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 B´ar´any et al.
(1980), by adding color constraints. It also provides an improved bound for the (topological) colored
Tverberg problem of B´ar´any & 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
