Galvin-Hajnal rank of an ordinal function $f: A \to Ord$ with respect to $I$ is defined as
$|f|_I=\sup_{g <_I f} (|g|_I+1)$
where $I$ is an ideal of $A$ and $g <_I f$ iff $\{a \in A: f(a) \leq g(a)\} \in I$.
What are the Galvin-Hajnal ranks with respect to the ideal of bounded subsets of $\omega_1$ of the functions $f,g: \omega_1 \to \omega_1$, $f(\alpha)=\omega$ and $g(\alpha)=\omega+1$ if $\alpha$ is $0$ or a limit ordinal, and $f(\alpha)=\omega+1$ and $g(\alpha)=\omega$ if $\alpha$ is a successor ordinal?
In comparison, I think that with respect to the ideal of nonstationary subsets the ranks are $|f|_I=\omega$ and $|g|_I=\omega+1$ because the subset of limit ordinals is club in $\omega_1$.
Also, is it possible that $|h|_I > \alpha$ for some ordinal $\alpha$ and at the same time $\{a \in A : h(a) \leq \alpha\} \not\in I$?