Optimal bounds for a colorful Tverberg-Vrećica type problem

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=⊎Ciℓ, |Cℓ|=(r−1)(d−k+1)+1, |Ciℓ|⩽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(Fjℓ), 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].