My favorite is the first problem I worked on, back in 1966 when I was an undergraduate. The question is: does every non separable Banach space have an uncountable biorthogonal system? Shelah constructed a counterexample under diamond; Kunen later gave a C(K) counterexample using CH. In 2005 Stevo Todorcevic gave a positive answer when mm > aleph_1. His paper "Biorthogonal Systems and Quotient spaces via Baire Category" appeared in Math. Annalen in the last couple of years.