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 universally Baire sets have either an uncountable path or an absolute impediment preventing one.