Recent zbMATH articles in MSC 05D40 https://www.zbmath.org/atom/cc/05D40 2021-04-16T16:22:00+00:00 Werkzeug A short proof of Bernoulli disjointness via the local lemma. https://www.zbmath.org/1456.37041 2021-04-16T16:22:00+00:00 "Bernshteyn, Anton" https://www.zbmath.org/authors/?q=ai:bernshteyn.anton Summary: Recently, \textit{E. Glasner} and al. [Bernoulli disjointness, \url{https://arxiv.org/pdf/1901.03406.pdf}] showed that if $$\Gamma$$ is an infinite discrete group, then every minimal $$\Gamma$$-flow is disjoint from the Bernoulli shift $$2^\Gamma$$. Their proof is somewhat involved; in particular, it invokes separate arguments for different classes of groups. In this note, we give a short and self-contained proof of their result using purely combinatorial methods applicable to all groups at once. Our proof relies on the Lovász Local Lemma, an important tool in probabilistic combinatorics that has recently found several applications in the study of dynamical systems. A remark on Hamilton cycles with few colors. https://www.zbmath.org/1456.05052 2021-04-16T16:22:00+00:00 "Balla, Ior" https://www.zbmath.org/authors/?q=ai:balla.ior "Pokrovskiy, Alexey" https://www.zbmath.org/authors/?q=ai:pokrovskiy.alexey "Sudakov, Benny" https://www.zbmath.org/authors/?q=ai:sudakov.benny Summary: \textit{S. Akbari} et al. [Australas. J. Comb. 37, 33--42 (2007; Zbl 1130.05024)] conjectured that every proper edge-coloring of $$K_n$$ with $$n$$ colors contains a Hamilton cycle with $$\le O(\log n)$$ colors. They proved that there is always a Hamilton cycle with $$\le 8\sqrt{n}$$ colors. In this note we improve this bound to $$O(\log^3 n)$$.