Radon partitions in convexity spaces

Understand

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.