In the 1970’s Erdos asked several questions about what kind of infinite structures can be found in every set of natural numbers with positive density. In recent joint work with Kra, Richter and Robertson we proved that every such set A can be shifted to contain a (restricted) sumset B+B for some infinite subset B of A. The proof involves a modification of the "correspondence principle" of Furstenberg to transfer the problem into a dynamical statement, followed by an analysis of the ergodic decomposition of self-products.