THE CONSISTENCY STRENGTH OF THE PERFECT SET PROPERTY FOR UNIVERSALLY BAIRE SETS OF REALS

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 virtually Shelah for supercompactness (VSS). These cardinals resemble Shelah cardinals and Shelah-for-supercompactness cardinals but are much weaker: if $0^\sharp $ exists then every Silver indiscernible is VSS in L. We also show that the statement $\operatorname {\mathrm {uB}} = {\boldsymbol {\Delta }}^1_2$ , where $\operatorname {\mathrm {uB}}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $\Sigma _2$ -reflecting VSS cardinal.