2010/08/31 by Jiří Matoušek, Matoušek, Jiří, Martin Tancer +3 · 1 citation
Mathematics · #52A35 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:52A35
paper · pdf · doi:10.48550/arxiv.1008.5275
19 pages, 9 figures
arxiv created 2011/06/01 · arxiv updated 2011/06/02
The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C in Rd of cardinality (d+1)t, partitioned into t-point subsets C1,C2,...,Cd+1 (which we think of as color classes; e.g., the points of C1 are red, the points of C2 blue, etc.), there exist r disjoint sets R1,R2,...,Rr ⊆ C that are "rainbow", meaning that |Ri ∩ Cj| < 2 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 Blagojević, 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.