Tverberg's theorem with constraints

2007/04/20 by Stephan Hell, Hell, Stephan
Mathematics · #05A18 #52A37 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A18 #msc:52A37

paper · pdf · doi:10.48550/arxiv.0704.2713

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

arxiv created 2008/02/25 · arxiv updated 2009/12/01

Abstract

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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