On hyperbolic Coxeter polytopes with mutually intersecting facets
  We prove that, apart from some well-known low-dimensional examples, any compact hyperbolic Coxeter polytope has a pair of disjoint facets. This is one of very few known general results concerning combinatorics of compact hyperbolic Coxeter polytopes. We also obtain a similar result for simple non-compact polytopes. 
Anna Felikson, Pavel Tumarkin