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