 Polychromatic coloring of integer translates

For finite $S\subset \mathbb{Z}$, use vertices $\mathbb{Z}$ and hyperedges $S+t$. Independent uniform $k$-coloring gives missing-color probability at most $k{e}^{-|S|/k}$, and the dependency graph of events has degree at most $|S|(|S|-1)$. For $k\ge 20$ and $|S|\ge 6k\log k$, the symmetric Lovász local lemma condition holds. The compactness extension of the Lovász local lemma gives a coloring in which every translate contains every color.