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:

1. Where the result is announced?
2. 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.".

• This question is not intended to be a debate on whether some result is true or not :) I am only looking for results whose proofs are not published.. – Praphulla Koushik Apr 13 '20 at 6:09
• I might object that a result without a proof cannot be known to be true as a matter of principle. What other method of verification do we have, other than a proof? Divine revelation? (A previous version of the question asked for results that were "believed" to be true, rather than "known" to be true, then I could generate a vast list from all theorems that assume the Riemann hypothesis.) – Carlo Beenakker Apr 13 '20 at 7:18
• I think the intent of the question would've been clearer if it asked (explicitly, especially in the title) for results which were announced to be true, but for which but whose proof has never appeared (yet). Anyway, clarifications considered, interesting question and +1 from me. – Wojowu Apr 13 '20 at 10:34
• @VladimirDotsenko If there is a result A published by some one which is used by others but there is no published proof of the result A, that will be an example for the question.. I do not see where opinion is coming here.. :) Please let me know if I am missing something.. I do not want anybody to spend time on something that is opinion based.. That is the whole point of this question.. :) Opinions are not appreciated.. evidence is appreciated.. – Praphulla Koushik Apr 13 '20 at 12:41
• Several famous results of Hugh Woodin fit the bill. – Monroe Eskew Apr 13 '20 at 14:28

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.

• Thanks for your answer... Do we know by any chance what are the reasons for no news after 2014? – Praphulla Koushik Apr 13 '20 at 16:30
• @PraphullaKoushik : I recently asked Geelen about this and he said that there's no specific reason, just that the proof is long and complicated and that the authors have had difficulty finding solid chunks of time to devote to writing it up. – Timothy Chow Apr 13 '20 at 16:40
• Ok.. I am also looking for some papers which cite this announcement and use that the result is true...If I see anything I will leave as a comment here, you can add it if you think it fits in the answer.. – Praphulla Koushik Apr 13 '20 at 16:44
• @PraphullaKoushik : I added an example. – Timothy Chow Apr 13 '20 at 17:06
• Thank you, this is now a complete answer for the question :) – Praphulla Koushik Apr 13 '20 at 17:12

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.

• This is the first example discussed in Buzzard's talk (though he doesn't go over where CFSG is applied) – Wojowu Apr 13 '20 at 21:07
• Isn't there a now complete proof after the publication of Aschbacher and Smith? The point of the book series is to have the proof in one place, and because it can be proven more efficiently once you already know the complete list of simple groups. – arsmath Apr 13 '20 at 22:48
• A complete proof exists. Just not in one place. – user6976 Apr 14 '20 at 1:28
• Well, Ron Solomon says they’re still finding gaps. I don’t think anyone expects dramatic changes, but it’s hard to argue that the proof is as solid and final as the typical published paper. – Alon Amit Apr 14 '20 at 1:58
• @MarkSapir would that contradict what Kevin Buzzard said in the slides? – Praphulla Koushik Apr 14 '20 at 3:36

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).

• This kind of known but whose proof is not published and you hesitate to ask on MO are what I find useful to Grad students :) :) Please add an outline of the proof of this result when you write your next paper. :) +1 – Praphulla Koushik Apr 14 '20 at 2:48

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.

• It's cited a lot, the massively general versions arrived decades after the original. As in, it's a key technical tool in intersection homology IIRC. – theHigherGeometer Apr 15 '20 at 10:24
• It is interesting how can some one cite a result more than once but there is no publication by the author where the result is mentioned... It must have been a very strong result :) :) – Praphulla Koushik Apr 15 '20 at 10:30
• Well, there's a number of famous examples. The Grothendieck–Riemann–Roch theorem, due to Grothendieck, appeared in a paper by Borel and Serre after being communicated in a letter. Grothendieck's proof was published about 14 years later, in SGA6. en.wikipedia.org/wiki/… – theHigherGeometer Apr 16 '20 at 0:21
• Ok. So, Grothendieck mentioned a theorem (with out proof) in a letter to Serre, what is now considered as a GRR theorem. Then, Serre and Borel ran a seminar to understand this, and published (Is it their own proof?) as “Le théorème de Riemann–Roch”.. Then, after some years Grothendieck published his first proof in the “book” SGA 6... Is it correct? – Praphulla Koushik Apr 16 '20 at 4:06
• Yes. It happened the other way around, too. For example: Grothendieck asked Serre a question, the latter proved it, then Grothendieck published the theorem and proof. This is not that unusual. It's the complete lack of proof of MacPherson's theorem by anyone that is weird, here. – theHigherGeometer Apr 16 '20 at 5:19

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.

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.

• Yes, and, indeed, sadly, this kind of top-heaviness is not good for the subject. – paul garrett May 4 '20 at 23:04
• I have edited the question. I could not find the slides of the talk in that YouTube link.. The slides are for the talk given by Kevin Buzzard along the same theme :) – Praphulla Koushik May 5 '20 at 4:09

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.

• Your answer highlights the distinction between "publicly available" and "published." My interpretation was that the proof needed to be unavailable, not just unpublished, but reading the question carefully, I see that it is ambivalent. Probably there are many more examples if "readily available but not formally published" counts. – Timothy Chow Apr 15 '20 at 13:21
• It is not clear how to respond to this answer... As mentioned by Timothy Chow, I am looking for "readily available but not formally published"... Thanks for your answer :) – Praphulla Koushik Apr 15 '20 at 13:31
• @PraphullaKoushik : You are? Or is that a typo? In most of the other answers, the proof is not readily available. – Timothy Chow Apr 15 '20 at 16:22
• @TimothyChow Sorry. It is a typo.. I am not looking for "readily available but not formally published". – Praphulla Koushik Apr 15 '20 at 16:32