arrow notation is preserved with n made smaller

Suppose I know $\kappa\to(\lambda)^n_m$ and I want to prove $\kappa\to(\lambda)^{n'}_m$ where $n'<n$. Given a partition $p$ of $[\kappa]^{n'}$ for which I want a homogeneous set of type $\lambda$, define a partition $q$ of $[\kappa]^n$ by "ignoring the last elements and using $p$". That is, two $n$-element sets are in the same $q$-class iff their initial segments of size $n'$ are in the same $p$-class. By assumption, $q$ has a homogeneous set $H$ of size $\lambda$. We can assume $H$ has order-type $\lambda$; if necessary just replace $H$ by an initial segment. In particular, the order-type of $H$ is a limit ordinal, and that suffices to ensure that $H$ is also homogeneous for $p$ (because every $n'$-element subset of $H$ is an initial segment of an $n$-element subset of $H$).