Nonrepetitive List Colorings of the Integers

A coloring of the integers is nonrepetitive if no two adjacent intervals have the same color sequence. A beautiful theorem of Thue asserts that there exists a nonrepetitive coloring of $${\mathbb {N}}$$ using only three colors. We obtain some generalizations of this result in which the adjacency of intervals is specified by more general graphs. We focus on the list variant of the problem, in which every integer gets a color from its own set of colors. For instance, we prove that there exists a coloring of $${\mathbb {N}}$$ from arbitrary lists of size 8, such that the following property holds for every $$n\ge 1$$ : among any $$2^n$$ consecutively adjacent intervals, each of length n, no two have the same color sequence. Another result is related to the possible extension of the famous Dejean’s conjecture to the list setting. It asserts that for every $$k\ge 1$$ , there is a coloring of $${\mathbb {N}}$$ from lists of size $$k+2\sqrt{k}$$ , such that no two among any k consecutively adjacent intervals have the same color sequence.