Radon partitions in convexity spaces

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Tverberg's theorem asserts that every (k-1)(d+1)+1 points in Rᵈ 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(k² log² k) points admits a Tverberg partition into k parts.