On strong chains of sets and functions

Shelah has shown that there are no chains of length $\omega_3$ increasing modulo finite in ${}^{\omega_2}\omega_2$. We improve this result to sets. That is, we show that there are no chains of length $\omega_3$ in $[\omega_2]^{\aleph_2}$ increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length $\omega_2$ increasing modulo finite in $[\omega_1]^{\aleph_1}$ as well as in ${}^{\omega_1}\omega_1$. More generally, we study the depth of function spaces ${}^\kappa\mu$ quotiented by the ideal $[\kappa]^{< \theta}$ where $\theta< \kappa$ are infinite cardinals.