We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.

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We prove the following optimal colorful Tverberg-Vrećica type transversal theorem: For prime r and for any k + 1 colored collections of points. . , k, there are partition of the collections C ℓ into colorful sets F ℓ 1 , . . . , F ℓ r such that there is a k-plane that meets all the convex hulls conv(F ℓ j ), 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 ( p) mequivariant bundles that generalizes results of Volovikov (1996Volovikov ( ) andŽivaljević (1999.