Abstrakt
The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C in R^(d) of cardinality (d+1)t, partitioned into t-point subsets C_1,C_2,...,C_(d+1) (which we think of as color classes; e.g., the points of C_1 are red, the points of C_2 blue, etc.), there exist r disjoint sets R_1, R_2, ... ,R_r subset of C that are rainbow, meaning that the size of the intersection of R_i and C_j is at most 1 for every i, j, and whose convex hulls all have a common point. All known proofs of this theorem are topological.
We present a geometric version of a recent beautiful proof by Blagojevic, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof more concrete and intuitive, and accessible to a wider audience.