Packings and coverings of groups: some results and open problems

I.V.Protasov

For a group $G$, we introduce and study the packing and covering spectra $\mathop{\rm Pack}(G)$ and $\mathop{\rm Cov}(G)$ as the sets of all possible packing and covering numbers of subsets of $G$. We show that $|S|\in \mathop{\rm Cov}(G)$ for every finite subgroup $S$ of $G$, and $\kappa\in \mathop{\rm Cov}(G)$ for every infinite group $G$ and every cardinal $\kappa$ such that $1\leqslant\kappa\leqslant |G|$ and $|[G]_{<\kappa}|\leqslant|G|$. We calculate or evaluate the packing and covering numbers of subsets of certain combinatorial size. The paper contains also six open problems.

1. Banakh T., Lyaskovska N. Weakly $P$-small not $P$-small subsets in groups // Internat. J. Algebra Comput. - 2008. - № 18. - P. 1-6.

2. Banakh T., Lyaskovska N., Repovs D. Packing index of subsets in Polish groups // Notre Dame J. Formal Logic. - 2009. - № 50. - P. 453-468.

3. Banakh T., Lyaskovska N. Completeness of translation-invariant ideals in groups // Ukr. Math. J. (to appear).

4. Banakh T. Thin subsets of groups. - Preprint, 2010.

5. Lutsenko Ie., Protasov I.V. Sparse, thin and other subsets of groups // Internat.J.Algebra Comput. - 2009. - № 19. - P. 491-510.

6. Lutsenko Ie. Thin systems of generators of groups // Algebra Discrete Math. - 2010. - № 2.

7. Lyaskovska N. Constructing subsets of given packing index in Abelian groups // Acta Univ. Carolin. Math. Phys. - 2007. - V.48, №2. - P. 69-80.

8. Ore O. Theory of Graphs. - American Math. Society Colloquium Publications, V.38, 1962.

9. Protasov I.V. Selective survey on subset combinatorics of groups // Ukr. Math. Bull. - 2010. - V.7, №2. - P. 204-241.

10. Protasov I.V. Partitions of groups into thin subsets. - Preprint, 2009.

11. Protasov I., Zarichniy M. General Asymptology. - Math. Stud. Monorg. Ser. V.12 VNTL Publishers, Lviv. - 2007.

	Packings and coverings of groups: some results and open problems
Author	I.V.Protasov I.V.Protasov@gmail.com Kyiv National University,Department of Cybernetics
Abstract	For a group $G$, we introduce and study the packing and covering spectra $\mathop{\rm Pack}(G)$ and $\mathop{\rm Cov}(G)$ as the sets of all possible packing and covering numbers of subsets of $G$. We show that $\|S\|\in \mathop{\rm Cov}(G)$ for every finite subgroup $S$ of $G$, and $\kappa\in \mathop{\rm Cov}(G)$ for every infinite group $G$ and every cardinal $\kappa$ such that $1\leqslant\kappa\leqslant \|G\|$ and $\|[G]_{<\kappa}\|\leqslant\|G\|$. We calculate or evaluate the packing and covering numbers of subsets of certain combinatorial size. The paper contains also six open problems.
Keywords	group, packing, spectra, covering
DOI	doi:10.30970/ms.33.2.115-119
Reference	1. Banakh T., Lyaskovska N. Weakly $P$-small not $P$-small subsets in groups // Internat. J. Algebra Comput. - 2008. - № 18. - P. 1-6. 2. Banakh T., Lyaskovska N., Repovs D. Packing index of subsets in Polish groups // Notre Dame J. Formal Logic. - 2009. - № 50. - P. 453-468. 3. Banakh T., Lyaskovska N. Completeness of translation-invariant ideals in groups // Ukr. Math. J. (to appear). 4. Banakh T. Thin subsets of groups. - Preprint, 2010. 5. Lutsenko Ie., Protasov I.V. Sparse, thin and other subsets of groups // Internat.J.Algebra Comput. - 2009. - № 19. - P. 491-510. 6. Lutsenko Ie. Thin systems of generators of groups // Algebra Discrete Math. - 2010. - № 2. 7. Lyaskovska N. Constructing subsets of given packing index in Abelian groups // Acta Univ. Carolin. Math. Phys. - 2007. - V.48, №2. - P. 69-80. 8. Ore O. Theory of Graphs. - American Math. Society Colloquium Publications, V.38, 1962. 9. Protasov I.V. Selective survey on subset combinatorics of groups // Ukr. Math. Bull. - 2010. - V.7, №2. - P. 204-241. 10. Protasov I.V. Partitions of groups into thin subsets. - Preprint, 2009. 11. Protasov I., Zarichniy M. General Asymptology. - Math. Stud. Monorg. Ser. V.12 VNTL Publishers, Lviv. - 2007.
Pages	115-119
Volume	33
Issue	2
Year	2010
Journal	Matematychni Studii
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