Extended VC-dimension, and Radon and Tverberg type theorems for unions of convex sets

We prove a new Radon type theorem for unions of convex sets, settling an open problem posed by Kalai in the 1970s. We also define and study an extension of the notion of the VC-dimension of a hypergraph and apply it to establish an extension of our Radon type theorem to a Tverberg type theorem for unions of convex sets.