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.
>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?
* 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)
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.
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!