Homogeneously Suslin sets in tame mice

Abstract

This paper studies homogeneously Suslin (hom) sets of reals in tame mice. The following results are established: In $0$^¶ the hom sets are precisely the $\underset{\widetilde{}}{\Pi^1_1}$ sets. In $M_n$ every hom set is correctly $\underset{\widetilde{}}{\Delta^1_{n+1}}$, and $(\delta+1)$-universally Baire where $\delta$ is the least Woodin. In $M_\omega$ every hom set is $< \lambda$-hom, where $\lambda$ is the supremum of the Woodins.