Optimal bounds for a colorful Tverberg-Vrećica type problem

Abstract

We prove the following optimal colorful Tverberg-Vrećica type transversal theorem: For prime r and for any k+1 colored collections of points Cl in Rd,Cl= Cili |Cli|=(r-1)(d-k+1)+1,|Cli| ≤ r-1, l=0...,k, there are partitions of the collections Cl into colorful sets F1l...,Frl such that there is a k-plane that meets all the convex hulls conv(Flj),under the assumption that r(d-k) is even or k=0 Along the proof we obtain three results of independent interest: We present two alternative proofs for the special case k=0 (our optimal colored Tverberg theorem (2009) [2]), calculate the cohomological index for joins of chessboard complexes, and establish a new Borsuk-Ulam type theorem for.(Zp)m-equivariant bundles that generalizes results of Volovikov (1996) [17] and Živaljević(1999)[21].