Geodesic
On a cardinal group invariant related to decompositions of Abelian groups
Matematičeskie zametki, Tome 64 (1998) no. 3, pp. 341-350.

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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 $n\le3$. 
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     author = {T. O. Banakh},
     title = {On a cardinal group invariant related to decompositions of {Abelian} groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {341--350},
     publisher = {mathdoc},
     volume = {64},
     number = {3},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_3_a1/}
}
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T. O. Banakh. On a cardinal group invariant related to decompositions of Abelian groups. Matematičeskie zametki, Tome 64 (1998) no. 3, pp. 341-350. http://geodesic.mathdoc.fr/item/MZM_1998_64_3_a1/

[1] Banakh T. O., Protasov I. V., “Asimmetrichnye razbieniya abelevykh grupp”, Matem. zametki (to appear)

[2] Protasov I. V., “Asimmetrichno razlozhimye abelevy gruppy”, Matem. zametki, 59:3 (1996), 468–471 | MR | Zbl

[3] Fuks L., Beskonechnye abelevy gruppy, T. I, Mir, M., 1970

[4] Spener E., Algebraicheskaya topologiya, Mir, M., 1971 | Zbl

[5] Lyusternik L. A., Vypuklye figury i mnogogranniki, Gostekhizdat, M., 1956