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We show that for any set S ⊆ Z, |S| = 4 there exists a 3-coloring of 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. We also consider related questions in Zd, d ≥ 2.
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Axenovich, Maria; Goldwasser, John; Lidicky, Bernard; Martin, Ryan R.; Offner, David; Talbot, John; and Young, Michael, "Polychromatic Colorings on the Integers" (2019). Mathematics Publications. 178.