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 $[{\omega }_2{]}^{{\mathrm{\aleph }}_2}$ 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 $[{\omega }_1{]}^{{\mathrm{\aleph }}_1}$ 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.