We propose a new proof technique that aims to be applied to the same problems as the Lov\'asz Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve upper-bounds relating different non-repetitive numbers to the maximal degree of a graph. It seems that there should be other interesting applications to the presented approach. In terms of upper-bound our approach seems to be as strong as entropy-compression, but the proofs are more elementary and shorter. The application we provide in this paper are upper bounds for graphs of maximal degree at most $\Delta$: a minor improvement on the upper-bound of the non-repetitive number, a $4.25\Delta +o(\Delta)$ upper-bound on the weak total non-repetitive number and a $ \Delta^2+\frac{3}{2^\frac{1}{3}}\Delta^{\frac{5}{3}}+ o(\Delta^{\frac{5}{3}})$ upper-bound on the total non-repetitive number of graphs. This last result implies the same upper-bound for the non-repetitive index of graphs, which improves the best known bound.

中文翻译：

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有界度图的非重复着色的另一种方法

我们提出了一种新的证明技术，旨在应用于与 Lov\'asz 局部引理或熵压缩方法相同的问题。我们在非重复着色的上下文中介绍了这种方法，并使用它来改进将不同非重复数字与图的最大程度相关联的上限。似乎所提出的方法应该有其他有趣的应用。就上限而言，我们的方法似乎与熵压缩一样强大，但证明更基本且更短。我们在本文中提供的应用是最大度数最多为 $\Delta$ 的图的上限：对非重复数的上限 $4 的微小改进。25\Delta +o(\Delta)$ 弱总非重复数的上限和 $ \Delta^2+\frac{3}{2^\frac{1}{3}}\Delta^{ \frac{5}{3}}+ o(\Delta^{\frac{5}{3}})$ 非重复图总数的上限。最后一个结果意味着图的非重复索引具有相同的上限，这改进了最佳已知边界。