Polychromatic Colorings on the Integers.

We show that for any set $Ssubseteq mathbb{Z}$, $|S|=4$ there exists a 3-coloring of $mathbb{Z}$ in which every translate of $S$ receives all three colors. This implies that $S$ has a codensity of at most $1/3$, proving a conjecture of Newman [D. J. Newman, Complements of finite sets of integers, Michigan Math. J. 14 (1967) 481--486]. We also consider related questions in $mathbb{Z}^d$, $dgeq 2$.