Tverberg-type theorems are a generalization of the theory of graph drawings to higher-dimensional cell complexes and spaces. In contrast to graphs one has to distinguish between results for affine maps (with straight faces) and continuous maps: In the topological case number-theoretic conditions on the multiplicity of intersections play a role. We will show that most Tverberg-type results, which were believed to require proofs using involved techniques from algebraic topology, follow from a simple combinatorial reduction via the pigeonhole principle.

Joint work with Pavle Blagojevic and Günter M. Ziegler.