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

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.