Recall that a cardinal $\kappa$ is $(\lambda,\infty)$-almost-strongly-compact if every $\kappa$-complete filter can be refined to a $\lambda$-complete ultrafilter. A cardinal $\mu$ has the tree property if every $\mu$-sized tree with $\mu$-small levels has a branch of length $\mu$. (If in addition $\mu$ is inaccessible then $\mu$ is weakly compact.)

Question: Can the following constellation occur?

• $\mu$ -- weakly inaccessible with the tree property

• $\kappa$ -- a $(\mu^+,\infty)$-strongly-compact cardinal

• every regular $\nu \in [\mu, \kappa)$ -- has the tree property.

I suspect this would be too good to be true. But I don't know much -- for all I know, maybe almost strong compactness implies inaccessibility, in which case of course the answer is no. But I'm having trouble tracking down even that information.

If $\kappa$ can be taken to be $(\mu,\infty)$-strongly-compact, that might be good enough for what I need.

I apologize for the repeated changes to the question.

Recall that a cardinal $\kappa$ is $(\lambda,\infty)$-almost-strongly-compact if every $\kappa$-complete filter can be refined to a $\lambda$-complete ultrafilter. A cardinal $\mu$ has the tree property if every $\mu$-sized tree with $\mu$-small levels has a branch of length $\mu$. (If in addition $\mu$ is inaccessible then $\mu$ is weakly compact.)

Question: Can the following constellation occur?

• $\mu$ -- weakly inaccessible with the tree property

• $\kappa$ -- a $(\mu^+,\infty)$-strongly-compact cardinal

• every regular $\nu \in [\mu, \kappa)$ -- has the tree property.

I suspect this would be too good to be true. But I don't know much -- for all I know, maybe almost strong compactness implies inaccessibility, in which case of course the answer is no. But I'm having trouble tracking down even that information.

If $\kappa$ can be taken to be $(\mu,\infty)$-strongly-compact, that might be good enough for what I need. Also it should suffice for only the successor cardinals in $(\mu,\kappa)$ to have the tree property.

I apologize for the repeated changes to the question.

Recall that a cardinal $\kappa$ is almost strongly compact if every $\kappa$-complete filter can be refined to a $\lambda$-complete ultrafilter for any $\lambda < \kappa$. A cardinal $\mu$ has the tree property if every $\mu$-sized tree with $\mu$-small levels has a branch of length $\mu$. (If in addition $\mu$ is inaccessible then $\mu$ is weakly compact.)

Question: Can the following constellation occur?

• $\nu$ - weakly inaccessible with the tree property

• $\nu^+$ - also has the tree property

• $\nu^{++}$ - almost strongly compact

I suspect this would be too good to be true. But I don't know much -- for all I know, maybe almost strong compactness implies inaccessibility, in which case of course the answer is no. But I'm having trouble tracking down even that information.

Recall that a cardinal $\kappa$ is $(\lambda,\infty)$-almost-strongly-compact if every $\kappa$-complete filter can be refined to a $\lambda$-complete ultrafilter. A cardinal $\mu$ has the tree property if every $\mu$-sized tree with $\mu$-small levels has a branch of length $\mu$. (If in addition $\mu$ is inaccessible then $\mu$ is weakly compact.)

Question: Can the following constellation occur?

• $\mu$ -- weakly inaccessible with the tree property

• $\kappa$ -- a $(\mu^+,\infty)$-strongly-compact cardinal

• every regular $\nu \in [\mu, \kappa)$ -- has the tree property.

I suspect this would be too good to be true. But I don't know much -- for all I know, maybe almost strong compactness implies inaccessibility, in which case of course the answer is no. But I'm having trouble tracking down even that information.

If $\kappa$ can be taken to be $(\mu,\infty)$-strongly-compact, that might be good enough for what I need.

I apologize for the repeated changes to the question.

Recall that a cardinal $\kappa$ is almost strongly compact if every $\kappa$-complete filter can be refined to a $\lambda$-complete ultrafilter for any $\lambda < \kappa$. A cardinal $\mu$ has the tree property if every $\mu$-sized tree with $\mu$-small levels has a branch of length $\mu$. (If in addition $\mu$ is inaccessible then $\mu$ is weakly compact.)

Question: Can there be a cardinal $\mu$ with the tree property such that $\mu^+$ is almost strongly compact? Ideally $\mu$ would be at least weakly inaccessible.

I suspect this would be too good to be true. But I don't know much -- for all I know, maybe almost strong compactness implies inaccessibility, in which case of course the answer is no. But I'm having trouble tracking down even that information.

Recall that a cardinal $\kappa$ is almost strongly compact if every $\kappa$-complete filter can be refined to a $\lambda$-complete ultrafilter for any $\lambda < \kappa$. A cardinal $\mu$ has the tree property if every $\mu$-sized tree with $\mu$-small levels has a branch of length $\mu$. (If in addition $\mu$ is inaccessible then $\mu$ is weakly compact.)

Question: Can the following constellation occur?

• $\nu$ - weakly inaccessible with the tree property

• $\nu^+$ - also has the tree property

• $\nu^{++}$ - almost strongly compact

I suspect this would be too good to be true. But I don't know much -- for all I know, maybe almost strong compactness implies inaccessibility, in which case of course the answer is no. But I'm having trouble tracking down even that information.