Polychromatic Colorings on the Integers

We show that for any set $S\subseteq \mathbb{Z}$, $\mathrm{\mid }S\mathrm{\mid }=4$ 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 $1\mathrm{/}3$, 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 ${\mathbb{Z}}^d$, $d\ge 2$.