Centerpole sets for colorings of Abelian groups

TL;DR

This paper investigates the minimal size of special symmetric subsets in Abelian groups that guarantee monochromatic symmetric structures under any coloring, extending combinatorial and topological understanding of such groups.

Contribution

It introduces and evaluates the cardinal characteristic $c_k(G)$ for Abelian groups, providing new bounds and exact values for the size of $k$-centerpole sets.

Findings

Calculated or estimated $c_k(G)$ for various Abelian groups.

Established bounds for the size of $k$-centerpole sets.

Connected topological group properties with combinatorial coloring results.

Abstract

Given a topological group $G$ we calculate or evaluate the cardinal characteristic $c_k(G)$ (and $c_k^B(G)$) equal to the smallest cardinality of a $k$-centerpole subset $C\subset G$ for (Borel) colorings of $G$. A subset $C\subset G$ of a topological group $G$ is called {\em $k$-centerpole} if for each (Borel) $k$-coloring of $G$ there is an unbounded monochromatic subset $G$, which is symmetric with respect to a point $c\in C$ in the sense that $S=cS^{-1}c$.