Optimal colored Tverberg theorems for prime powers

The type A colored Tverberg theorem of Blagojevic, Matschke, and Ziegler provides optimal bounds for the problem, under condition that number intersecting rainbow simplices is a prime number. We extend this result to an optimal, multisets points, which valid each power $r=p^k$. One principal new ideas replace ambient simplex $\Delta^N$, used in original theorem, by abridged smaller dimension, compensate reduction allowing vertices repeatedly appear controlled times different simplices. Configuration spaces, proof, are combinatorial pseudomanifolds can be represented as multiple chessboard complexes. Our main topological tool Eilenberg-Krasnoselskii theory degrees equivariant maps non-free actions.