We propose a new proof technique that applies to the same problems as the  Lovász 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 chromatic numbers to the maximal degree of a graph. It seems that there should be other interesting applications of the presented approach. In terms of upper-bounds our approach seems to be as strong as entropy-compression, but the proofs are more elementary and shorter. The applications 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 chromatic number, a $4.25\Delta +o(\Delta)$ upper-bound on the weak total non-repetitive chromatic number, and a $ \Delta^2+\frac{3}{2^{1/3}}\Delta^{5/3}+ o(\Delta^{5/3})$ upper-bound on the total non-repetitive chromatic number of graphs. This last result implies the same upper-bound for the non-repetitive chromatic index of graphs, which improves the best known bound.

Monadic second order logic can be used to express many classical notions of sets of vertices of a graph as for instance: dominating sets, induced matchings, perfect codes, independent sets or irredundant sets. Bounds on the number of sets of any such family of sets are interesting from a combinatorial point of view and have algorithmic applications. Many such bounds on different families of sets over different classes of graphs are already provided in the literature. In particular, Rote recently showed that the number of minimal dominating sets in trees of order n is at most 95 n 13 and that this bound is asymptotically sharp up to a multiplicative constant. We build on his work to show that what he did for minimal dominating sets can be done for any family of sets definable by a monadic second order formula.We first show that, for any monadic second order formula over graphs that characterizes a given kind of subset of its vertices, the maximal number of such sets in a tree can be expressed as the growth rate of a bilinear system. This mostly relies on well known links between monadic second order logic over trees and tree automata and basic tree automata manipulations. Then we show that this "growth rate" of a bilinear system can be approximated from above. We then use our implementation of this result to provide bounds (some sharp and some almost sharp) on the number of independent dominating sets, total perfect dominating sets, induced matchings, maximal induced matchings, minimal perfect dominating sets, perfect codes and maximal irredundant sets on trees. We also solve a question from D. Y. Kang et al. regarding r-matchings and obtain a sharp upper-bound on the number of maximal matchings on trees. Remark that this approach is easily generalizable to graphs of bounded tree width or clique width (or any similar class of graphs where tree automata are meaningful).

We study the avoidability of long k-abelian-squares and k-abeliancubes on binary and ternary alphabets. For k = 1, these are Mäkelä's questions. We show that one cannot avoid abelian-cubes of abelian period at least 2 in infinite binary words, and therefore answering negatively one question from Mäkelä. Then we show that one can avoid 3-abelian-squares of period at least 3 in infinite binary words and 2-abelian-squares of period at least 2 in infinite ternary words. Finally we study the minimum number of distinct k-abelian-squares that must appear in an infinite binary word.

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