76 research outputs found
Meunier Conjecture
Fr\'ed\'eric Meunier's question about a multicolored Sperner lemma is
addressed, leaving the question of connectivity for the color hypergraphs of
such a multicolored simplex. Sperner's lemma asserts the existence of a simplex
using all the colors for any vertex coloring of a subdivision of a large
simplex with appropriate boundary conditions. Meunier's questions generalizes
this to the situation of having several such colorings and asserts the
existence of a simplex using enough different colors from each coloring
Fundamental Groups of Random Clique Complexes
Clique complexes of Erd\H{o}s-R\'{e}nyi random graphs with edge probability
between and are shown to be aas not simply
connected. This entails showing that a connected two dimensional simplicial
complex for which every subcomplex has fewer than three times as many edges as
vertices must have the homotopy type of a wedge of circles, two spheres and
real projective planes. Note that is a threshold for simple
connectivity and is one for vanishing first \F_2 homology.Comment: 7 pages statement of Theorem 1.2 correcte
Random Walks and Mixed Volumes of Hypersimplices
Below is a method for relating a mixed volume computation for polytopes
sharing many facet directions to a symmetric random walk. The example of
permutahedra and particularly hypersimplices is expanded upon.Comment: 6 page
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