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December 27

On two (supposedly solved) problems

I originally posted both of these questions on the talk page for list of unsolved problems in mathematics and was kindly advised that they might get a good answer more swiftly here; I apologize in advance if this is formatted improperly or not the right way to ask, but I had these questions:

1. A counterexample of the Vaught conjecture was announced by Robin Knight in 2002; however, it is also stated that said counterexample has not yet been verified as of this year. As such I was wondering if it is widely believed that the counterexample is valid or not? A quick search online indicates that at least some people believe that the counterexample/disproof is invalid, but I would need to get a second opinion before touching the listed status of the problem.

2. The Sidon set problem is listed on the unsolved problems page as having been solved by J. Cilleruelo, I. Ruzsa, and C. Vinuesa in 2010, but the result is not mentioned in the problem's main article and there is no indication on either article of whether or not the problem solved in 2010 is indeed the "main" Sidon sequence problem. Can anyone confirm what the specific result proved is and whether or not the Sidon sequence article needs to be updated?

Cheers, GalacticShoe (talk) 23:28, 27 December 2021 (UTC)[reply]

No one seems to be able to follow the purported counterexample to the Vaught conjecture, and my impression is that the general attitude is skeptical. I think Julia Knight (no relationship I know of to Robin Knight) had some of her grad students put some very significant effort into trying to work through it and they didn't really get anywhere. However the latest news I have on this is quite a few years old, and it was always second- or third-hand at best. --Trovatore (talk) 23:34, 27 December 2021 (UTC)[reply]

Hey Trovatore, thanks so much for the info! Even if the latest news you have is a bit old and indirect, I think it's safe to say, given the lack of verification, as well as the lack of apparent faith towards the counterexample, that the problem can be unlisted from the proved section (and it turns out that it was already accidentally double-listed anyhow, once as solved and once as not, so it won't be much of a problem.) Once again thanks so much for taking the time to help! GalacticShoe (talk) 14:31, 28 December 2021 (UTC)[reply]

I know Sacks and a few students put some serious work into understanding the argument, as did the Berkeley recursion theory group. The general consensus was that this was not a proof.--72.80.81.137 (talk) 19:28, 28 December 2021 (UTC)[reply]

Understood, thanks for the information! I think this confirms to a reasonable degree that the problem should not be listed as solved. GalacticShoe (talk) 05:28, 30 December 2021 (UTC)[reply]

On 2, It appears we don't actually have an article on the Sidon set problem. One option would be to create a redirect page. But if the article is the target of a redirect then the redirected term should be mentioned in the article. So just from an editor's POV that should be fixed. On the other hand there is a lot of red ink in that section of the unsolved problems article, so perhaps the Sidon set problem isn't the highest priority issue there. --RDBury (talk) 10:28, 28 December 2021 (UTC)[reply]

Hey RDBury, thanks for the heads-up; I'll see what I can do about the formatting, and also about addressing some of the redlinks. Thanks for taking the time to help! GalacticShoe (talk) 14:31, 28 December 2021 (UTC)[reply]