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We consider the hard-core lattice gas model on ${\mathbb{Z}}^d$ and investigate its phase structure in high dimensions. We prove that when the intensity parameter exceeds $C{d}^{-1/3}{(logd)}^2$, the model exhibits multiple hard-core measures, thus improving the previous bound of $C{d}^{-1/4}{(logd)}^{3/4}$ given by Galvin and Kahn. At the heart of our approach lies the study of a certain class of edge cutsets in ${\mathbb{Z}}^d$, the so-called odd cutsets, that appear naturally as the boundary between different phases in the hard-core model. We provide a refined combinatorial...

We consider a planar Poisson process and its associated Voronoi map. We show that there is a proper coloring with 6 colors of the map which is a deterministic isometry-equivariant function of the Poisson process. As part of the proof we show that the 6-core of the corresponding Delaunay triangulation is empty. Generalizations, extensions and some open questions are discussed.