The consistency strength of the perfect set property for universally Baire sets of reals

Abstract

We show that the statement "every universally Baire set of reals has the perfect set property" is equiconsistent modulo ZFC with the existence of a cardinal that we call a virtually Shelah cardinal. These cardinals resemble Shelah cardinals but are much weaker: if $0^\sharp$ exists then every Silver indiscernible is virtually Shelah in $L$. We also show that the statement $\text{uB} = {\bf\Delta}^1_2$, where $\text{uB}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $\Sigma_2$-reflecting virtually Shelah cardinal.