Journal of Algebraic Combinatorics

, Volume 34, Issue 2, pp 267–300

# 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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