  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 $q=3$.
