Abstract
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 (ℤm) if G is an abelian group of free rank m≥k. Also we prove that c 1(ℤn+1)=1, c 2(ℤn+2)=3, c 3(ℤn+3)=6, 8≤c 4(ℤn+4)≤c 4(ℤ4)=12 for all n∈ω, 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.
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Banakh, T., Chervak, O. Centerpole sets for colorings of abelian groups. J Algebr Comb 34, 267–300 (2011). https://doi.org/10.1007/s10801-010-0271-3
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DOI: https://doi.org/10.1007/s10801-010-0271-3