In descriptive set theory, a set ${\displaystyle S}$ is said to be homogeneously Suslin if it is the projection of a homogeneous tree. ${\displaystyle S}$ is said to be ${\displaystyle \kappa }$-homogeneously Suslin if it is the projection of a ${\displaystyle \kappa }$-homogeneous tree.

If ${\displaystyle A\subseteq {}^{\omega }\omega }$ is a ${\displaystyle \mathbf {\Pi } _{1}^{1}}$ set and ${\displaystyle \kappa }$ is a measurable cardinal, then ${\displaystyle A}$ is ${\displaystyle \kappa }$-homogeneously Suslin. This result is important in the proof that the existence of a measurable cardinal implies that ${\displaystyle \mathbf {\Pi } _{1}^{1}}$ sets are determined.