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.

如果没有两个相邻的间隔具有相同的颜色顺序，则整数的着色是非重复的。Thue的一个美丽定理断言，仅使用三种颜色存在\（{\ mathbb {N}} \）的非重复着色。我们得到了该结果的一些概括，其中间隔的邻接由更一般的图指定。我们关注问题的列表变体，其中每个整数都从其自己的颜色集中获取颜色。例如，我们证明存在大小为8的任意列表中\（{\ mathbb {N}} \）的着色，从而对于每个\（n \ ge 1 \）都具有以下属性：在任何\（ 2 ^ n \）个连续相邻的间隔，每个间隔n，没有两个具有相同的颜色顺序。另一个结果与著名的Dejean猜想可能扩展到列表设置有关。它断言，对于每个\（k \ ge 1 \），大小为\（k + 2 \ sqrt {k} \）的列表都有\（{\ mathbb {N}} \）的着色，因此没有在任何k个连续相邻的间隔中，有两个具有相同的颜色顺序。