Tverberg's theorem with constraints

Details Tverberg's theorem with constraints Tverberg's theorem with constraints

2007-04-20

arxiv.org/abs/0704.2713v2

Mathematics

Combinatorics

16 pages, 12 figures. Accepted for publication in JCTA. Substantial revision due to the referees

Scientific paper

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 $q$ 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 $d=2$, and $q=3$.

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