There is a good amount of second-order evidence that the proof is wrong. Further, I skimmed the introduction, and it seems the indicated approach cannot possibly work. My guess is that both authors are senile. (They're quite old.)
If you give me decent odds, I'd be happy to bet against you regarding the proof's correctness. Would you take 1:1?
I have no idea whether the proof is right or wrong, and I'm not so sentimental as to believe that the harsher person is more likely to be wrong. (Besides, naysayers are usually right, just like they are when predicting that any given startup will fail.) However, the HN guidelines (https://news.ycombinator.com/newsguidelines.html) ask you not to post like this and that's important too. Other users are making their case against the OP while following the guidelines just fine. Please be more like them.
There's no personal attack here. I'm sorry if that comment comes across as curt, but the post I responded is (in my opinion) grossly misleading and deserves pushback. For example, consider the statement: "If the proof has a flaw, the issue will be technical and difficult to uncover," made by someone claiming years of experience as a math professor thinking about this question. This is easily identifiable as wrong, because the purported proof is just 7 pages (actually less than that due to extraneous material) and does not use any particularly technical mathematics (a strong undergraduate major probably knows enough to comprehend what's going on). Indeed, people found the error in (less than) 12 hours after that comment was made.
The comment about senility is not a "swipe." It is a very real problem in the mathematics community. Mathematicians get old and are sometimes afflicted by dementia, and this unfortunately can manifest as hopeless attempts at famous problems. Atiyah's "proof" of the Riemann Hypothesis right before his death is perhaps the most well known example, but there are others that (thankfully) aren't disseminated publicly. In the pre-internet age, journal editors were able to make such things quietly disappear. But now we have the arxiv. This is valuable context for understanding why two mathematicians with distinguished publication histories might be posting an incorrect proof, which had not been provided by other comments.
That being said, I will endeavor to phrase such comments more carefully in the future. Thank you for your message.
Are both of them senile?
Second-order evidence: Old mathematicians (~80). Famous problem. Weird, imprecise writing style. No mention of the proof by mainstream mathematical news sources (breakthroughs are usually accompanied by excited blogging/tweeting). No acknowledgements directed at other mathematicians who have checked or commented on the proof.
[deleted argument here; replaced by more precise comment below]
My tone is harsh because I think the best thing to do is to quietly ignore it, similar to how the community treated Atiyah's claims of a RH proof at the end of his life.
Have you seen this comment? https://news.ycombinator.com/item?id=34083099
> the argument looks like it's based on large n asymptotics, so even assuming everything works correctly the strongest statement they can hope to show is that the theorem is true for all n > n_0, where n_0 is some large constant. but there is no mention of this fact. the theorem is claimed for all n.
They're claiming:
> Theorem 7. If there is a map L which cannot be 4-coloured then only an exponentially small fraction of the maps with n edges can be 4-coloured.
These claims appear mutually exclusive.
>The argument looks like it's based on large n asymptotics, so even assuming everything works correctly the strongest statement they can hope to show is that the theorem is true for all n > n_0, where n_0 is some large constant. But there is no mention of this fact. The theorem is claimed for all n.
This is completely wrong. A proof for all n>n_0 is a proof for all n, since any counter-examples have to exist as subgraphs of arbitrarily large graphs.
Specifically I think section 4 is basically nonsense. (I see Sniffnoy has already pointed this out below.)
(Re: your comment, Theorem 7 is going to fail below the smallest counterexample, right? This is bad, imprecise writing - a red flag.)
If you think that's the best thing to do then why not do it?
To be clear, I don't like this state of affairs. As suggested above, the best course of action seems to be to ignore the posting.
It's so inappropriate that in the field that's actually trained for this, they're disallowed by compact, even with extensive evidence.
Please stop.
The field trained for this is not refraining from this because it's hard to get right, but because it undermines the privacy promise they give their clients! not an argument applicable to people who do not have that professional reputation to uphold.
Of course you should speculate about mental health of people when it's relevant to the topic - it's a factor heavily shaping many people's behaviour!
* No discussion of why they were not found by other people in the past.
* No discussion of how these techniques could be used on other problems.
* A lot of calculations which would be left out as trivial by most graph theory papers (for example, calculations about the edge counts of subgraphs)