TITLE: Results that are widely accepted but no proof has appeared
QUESTION [75 upvotes]: 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.".

REPLY [6 votes]: 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 Erdős–Sós conjecture for sufficiently large trees."