The relation between the possible sizes of a biorthogonal system in a Banach space and its density is not completely understood. In this talk, we discuss this problem and present a result stating that, under the P-ideal dichotomy, the existence of uncountable biorthogonal systems in all nonseparable Banach spaces of a certain class is equivalent to the cardinal assumption $\mathfrak{b} >\omega_1$. This is a joint work with Stevo Todorcevic.