If $\kappa>\omega_1$ is a regular cardinal, $FP_\kappa$ is the assertion that every stationary subset of $\kappa$ consisting of ordinals of countable cofinality has a closed subset of order type $\omega_1$.

Question: If $\kappa$ is supercompact or has a supercompact cardinal below it, is it consistent that $FP_\kappa$ fails?

For context: I'm trying to show that a particular strengthening of PFA does not imply MM. Friedman's property (which under MM holds at almost all regular cardinals) seems like a natural candidate for separating them. My understanding is that the standard way of forcing a failure of FP is to add a nonreflecting stationary set; unfortunately, this preserves neither my axiom nor supercompactness. It is not too hard to show that if $FP_\kappa$ fails at a supercompact cardinal $\kappa$ (or at some larger cardinal), it still holds after performing the iteration to force the axiom, but I'm having trouble showing that that premise is consistent.

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    $\begingroup$ Problem 72 in Friedman's "One Hundred and Two Problems in Mathematical Logic" states "If $\kappa$ is a measurable cardinal then every subset of $\kappa$ either contains or is disjoint from a closed subset of $\kappa$ ordertype $\omega_1$". So if a solution exists it might shed light on your problem or the problem has been open for 50 years. $\endgroup$ Jan 20 at 14:06


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