Tverberg's theorem with constraints

The topological Tverberg theorem claims that for any continuous map of the (q−1)(d+1)-simplex σ(d+1)(q−1) to Rd there are q disjoint faces of σ(d+1)(q−1) 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.