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.

翻译：我们证明了一种关于凸集并集的新Radon型定理，解决了Kalai于1970年代提出的一个公开问题。同时，我们定义并研究了超图VC维数概念的扩展，并应用该扩展将我们的Radon型定理推广至凸集并集的Tverberg型定理。