On a fragment of the universal Baire property for $\Sigma ^1_2$ sets

Abstract:

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