Saturation, Suslin trees and meager sets

We show, using a variation of Woodin’s partial order ℙ_max, that it is possible to destroy the saturation of the nonstationary ideal on ω₁ by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ_max variations where one does not try to arrange it otherwise. In the last section, we show that it is possible to have a nonmeager set of reals of size ℵ₁, saturation of the nonstationary ideal, and no weakly Lusin sequences, answering a question of Shelah and Zapletal.Supported by the Japan Society for the Promotion of Science, the Mittag-Leffler Institute and the São Paulo State Research Support Foundation (FAPESP, Grant # 02/11551-3).