Correct(?) proofsAlthough you might expect otherwise, the fact that this page called "Correct(?) proofs" is part of this website does not imply that a correct proof of SSNC exists. Indeed, an errorless proof of SSNC has not been found yet. At least, I could not find a correct proof of SSNC on the Internet and my presumption was confirmed by Paul Seymour, who originally came up with the problem.
Or is there...?However, in my search for a correct proof of SSNC I encountered a document about a student named Brett Harrison. In 2005, at the age of 17, he received $ 25,000 from the Davidson Institute for Talent Development for developing a proof of SSNC which is supposedly even more precise than the conjecture itself.
Yet Brett's proof cannot be found anywhere on the Internet. I must admit that I have not tried to contact him so far, but perhaps I will do so in the near future, since I am very curious if his proof is right – or rather, I am very curious to see where he made a mistake.
In short, until Brett Harrison's proof has been published, reviewed, and approved, Seymour's Second Neighborhood Conjecture remains unsolved.