In what follows, $\|\Phi\|_I$ means the Galvin-Hajnal rank of a function $\Phi:\kappa \to Ord$ with respect to a $\omega_1$-complete ideal $I$ of $\kappa$.
Lemma 2.2.5 of Introduction to Cardinal Arithmetic by Holz, Steffens and Weitz says:
Assume that $\kappa > \omega$ is a regular cardinal, $\Phi \in \,^\kappa Ord$, and $I$ is a $\kappa$-complete normal ideal on $\kappa$ with $\bigcup I=\kappa$. Then $\|\Phi\|_I \leq \sigma \Leftrightarrow \{\xi < \kappa : \Phi(\xi) \leq \sigma\} \notin I$ for every ordinal $\sigma < \kappa$.
The assumption on $\kappa$-completeness cannot be dropped. But can you drop one or more of the following assumptions:
- $\kappa$ is regular,
- $\bigcup I=\kappa$ and
- $I$ is normal?