Colored Tverberg problem, extensions and new results

Abstract

We prove a "multiple colored Tverberg theorem" and a "balanced colored Tverberg theorem", by applying different methods, tools and ideas. The proof of the first theorem uses multiple chessboard complexes (as configuration spaces) and Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions. The proof of the second result relies on high connectivity of the configuration space, established by discrete Morse theory.1.1 Brief overview of Tverberg's theorem and its relatives "Tverberg type theorems" is a common name for a growing family of theorems, conjectures and problems about special partitions (patterns) of finite sets of points (point clouds) in the affine euclidean space R d .