  The colorful Helly theorem and Tverberg's theorem are fundamental results in
discrete geometry. We prove a theorem which interpolates between the two. In
particular, we show the following for any integers $d \geq m \geq 1$ and $k$ a
prime power. Suppose $F_1, F_2, \dots, F_m$ are families of convex sets in
$\mathbb{R}^d$, each of size $n > (\frac{d}{m}+1)(k-1)$, such that for any
choice $C_i\in F_i$ we have $\bigcap_{i=1}^mC_i\neq \emptyset$. Then, one of
the families $F_i$ admits a Tverberg $k$-partition. That is, one of the $F_i$
can be partitioned into $k$ nonempty parts such that the convex hulls of the
parts have nonempty intersection. As a corollary, we also obtain a result
concerning $r$-dimensional transversals to families of convex sets in
$\mathbb{R}^d$ that satisfy the colorful Helly hypothesis, which extends the
work of Karasev and Montejano.
