Colored Tverberg Theorems for non-prime powers

We prove a relative of the Optimal (Type B)} Colored Tverberg theorem of Živaljević and Vrećica which modifies this results in two different ways. (1) Our result is valid if the number of rainbow faces is $q=p^n-1$, where $p$ is a prime. (2) The size of rainbow simplices satisfies the condition $\mathrm{\mid }{C}_i\mathrm{\mid }\in \{2q-2,2q+1\}$ while in the original theorem $\mathrm{\mid }{C}_i\mathrm{\mid }=2q-1$ for all $i$.