Colored Tverberg Theorems for Non-prime Powers

Abstract

We prove a relative of both the original and the optimal (Type B) version of the Colored Tverberg theorem of Živaljević and Vrećica (Theorems 2.2 and 2.3), which modifies these results in two different ways. (1) We extend the original theorems beyond the prime powers by showing that the theorem is valid if the number of rainbow faces is q=p^n-1. (2) The size of some rainbow simplices may be smaller than in the original theorems. More precisely |C_i|∈{2q-2,2q+1} while (for comparison) in the original theorems it is |C_i|=2q-1. The proof relies on equivariant index theory and a result of Volovikov [17] about partial coincidences of maps f:X→ℝ^d, from a G-space into the Euclidean space.