TITLE: Failure of diamond at large cardinals
QUESTION [14 upvotes]: What is known about the failure of $\Diamond_{\kappa}$ (diamond at $\kappa$) for $\kappa$ (the least) inaccessible, (the least) Mahlo and (the least) weakly compact. 
Remark. The problem of forcing the failure of diamond at a weakly compact cardinal is solved recently by Gitik's student Omer Ben-Neria (see  The failure of diamond at large cardinals and GCH). This solves a longstanding open problem.
Does anyone know the main ideas of the proof?

REPLY [10 votes]: The failure of diamond at large cardinals is an interesting and
difficult problem. Let me just mention a few references to
supplement those mentioned by Andres.
Much of the interest in diamond principles at large cardinals lies
in the case of $\Diamond_\kappa(\text{REG})$, and in this case
somewhat more is known than in the general case.

Woodin proved the consistency of the failure of
$\Diamond_\kappa(\text{REG})$ at a weakly compact cardinal.
Kai Hauser extended this result to the
case of indescribable cardinals.
Mirna Džamonja and I extended this result further to the case of
strongly unfoldable cardinals, which can be viewed as a
transfinite extensions of the indescribability hierarchy. Our
paper is M. Džamonja, J. D. Hamkins, Diamond (on the regulars) can fail at any strongly unfoldable cardinal, Ann. Pure Appl. Logic, vol. 144, iss. 1-3, pp. 83-95, 2006..

To my way of thinking, the problem is deeply connected with the
Laver diamond principle, which I introduced in A class of strong diamond principles. This develops the concept of the Laver diamond for a large
variety of large cardinal concepts, where one has a Laver function $\ell:\kappa\to V_\kappa$, such that one can obtain the Laver anticipation property $j(\ell)(\kappa)=x$ for a variety of sets $x$ with embeddings $j$ witnessing that large cardinal property. (Hellsten had around the same time developed independently the case of the Laver diamond for weakly
compact cardinals, so there is some overlap between Hellsten's work
and my paper.) One of the interesting things that Mirna and I
proved was that one can have an ordinal-anticipating Laver
function for strong unfoldability, even when there is no
set-anticipating Laver function. And there can be pseudo Laver
functions for strong unfoldability, that anticipate every ground
model set on a stationary set, but still
$\Diamond_\kappa(\text{REG})$ fails.
There are some extremely interesting open questions about this,
such as:

Can there be a $\theta$-supercompact cardinal having no
$\theta$-supercompactness Laver function?
Can there be an uplifting cardinal with no uplifting Laver
function?