat. Zametki \yr 1998 \vol 64 \issue 3 \pages 341--350 \mathnet{http://mi.mathnet.ru/mz1404} \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} 

• http://mi.mathnet.ru/eng/mz1404
• https://doi.org/10.4213/mzm1404
• http://mi.mathnet.ru/eng/mz/v64/i3/p341

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Banakh T.O., Kmit I.Y., Verbitsky O.V., “On asymmetric colorings of integer grids”, Ars Combinatoria, 62 (2002), 257–271
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
4. Banakh T., Chervak O., “Centerpole Sets for Colorings of Abelian Groups”, J. Algebr. Comb., 34:2 (2011), 267–300
•  Number of views: This page: 150 Full text: 57 References: 14 First page: 1