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Selective survey on Subset Combinatorics of Groups

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Abstract

We survey recent results concerning the combinatorial size of subsets of groups. For a cardinal κ, according to its arrangement in a group G, a subset of G is distinguished as κ-large, κ-small, κ-thin, κ-thick, and P κ -small. By analogy with topology, there arise the following combinatorial cardinal invariants of a group: density, cellularity, resolvability, spread, etc. The paper consists of 7 sections: Ballean context, Amenability, Ideals, Partitions, Packings, Around thin subsets, and Colorings.

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References

  1. T. Banakh and N. Lyaskovska, “Weakly P-small not P-small subsets in groups,” Int. J. Alg. Comput., 18, 1–6 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  2. T. Banakh and N. Lyaskovska, “Completeness of translation invariant ideals in groups,” Ukr. Math. J., (to appear).

  3. T. Banakh and N. Lyaskovska, and D. Repovš, “Packing index of subsets in Polish groups,” Notre Dame J. Formal Logic, 50:4, 453–468 (2009).

    Article  MATH  Google Scholar 

  4. T. Banakh, O. Petrenko, and I. Protasov, “Kaleidoscopical configurations,” manuscript.

  5. T. Banakh and S. Pikhurko, “A game characterization of limit-detecting sequences in locally compact G-space,” Math. Stud., 21, No. 2, 115–132 (1999).

    Google Scholar 

  6. T. O. Banakh and I. V. Protasov, “Asymmetric colorings of Abelian groups,” Mat. Zam., 66, 17–30 (1999).

    MathSciNet  Google Scholar 

  7. T. O. Banakh and I. V. Protasov, “Symmetry and colorings: some results and open problems,” Izv. Gomel Univ., Vopr. Alg., 17, 4–15 (2001).

    Google Scholar 

  8. T. Banakh, I. Protasov, and O. Verbitskiy, Symmetry and Colorings, monograph in preparation.

  9. A. Bella and V. Malykhin, “Certain subsets of a group,” Quest. Answ. Gen. Topology, 17, 183–187 (1999).

    MATH  MathSciNet  Google Scholar 

  10. T. Carlson, N. Hindman, J. McLeod, and D. Strauss, “Almost disjoint large subsets of semigroups,” Topology Appl., 155, 433–444 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  11. C. Chou, “On the size of the set of left invariant means on a semigroup,” Proc. Amer. Math. Soc., 23, 199–205 (1969).

    MATH  MathSciNet  Google Scholar 

  12. W. W. Comfort and S. Garsia-Ferreira, “Resolvability: a selective survey and some results,” Topology Appl., 74, 149–167 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  13. D. Dikranjan and I. Protasov, “Every infinite group can be generated by P-small subset,” Appl. Gen. Topology, 7, 265–268 (2006).

    MATH  MathSciNet  Google Scholar 

  14. D. Dikranjan and I. Protasov, “Counting maximal topologies on countable groups and rings,” Topol. Appl., 156, 322–325 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  15. P. C. Eklof, Set Theoretical Methods in Homological Algebra and Abelian Group, Les Press de L’Univ. de Montreal, Montreal, 1980.

    Google Scholar 

  16. M. Filali, Ie. Lutsenko, and I. V. Protasov, “Boolean group ideals and the ideal structure of βG,” Math. Stud., 31, 19–28 (2009).

    MATH  MathSciNet  Google Scholar 

  17. M. Filali and I. V. Protasov, “Spread of balleans,” Appl. Gen. Topology, 9, 161–175 (2008).

    MathSciNet  Google Scholar 

  18. V. Gavrylkiv, Algebraic-Topological Structures on Superextensions, Dissertation, Lviv, 2009.

  19. R. Graham, B. Rotschild, and J. Speneer, Ramsey Theory, Wiley, New York, 1980.

    MATH  Google Scholar 

  20. F. Greenleaf, Invariant Means on Topological Groups and Their Applications, Van Nostrand, New York, 1966.

    Google Scholar 

  21. E. Hewitt, “A problem in set-theoretical topology,” Duke Math. J., 10, 309–333 (1943).

    Article  MATH  MathSciNet  Google Scholar 

  22. N. Hindman and D. Strauss, Algebra in the Stone- Čech Compactification — Theory and Applications,Walter de Gruyter, Berlin, New York, 1998.

    MATH  Google Scholar 

  23. Ie. Lutsenko, “Thin systems of generators of groups,” Alg. Discrete Math., to appear.

  24. Ie. Lutsenko and I. V. Protasov, “Sparse, thin and other subsets of groups,” Int. J. Alg. Computation, 19, 491–510 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  25. Ie. Lutsenko and I. V. Protasov, “Sketch of vector balleans,” Math. Stud., 31, 219–224 (2009).

    MATH  MathSciNet  Google Scholar 

  26. Ie. Lutsenko and I. V. Protasov, “Thin subsets of balleans,” manuscript.

  27. N. Lyaskovska, “Constructing subsets of given packing index in Abelian groups,” Acta Univ. Carolinae, Math. et Phis., 48, 69–80 (2007).

    MATH  MathSciNet  Google Scholar 

  28. V. Malykhin and R. Moresko, “Small generated groups,” Quest. Answ. Gen. Topology, 19, 47–53 (2001).

    MATH  Google Scholar 

  29. V. I. Malykhin and I. V. Protasov, “Maximal resolvability of bounded groups,” Topol. Appl., 20, 1–6 (1996).

    MathSciNet  Google Scholar 

  30. I. Prodanov, “Some minimal group topologies are precompact,” Math. Ann., 227, 117–125 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  31. I. V. Protasov, “Resolvability of τ -bounded groups,” Math. Stud., 5, 17–20 (1995).

    MATH  MathSciNet  Google Scholar 

  32. I. V. Protasov, “Resolvability of groups,” Math. Stud., 9, 130–148 (1998).

    MATH  MathSciNet  Google Scholar 

  33. I. V. Protasov, “Partitions of groups onto large subsets,” Math. Notes, 73, 271–281 (2003).

    Article  MathSciNet  Google Scholar 

  34. I. V. Protasov, “Small systems of generators of groups,” Math. Notes, 76, 420–426 (2004).

    Article  MathSciNet  Google Scholar 

  35. I. V. Protasov, “Resolvability of ball structures,” Appl. Gen. Topology, 5, 191–198 (2004).

    MATH  MathSciNet  Google Scholar 

  36. I. V. Protasov, “Cellularity and density of balleans,” Appl. Gen. Topology, 8, 283–291 (2007).

    MATH  MathSciNet  Google Scholar 

  37. I. V. Protasov, “Extraresolvability of balleans,” Comment. Math. Univ. Carolinae, 48, 161–175 (2007).

    MathSciNet  Google Scholar 

  38. I. V. Protasov, “Balleans of bounded geometry and G-spaces,” Math. Stud., 30, 61–66 (2008).

    MATH  MathSciNet  Google Scholar 

  39. I. V. Protasov, “Small subsets of groups,” Topology Appl., 156, 2651–2658 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  40. I. Protasov, “Counting Ω-ideals,” Alg. Universalis, 62, 339–343 (2010).

    Article  MathSciNet  Google Scholar 

  41. I. V. Protasov, Combinatorics of Numbers, Math. Stud. Monogr. Ser., Vol. 2, VNTL, Lviv, 1997.

  42. I. Protasov and E. Zelenyuk, Topologies on Groups Determined by Sequence, Math. Stud. Monogr. Ser., Vol. 4, VNTL, Lviv, 1999.

  43. I. Protasov and T. Banakh, Ball Structures and Colorings of Graphs and Groups, Math. Stud. Monogr. Ser., Vol. 11, VNTL, Lviv, 2003.

  44. I. Protasov and M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., Vol. 12, VNTL, Lviv, 2007.

  45. I. Protasov and M. Filaly, Ultrafilters and Topologies on Groups, Math. Stud. Monorg. Ser., (to appear).

  46. I. V. Protasov and O. I. Protasova, “Color-detectors of hypergraphs,” Alg. Discrete Math., No. 1, 91–98 (2005).

  47. I. V. Protasov and O. I. Protasova, “On closed ideals of βG,” Semigroup Forum, 75, 237–240 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  48. J. Roe, Lectures on Coarse Geometry, Amer. Math. Soc., Providence, RI, 2003.

    MATH  Google Scholar 

  49. W. Serediński, “Some operations related to translations,” Colloq. Math., 57, 203–219 (1989).

    MathSciNet  Google Scholar 

  50. S. Solecki, “Translation invariant ideals,” Israel J. Math., 135, 93–110 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  51. S. Solecki, “Size of subsets of groups and Haar null sets,” GAFA, 15, 246–273 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  52. T. Tao and V. Vu, Additive Combinatorics, Cambridge Univ. Press, Cambridge, 2006.

    Book  MATH  Google Scholar 

  53. V. O. Vasil’eva and I. V. Protasov, “Around van der Waerden’s Theorem,” U sviti mat., 17–19 (2000).

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  1. Taras Shevchenko National University of Kyiv, 64, Volodymyrs’ka Str., Kyiv, 01601, Ukraine

    Igor V. Protasov

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  1. Igor V. Protasov
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Correspondence to Igor V. Protasov.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 7, No. 2, pp. 220–257, April–May, 2010.

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Protasov, I.V. Selective survey on Subset Combinatorics of Groups. J Math Sci 174, 486–514 (2011). https://doi.org/10.1007/s10958-011-0314-x

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