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Colouring polytopic partitions in d

Michal Křížek (2002)

Mathematica Bohemica

We consider face-to-face partitions of bounded polytopes into convex polytopes in d for arbitrary d 1 and examine their colourability. In particular, we prove that the chromatic number of any simplicial partition does not exceed d + 1 . Partitions of polyhedra in 3 into pentahedra and hexahedra are 5 - and 6 -colourable, respectively. We show that the above numbers are attainable, i.e., in general, they cannot be reduced.

Combinatorial and group-theoretic compactifications of buildings

Pierre-Emmanuel Caprace, Jean Lécureux (2011)

Annales de l’institut Fourier

Let X be a building of arbitrary type. A compactification 𝒞 sph ( X ) of the set Res sph ( X ) of spherical residues of X is introduced. We prove that it coincides with the horofunction compactification of Res sph ( X ) endowed with a natural combinatorial distance which we call the root-distance. Points of 𝒞 sph ( X ) admit amenable stabilisers in Aut ( X ) and conversely, any amenable subgroup virtually fixes a point in 𝒞 sph ( X ) . In addition, it is shown that, provided Aut ( X ) is transitive enough, this compactification also coincides with the group-theoretic...

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