C O ] 3 A ug 2 01 8 Polychromatic Colorings on the Integers

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 [D. J. Newman, Complements of finite sets of integers, Michigan Math. J. 14 (1967) 481–486]. We also consider related questions in Zd, d ≥ 2. Disciplines Discrete Mathematics and Combinatorics | Mathematics Comments This is a manuscript made available through arxiv: https://arxiv.org/abs/1704.00042. Authors Maria Axenovich, John Goldwasser, Bernard Lidicky, Ryan R. Martin, David Offner, John Talbot, and Michael Young This article is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/math_pubs/178 ar X iv :1 70 4. 00 04 2v 2 [ m at h. C O ] 3 A ug 2 01 8 Polychromatic Colorings on the Integers Maria Axenovich John Goldwasser Bernard Lidický Ryan R. Martin § David Offner John Talbot Michael Young