  Tverberg's theorem asserts that every (k-1)(d+1)+1 points in R^d can be
partitioned into k parts, so that the convex hulls of the parts have a common
intersection. Calder and Eckhoff asked whether there is a purely combinatorial
deduction of Tverberg's theorem from the special case k=2. We dash the hopes of
a purely combinatorial deduction, but show that the case k=2 does imply that
every set of O(k^2 log^2 k) points admits a Tverberg partition into k parts.
