Centerpole sets for colorings of abelian groups
Taras Banakh
and Ostap Chervak
DOI: 10.1007/s10801-010-0271-3
Abstract
A subset C\subset G of a group G is called k-centerpole if for each k-coloring of G there is an infinite monochromatic subset G, which is symmetric with respect to a point c\in C in the sense that S= cS - 1 c. By c k ( G) we denote the smallest cardinality c k ( G) of a k-centerpole subset in G. We prove that c k ( G)= c k (\Bbb Z m ) if G is an abelian group of free rank m\geq k. Also we prove that c 1(\Bbb Z n+1)=1, c 2(\Bbb Z n+2)=3, c 3(\Bbb Z n+3)=6, 8\leq c 4(\Bbb Z n+4)\leq c 4(\Bbb Z 4)=12 for all n\in ω , and \frac12( k 2+3 k -4) \sterling c k(\mathbb Z n) \sterling 2 k -1 -max s \sterling k -2\binom k -1 s -1 {\frac{1}{2}(k^{2}+3k-4)\le c_{k}(\mathbb{Z}^{n})\le2^{k}-1-\max_{s\le k-2}\binom {k-1}{s-1}} for all n\geq k\geq 4.
Pages: 267–300
Keywords: keywords abelian group; centerpole set; coloring; symmetric subset; monochromatic subset
Full Text: PDF
References
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2. Banakh, T., Bokalo, B., Guran, I., Radul, T., Zarichnyi, M.: Problems from the Lviv topological seminar. In: Pearl, E. (ed.) Open Problems in Topology, vol. II, pp. 655-667. Elsevier, Amsterdam (2007)
3. Banakh, T., Dudko, A., Repovs, D.: Symmetric monochromatic subsets in colorings of the Lobachevsky plane. Discrete Math. Theor. Comput. Sci. 12(1), 12-20 (2010)
4. Banakh, T., Protasov, I.V.: Asymmetric partitions of Abelian groups. Mat. Zametki 66(1), 17-30 (1999)
5. Banakh, T., Protasov, I.: Symmetry and colorings: some results and open problems. Izv. Gomel Univ. Vopr. Algebry 4(17), 5-16 (2001)
6. Banakh, T., Verbitski, O., Vorobets, Ya.: Ramsey treatment of symmetry. Electron. J. Comb. 7(1), R52 (2000), 25 pp.
7. Engelking, R.: General topology. Sigma Series in Pure Mathematics, vol.
6. Heldermann Verlag, Berlin (1989)
8. Gryshko, Yu.: Monochrome symmetric subsets in 2-colorings of groups. Electron. J. Comb. 10, R28 (2003), 8 pp.
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