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Reading some materials on set theory, I know the generalisation of Fodor's theorem:

A filter $\cal F$ on a regular uncountable cardinal $\kappa$ is normal if and only if for every regressive function $f$ on a $\cal F$-positive set $S$, there exists an $\cal F$-positive set $S'\subseteq S$ on which $f$ is constant.

A subsequent lemma states that if follows immediately from this that

An ultrafilter $\cal U$ on a regular uncountable cardinal $\kappa$ is normal if and only if for every regressive function $f$ on a set $S\in \cal U$, there exists a set $S'\in \cal U$ on which $f$ is constant.

I do not see it as "immediate"... There are some differences in the theorem, mainly the lemma does not require the sets $S$ ans $S'$ to be $\cal U$-positive, which make the direct application of the theorem tricky. Would anyone have a detailled source for that?

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Assuming "$\mathcal{F}$-positive" means "complement is not in $\mathcal{F}$," the key point is that for an ultrafilter $\mathcal{U}$ the notions "$\mathcal{U}$-positive" and "in $\mathcal{U}$" coincide.

For any filter $\mathcal{F}$, $X\in\mathcal{F}$ implies that $X$ is $\mathcal{F}$-positive. I claim that the converse holds for ultrafilters. Suppose $\mathcal{U}$ is an ultrafilter on $\kappa$ and suppose $X$ is $\mathcal{U}$-positive. Then we must have $\kappa\setminus X\not\in\mathcal{U}$; but since $\mathcal{U}$ is an ultrafilter, this means $X\in\mathcal{U}$.

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