22 research outputs found
Tverberg's theorem with constraints
The topological Tverberg theorem claims that for any continuous map of the
(q-1)(d+1)-simplex to R^d there are q disjoint faces such that their images
have a non-empty intersection. This has been proved for affine maps, and if
is a prime power, but not in general.
We extend the topological Tverberg theorem in the following way: Pairs of
vertices are forced to end up in different faces. This leads to the concept of
constraint graphs. In Tverberg's theorem with constraints, we come up with a
list of constraints graphs for the topological Tverberg theorem.
The proof is based on connectivity results of chessboard-type complexes.
Moreover, Tverberg's theorem with constraints implies new lower bounds for the
number of Tverberg partitions. As a consequence, we prove Sierksma's conjecture
for , and .Comment: 16 pages, 12 figures. Accepted for publication in JCTA. Substantial
revision due to the referee
A Tverberg type theorem for matroids
Let b(M) denote the maximal number of disjoint bases in a matroid M. It is
shown that if M is a matroid of rank d+1, then for any continuous map f from
the matroidal complex M into the d-dimensional Euclidean space there exist t
\geq \sqrt{b(M)}/4 disjoint independent sets \sigma_1,\ldots,\sigma_t \in M
such that \bigcap_{i=1}^t f(\sigma_i) \neq \emptyset.Comment: This article is due to be published in the collection of papers "A
Journey through Discrete Mathematics. A Tribute to Jiri Matousek" edited by
Martin Loebl, Jaroslav Nesetril and Robin Thomas, due to be published by
Springe
Geometric and Algebraic Combinatorics
The 2015 Oberwolfach meeting “Geometric and Algebraic Combinatorics” was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) counterexamples to the topological Tverberg conjecture, and (2) the latest results around the Heron-Rota-Welsh conjecture