Results that are widely accepted but no proof has appeared The background of this question is the talk given by Kevin Buzzard.
I could not find the slides of that talk. The slides of another talk given by Kevin Buzzard along the same theme are available here.
One of the points in the talk is that, people accept some results but whose proofs are not publicly available. (He says this leads to wrong conclusions, but, I am not interested in wrong conclusions as of now. All I am interested is are results which are accepted as true but without a detailed proof, or with only a partial proof.)

What are results that are widely accepted to be true with no detailed proof, or only a partial proof?

I am looking for situations where $A$ has asserted in print that he/she has a proof of $X$, but hasn't published a proof of $X$, and then $B$ publishes a proof of $Y$, where the proof depends on the validity of $X$. For example as in page 20,21,22 of the slides mentioned above.
Edit: Please give reference for the following: 


*

*Where the result is announced?

*Where the result is used?


Edit (made after Per Alexandersson's answer) : I am not looking for "readily available but not formally published". As mentioned by Timothy Chow, "there are many more examples if "readily available but not formally published" counts.". 
 A: A long time ago, M. Ajtai, J. Komlós, M. Simonovits, and E. Szemerédi announced a proof (for large $k$) of the Erdős–Sós conjecture that every graph with average degree
more than $k-1$ contains all trees with $k$ edges as subgraphs, but the proof has not yet appeared as of this writing (2022).
What do I mean by "a long time ago"?  Reinhard Diestel, in the notes to Chapter 7 of Graph Theory (5th edition), gives a date of 2009.  But Václav Rozhoň, in A local approach to the Erdős-Sós conjecture, says that the result was announced in the early 1990's.

EDIT: I found another reference, Local and mean Ramsey numbers for trees, by B. Bollobás, A. Kostochka, and R. H. Schelp (J. Combin. Theory Ser. B 79 (2000), 100–103), which says, "It was
announced recently that M. Ajtai, J. Komlós, and E. Szemerédi confirmed
the Erd&odblac;s–Sós conjecture for sufficiently large trees."
A: I'm going to interpret this as a request for examples of results that were announced a while ago but whose proofs have not yet appeared.  In other words, people don't doubt that the result is correct and that the author(s) can prove it, and there is an expectation that the current lack of a proof will not be a permanent state of affairs (i.e., a paper with the proof will be written and made public eventually).
One example of this is Rota's Conjecture on excluded minor characterizations of matroids representable over a given finite field.  This was announced in 2014 by Geelen, Gerards, and Whittle, but apart from the sketch in that Notices article, no further details have yet appeared.
EDIT: An example of a paper that cites this unpublished work, and relies on it in an essential way, is The matroid secretary problem for minor-closed classes and random matroids by Tony Huynh and Peter Nelson.  After stating Theorem 2, Huynh and Nelson write:

To be forthright with the reader, we stress that Theorem 2 relies on a structural hypothesis communicated to us by Geelen, Gerards, and Whittle, which has not yet appeared in print. This hypothesis is stated as Hypothesis 1. The proof of Hypothesis 1 will stretch to hundreds of pages, and will be a consequence of their decade-plus ‘matroid
minors project’. This is a body of work generalising Robertson and Seymour’s graph minors structure theorem  to matroids representable over a ﬁxed ﬁnite ﬁeld, leading to a solution of Rota’s Conjecture.

Another example is On the existence of asymptotically good linear codes in minor-closed classes by Peter Nelson and Stefan H. M. van Zwam, IEEE Trans. Info. Theory 61 (2015), 1153–1158.  The results of Nelson and van Zwamm have in turn been used in an essential way to prove Theorem 1.4 of Girth conditions and Rota's basis conjecture by Benjamin Friedman and Sean McGuinness.
A: The comments by Monroe Eskew and Andrés E. Caicedo concerning unpublished results of Hugh Woodin deserve to be made into an answer IMO. As a concrete example, Caicedo wrote:

There is the fact that Turing determinacy implies Suslin-coSuslin determinacy (in the presence of DC?), which gives L(R)-determinacy.

There are various other results by Woodin that may or may not fit the bill; in many (though maybe not all) cases, proofs have been provided by other authors.  For more details, see Woodin's unpublished proof of the global failure of GCH and Unpublished works of Woodin on SCH and Radin forcing.
A: Well, in some sense the Classification of Finite Simple Groups is in this state. It most certainly satisfies your second requirement: lots of papers have been published which rely on CFSG. However, a complete proof is (at least in some sense) still work in progress by Lyons, Solomon, Ashbacher, Smith and others. 
A: The Schur positivity of LLT polynomials by I. Grojnowski and M. Haiman is widely accepted in the community of algebraic combinatorics, but their preprint has not been published.
It is still a major open problem to give a combinatorial formula for the coefficients in the Schur expansion, which is manifestly positive.
A: I think one example is given in this MO question of mine: a quartic in $\mathbb{P}^3$ with at worst Du Val singularities is a K3 surface (and similar statements for two types of complete intersections in higher-dimensional projective spaces). 
Using the excellent answer and comments I was able to piece together a proof, but I could not locate one in the literature, whereas of course the result was "well-known to experts" (to such an extent that I even felt embarrassed for asking about the proof in the first place).
A: The proof of the theorem of MacPherson that functors out of the exit path category are equivalent to constructible sheaves was not written down, just claimed. Others have since given much more general theorems, but whose reduction to MacPherson's result is not immediate.
A: I just realized that the OP links to a YouTube video and to some slides, but the two don't match—they're two different talks by Buzzard.
For completeness, let me therefore mention some results by James Arthur, which are mentioned in the linked slides but not the linked YouTube video.
On page 13 of Abelian Surfaces over totally real fields are Potentially Modular by George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni, there is the following remark.

It should be noted that we use Arthur’s multiplicity formula for the discrete spectrum of GSp4, as announced in [Art04]. A proof of this (relying on Arthur’s work for symplectic and orthogonal groups in [Art13]) was given in [GT18], but this proof is only as unconditional as the results of [Art13] and [MW16a, MW16b]. In particular, it depends on cases of the twisted weighted fundamental lemma that were announced in [CL10], but whose proofs have not yet appeared, as well as on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time of writing have not appeared publicly.

Arthur's (unavailable) references [A24] through [A27] are:
[A24] Endoscopy and singular invariant distributions, in preparation.
[A25] Duality, Endoscopy, and Hecke operators, in preparation.
[A26] A nontempered intertwining relation for $GL(N)$, in preparation.
[A27] Transfer factors and Whittaker models, in preparation.
A: In 1999, Robertson, Sanders, Seymour, and Thomas announced a proof of Tutte's "snark conjecture" (that every snark has a Petersen graph minor), but as far as I am aware the full proof still has not appeared: see this MO question. I don't know if this result has ever been applied anywhere, though. The proof was announced in "Recent Excluded Minor Theorems for Graphs" by Thomas (available as a preprint online here; with citation information at MR1725004): see Theorem 10.2 of that paper specifically. More information about the status of these results seems available on Thomas's webpage.
