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We show that for any set $S\subseteq \mathbb{Z}$, $|S|=4$ there exists a3-coloring of $\mathbb{Z}$ in which every translate of $S$ receives all threecolors. This implies that $S$ has a codensity of at most $1/3$, proving aconjecture of Newman [D. J. Newman, Complements of finite sets of integers,Michigan Math. J. 14 (1967) 481--486]. We also consider related questions in$\mathbb{Z}^d$, $d\geq 2$.

Polychromatic Colorings on the Integers