Monochromatic generating sets in groups and other algebraic structures

TL;DR

This paper introduces the concept of the generating chromatic number in groups, characterizes it for various classes, and explores its applications to vector spaces and fields, revealing differences between nilpotent and solvable groups.

Contribution

It provides new characterizations of the generating chromatic number for groups, especially nilpotent ones, and extends the concept to algebraic structures like vector spaces and fields.

Findings

Characterization of groups with infinite generating chromatic number

Complete characterization for nilpotent groups' chromatic numbers

Examples showing non-generalizability to solvable groups

Abstract

The \emph{generating chromatic number} of a group $G$, $\chigen(G)$, is the maximum number of colors $k$ such that there is a monochromatic generating set for each coloring of the elements of $G$ in $k$ colors. If no such maximal $k$ exists, we set $\chigen(G)=\infty$. Equivalently, $\chigen(G)$ is the maximal number $k$ such that there is no cover of $G$ by proper subgroups ($\infty$ if there is no such maximal $k$). We provide characterizations, for arbitrary gruops, in the cases $\chigen(G)=\infty$ and $\chigen(G)=2$. For nilpotent groups (in particular, for abelian ones), all possible chromatic numbers are characterized. Examples show that the characterization for nilpotent groups do not generalize to arbitrary solvable groups. We conclude with applications to vector spaces and fields.