# Optimal bounds for a colorful Tverberg--Vrecica type problem

Pavle Blagojevic, Benjamin Matschke, Gunter Ziegler

Arxiv ID: 0911.2692•Last updated: 3/25/2022

We prove the following optimal colorful Tverberg-Vrecica type transversal
theorem: For prime r and for any k+1 colored collections of points C^l of size
|C^l|=(r-1)(d-k+1)+1 in R^d, where each C^l is a union of subsets (color
classes) C_i^l of size smaller than r, l=0,...,k, there are partition of the
collections C^l into colorful sets F_1^l,...,F_r^l such that there is a k-plane
that meets all the convex hulls conv(F_j^l), 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)), calculate the cohomological index for joins of chessboard
complexes, and establish a new Borsuk-Ulam type theorem for (Z_p)^m-equivariant
bundles that generalizes results of Volovikov (1996) and Zivaljevic (1999).

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