Centerpole sets for colorings of Abelian groups

Banakh, Taras ; Chervak, Ostap

Journal of Algebraic Combinatorics, Tome 34 (2011) no. 2, pp. 267-300.

Résumé

Summary: A subset $C\subset G$ 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 $c\in C$ 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 $c _{ k }( G)= c _{ k }(\Bbb Z ^{ m })$ if $G$ is an abelian group of free rank $m\geq k$. Also we prove that $c _{1}(\Bbb Z ^{ n+1})=1, c _{2}(\Bbb Z ^{ n+2})=3, c _{3}(\Bbb Z ^{ n+3})=6, 8\leq c _{4}(\Bbb Z ^{ n+4})\leq c _{4}(\Bbb Z ^{4})=12$ for all $n\in \omega $, and $\frac12( k ^{2}+3 k -4) \sterling c _{ k}(\mathbb Z ^{ n}) \sterling 2 ^{ k} -1$ -max $_{ s \sterling k -2}\binom k -1 s -1$ frac12(k^2+3k-4)$\le $c_k(mathbbZ^n)le2^k-1-max_s$\le $k-2$\binom $k-1s-1 for all $n\geq k\geq 4$.

Détail

Keywords: keywords abelian group, centerpole set, coloring, symmetric subset, monochromatic subset