Centerpole sets for colorings of abelian groups

A subset C?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??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 (? ^m ) if G is an abelian group of free rank m??k. Also we prove that c ₁(? ⁿ⁺¹)=1, c ₂(? ⁿ⁺²)=3, c ₃(? ⁿ⁺³)=6, 8??c ₄(? ⁿ⁺⁴)??c ₄(?⁴)=12 for all n????, 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 n??k??4.