We show that under the assumptions of the P-ideal dichotomy and that the bounding number b is larger than ω1, every Banach space of density ω1 with weak* sequentially compact dual ball has a quotient of density ω1 with a Schauder basis. Together with an example of Todorcevic, it follows that under the PID, b=ω1 is equivalent to the existence of a nonseparable Asplund space with no uncountable biorthogonal systems.

This is a joint work with S. Todorcevic.