Journal of Algebraic Combinatorics

, Volume 34, Issue 2, pp 267–300 | Cite as

Centerpole sets for colorings of abelian groups

Article

Abstract

A subset CG 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 cC in the sense that S=cS−1c. By ck(G) we denote the smallest cardinality ck(G) of a k-centerpole subset in G. We prove that ck(G)=ck(ℤm) if G is an abelian group of free rank mk. Also we prove that c1(ℤn+1)=1, c2(ℤn+2)=3, c3(ℤn+3)=6, 8≤c4(ℤn+4)≤c4(ℤ4)=12 for all nω, and \({\frac{1}{2}(k^{2}+3k-4)\le c_{k}(\mathbb{Z}^{n})\le2^{k}-1-\max_{s\le k-2}\binom {k-1}{s-1}}\) for all nk≥4.

Keywords

Abelian group Centerpole set Coloring Symmetric subset Monochromatic subset 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Ivan Franko National University of LvivLvivUkraine
  2. 2.Uniwersytet Humanistyczno-Przyrodniczy Jana KochanowskiegoKielcePoland

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