The Polychromatic Number of Small Subsets of the Integers Modulo $n$

TL;DR

This paper determines the maximum number of colors in a coloring of integers modulo n such that every translate of small subsets (size 2 or 3) contains all colors, providing exact polychromatic numbers for these sets.

Contribution

It explicitly computes the polychromatic number for all small subsets of size 2 or 3 in the cyclic group of integers modulo n, filling a gap in combinatorial coloring theory.

Findings

Exact polychromatic numbers for all size 2 subsets.

Exact polychromatic numbers for all size 3 subsets.

Results applicable to combinatorial coloring problems in cyclic groups.

Abstract

If $S$ is a subset of an abelian group $G$, the polychromatic number of $S$ in $G$ is the largest integer $k$ so that there is a $k-$coloring of the elements of $G$ such that every translate of $S$ in $G$ gets all $k$ colors. We determine the polychromatic number of all sets of size 2 or 3 in the group of integers mod n.