Rainbow-free 3-colorings of Abelian Groups

TL;DR

This paper characterizes rainbow-free 3-colorings of abelian groups, proving a conjecture about the minimal size of a color class in cyclic groups and providing structural insights.

Contribution

It offers a structural characterization of rainbow-free colorings and confirms a conjecture regarding the minimal chromatic class size in cyclic groups.

Findings

Structural characterization of rainbow-free colorings

Proof of the conjecture on minimal chromatic class size

Insights into 3-term arithmetic progressions in abelian groups

Abstract

A 3-coloring of the elements of an abelian group is said to be rainbow--free if there is no 3-term arithmetic progression with its members having pairwise distinct colors. We give a structural characterization of rainbow--free colorings of abelian groups. This characterization proves a conjecture of Jungi\'c et al. on the size of the smallest chromatic class of a rainbow-free 3-coloring of cyclic groups.