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I'm a math professor who has lost years to this question. I've met various authors involved in successful proofs. My 1976 grad school application from the Unive

I'm a math professor who has lost years to this question. I've met various authors involved in successful proofs. My 1976 grad school application from the University of Illinois came postmarked "Four Colors Suffice" (Appel, Haken). I've also been inundated with well-intended crank proofs of every tar pit like this. One can tell the difference without focusing one's eyes. These authors have long, established careers, with 2,000 MathSciNet citations between them. Waterloo is a hotbed of combinatorics; I doubt that they posted this before passing the paper around a bit. Graph theory as a branch of mathematics is like photography as an art form: Anyone can point a camera, and one sees a lot of incredibly weak graph theory if one digs deep enough into the regional conference circuit. This work is deep, relying on an understanding of generating functions that many casual practitioners don't possess. If the proof has a flaw, the issue will be technical and difficult to uncover. I love how people take the obvious bet that this is likely wrong because so many proof attempts have failed. In a prediction market, I'd bet on this proof being right.

ngvrnd 4y ago
well put.
ruuda 4y ago
> In a prediction market, I'd bet on this proof being right. Let's create a question on Metaculus!
- bakkoting 4y ago
  How about Manifold? https://manifold.markets/ZviMowshowitz/will-the-new-nonconstructed-proof-o https://manifold.markets/ZviMowshowitz/will-the-new-nonconst...
ColinWright 4y ago
I caught a rumour of this paper about 6 months ago, so I'm pretty sure that in the meantime it's been evaluated and assessed by colleagues. So I'm pretty sure it's going to be right, with only small fixable things that need tidying up.
spekcular 4y ago
Your post feels like a dispatch from bizarro-world. 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?
- ogogmad 4y ago
  The tone of your comment is pretty harsh. What kind of 2nd-order evidence do you see? And why did you learn from the introduction that makes you so sure? It's only an introduction, which might only give you an oversimplified summary of their ideas. Are both of them senile?
  - spekcular 4y ago
    Yeah, my guess is that both are senile if they're putting this on the arxiv. 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.
    - ogogmad 4y ago
      > 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. Have you seen this comment? https://news.ycombinator.com/item?id=34083099 https://news.ycombinator.com/item?id=34083099
      - 4y ago
wikfwikf 4y ago
Just as the incorrect proof by the amateur mathematician or crank is likely to be in a predictable form - a bunch of drawings of graphs with increasingly obtuse notation, reference to special cases within special cases, etc., isn't the incorrect proof by working combinatorialists likely to look exactly like this paper? An argument from complex analysis relating to either generating functions, Tutte polynomials, or some similar concept, together with some difficult arguments about how the asymptotics should be correctly used? The fact that this includes arguments which casual practitioners would not be able to confidently use in a crank proof does not evidence its correctness. It just shows us that it's from a different type of person, susceptible to making a different type of error.

ngvrnd 4y ago

well put.

ruuda 4y ago

> In a prediction market, I'd bet on this proof being right. Let's create a question on Metaculus!

bakkoting 4y ago
How about Manifold? https://manifold.markets/ZviMowshowitz/will-the-new-nonconstructed-proof-o https://manifold.markets/ZviMowshowitz/will-the-new-nonconst...

ColinWright 4y ago

I caught a rumour of this paper about 6 months ago, so I'm pretty sure that in the meantime it's been evaluated and assessed by colleagues. So I'm pretty sure it's going to be right, with only small fixable things that need tidying up.

spekcular 4y ago

Your post feels like a dispatch from bizarro-world. 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?

ogogmad 4y ago
The tone of your comment is pretty harsh. What kind of 2nd-order evidence do you see? And why did you learn from the introduction that makes you so sure? It's only an introduction, which might only give you an oversimplified summary of their ideas. Are both of them senile?
- spekcular 4y ago
  Yeah, my guess is that both are senile if they're putting this on the arxiv. 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.
  - ogogmad 4y ago
    > 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. Have you seen this comment? https://news.ycombinator.com/item?id=34083099 https://news.ycombinator.com/item?id=34083099
    - 4y ago

wikfwikf 4y ago

Just as the incorrect proof by the amateur mathematician or crank is likely to be in a predictable form - a bunch of drawings of graphs with increasingly obtuse notation, reference to special cases within special cases, etc., isn't the incorrect proof by working combinatorialists likely to look exactly like this paper? An argument from complex analysis relating to either generating functions, Tutte polynomials, or some similar concept, together with some difficult arguments about how the asymptotics should be correctly used? The fact that this includes arguments which casual practitioners would not be able to confidently use in a crank proof does not evidence its correctness. It just shows us that it's from a different type of person, susceptible to making a different type of error.