What is the consistency strength of the following statement:



> 
> $\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$ where $\theta> \kappa$ is some fixed inaccessible cardinal.
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> 


**Remark 1.** By Laver's result, the above statement is consistent relative to the existence of a supercompact cardinal (and an inaccessible above it).


**Remark 2.** By results of Mitchell, Schimmerling and Steel, ``the covering lemma up to a Woodin cardinal'', if $\kappa$ remains measurable after collapsing $\kappa^+$ to $\kappa,$
then there is an inner model with a Woodin cardinal.


**Remark 3.** Apter's paper ``Strong cardinals can be fully indestructible'' have some positive consistency results related to the question.



Consistency strength of being strong cardinal and indestructible under collapses
Apter and I proved that the only way to force this situation by
any forcing that resembles anything like the Laver preparation is
to begin with a supercompact cardinal (or a partially supercompact
cardinal for $\theta$ fixed). Specifically, in our article [A. W.
Apter and J. D. Hamkins, “Indestructible weakly compact cardinals
and the necessity of supercompactness for certain proof schemata,”
MLQ Math. Log. Q., vol. 47, iss. 4, pp. 563-571,
2001](http://jdh.hamkins.org/indestructiblewc/), we prove:


**Theorem 2.** If after forcing with a closure point below
$\kappa$ the weak compactness of $\kappa$ becomes indestructible
by the forcing to collapse cardinals to $\kappa$, then $\kappa$
was supercompact in the ground model.


**Theorem 3.** Suppose $\kappa$ is weakly compact in a forcing
extension that admits a closure point below $\kappa$ and that
collapses $(2^{\theta^{<\kappa}})^V$ to $\kappa$. Then $\kappa$ was
$\theta$-supercompact in $V$.


The proof really uses only that the universe has the
$\delta$-approximation and cover properties over the ground model,
for some $\delta<\kappa$.


This result does not settle the exact consistency strength,
because of the assumption on the forcing, but provides fairly
strong evidence that a supercompact cardinal is required for
arbitrary collapses and a $\theta$-supercompact is required for
collapsing $2^{\theta^{<\kappa}}$.


Meanwhile, as in your remark 2, I am given to understand from the inner model theory
experts that having a weakly compact cardinal that is
indestructible by collapse forcing to $\kappa$ outstrips the inner
model theory, going past Woodin cardinals and more. But we'll have
to hear the details from the inner model theory experts.


