Radon’s theorem asserts that any set P of at least d+2 points in R^d can be partitioned into two parts whose convex hulls intersect. In this talk, I will present our recent result that settles a long-standing open problem posed by Gil Kalai in the 1970s. We obtain a near-tight estimate, for every dimension d and integer s≥1, the smallest number f(d,s) such that any set of f(d,s) points in R^d admits a partition P=A∪B with the property that any union of s convex sets containing A must intersect any union of s convex sets containing B. The classical Radon theorem corresponds to the case s=1 and is equivalent to the fact that f(d,1)=d+2 . We also extend our framework to a Tverberg-type theorem for unions of convex sets. Our proofs are combinatorial and our bounds extend to so-called abstract convexity spaces with certain separability conditions.

Joint work with Noga Alon.