On a Fragment of the Universal Baire Property for Sigma^1_2 Sets

Abstract

There is a well-known global equivalence between Σ^1_2 sets having the Universal Baire property, two-step Σ^1_3 generic absoluteness, and the closure of the universe under the sharp operation. In this note, we determine the exact consistency strength of Σ^1_2 sets being (2^ω)^{+}-cc Universally Baire, which is below 0^{#}. In a model obtained, there is a Σ^1_2 set which is weakly ω_2-Universally Baire but not ω_2-Universally Baire.