691 research outputs found
Polytopal complexes: maps, chain complexes and... necklaces
The notion of polytopal map between two polytopal complexes is defined.
Surprisingly, this definition is quite simple and extends naturally those of
simplicial and cubical maps. It is then possible to define an induced chain map
between the associated chain complexes. Finally, we use this new tool to give
the first combinatorial proof of the splitting necklace theorem of Alon. The
paper ends with open questions, such as the existence of Sperner's lemma for a
polytopal complex or the existence of a cubical approximation theorem.Comment: Presented at the TGGT 08 Conference, May 2008, Paris. The definition
of a polytopal map has been modifie
The chromatic number of almost stable Kneser hypergraphs
Let be the set of -subsets of such that for all
, we have We define almost -stable Kneser hypergraph
to be the
-uniform hypergraph whose vertex set is and whose edges are the
-uples of disjoint elements of .
With the help of a -Tucker lemma, we prove that, for prime and for
any , the chromatic number of almost 2-stable Kneser hypergraphs
is equal
to the chromatic number of the usual Kneser hypergraphs ,
namely that it is equal to
Defining to be the number of prime divisors of , counted with
multiplicities, this result implies that the chromatic number of almost
-stable Kneser hypergraphs is equal to the
chromatic number of the usual Kneser hypergraphs for any
, namely that it is equal to $\lceil\frac{n-(k-1)r}{r-1}\rceil.
Envy-free cake division without assuming the players prefer nonempty pieces
Consider players having preferences over the connected pieces of a cake,
identified with the interval . A classical theorem, found independently
by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it
is possible to divide the cake into connected pieces and assign these
pieces to the players in an envy-free manner, i.e, such that no player strictly
prefers a piece that has not been assigned to her. One of these conditions,
considered as crucial, is that no player is happy with an empty piece. We prove
that, even if this condition is not satisfied, it is still possible to get such
a division when is a prime number or is equal to . When is at most
, this has been previously proved by Erel Segal-Halevi, who conjectured that
the result holds for any . The main step in our proof is a new combinatorial
lemma in topology, close to a conjecture by Segal-Halevi and which is
reminiscent of the celebrated Sperner lemma: instead of restricting the labels
that can appear on each face of the simplex, the lemma considers labelings that
enjoy a certain symmetry on the boundary
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