Absoluteness for universally Baire sets and the uncountable II

Abstract. Using ♦ and large cardinals we extend results of Magidor–Malitz and Farah–Larson to obtain models correct for the existence of uncountable homogeneous sets for finite-dimensional partitions and universally Baire sets. Furthermore, we show that the constructions in this paper and its predecessor can be modified to produce a family of 2ω1-many such models so that no two have a stationary, costationary subset of ω1 in common. Finally, we extend a result of Steel to show that trees on reals of height ω1 which are coded by uni-versally Baire sets have either an uncountable path or an absolute impediment preventing one. In [4] it was shown (using large cardinals) that if a model of a theory T satisfying a certain second-order property P can be forced to exist, then a model of T satisfying P exists already. The properties P considered in [4] included the following. (1) Containing any specified set of ℵ1-many reals. (2) Correctness about NSω1. (3) Correctness about any given universally Baire set of reals (with a predicat