Noga Alon, speaking in the Institute for Advanced Study's Computer Science and Discrete Mathematics seminar, presents joint work with Shakhar Smorodinsky on an extension of VC-dimension, a concept from combinatorics and learning theory that measures the complexity of a hypergraph. Alon shows how this extended notion can be used to prove a Tverberg type theorem for unions of convex sets, a generalization of classical convexity results to collections built from multiple convex pieces rather than single points. He also presents a new Radon type theorem for unions of convex sets, resolving a question Gil Kalai posed in the 1970s. The talk moves through the definitions, the combinatorial machinery connecting VC-dimension to geometric partition theorems, and the proof ideas behind the new Radon result, aimed at an audience with background in discrete mathematics and combinatorial geometry.