Written: Dec. 30, 2022

My Rutgers colleague, the great evolutionary biologist Robert Trivers wrote a whole book about the propensity of humans for self-deception, and mathematicians are no exception. Even the great Sir Michael Atiyah believed that he had a proof of the Riemann Hypothesis.

Of course the arxiv is full of claimed proofs of Collatz and other major open problems, but there is a special section for them called GM (General Mathematics).

It really damages the credibility of the arxiv if a non-GM department, in this case CO (combinatorics) has a false proof of a long-standing open problem. I am talking about the recent Erroneous computer-less proof of the Four Color Theorem posted by distinguished and accomplished combinatorialists David Jackson and Bruce Richmond, who hail from the Mecca of combinatorics, University of Waterloo. Their proof is definitely flawed in its current form, and very unlikely to be fixable. (See here).

It would have made more sense, before rushing to make it public, to do the following:

- Write it in a crystal-clear style, broken up to lemmas, and if necessary, sublemmas, subsublemmas, etc., as in my structured paper,
facilitating easy verification by multiple experts.
- Ask some of their brilliant colleagues and collaborators (e.g. Jeff Shallit, Ian Goulden, Andrew Granville, all very competent and reliable people) to carefully critique it, and even offer prizes (or pledge donations to the OEIS) for finding issues with the proof.

I am surprised, and disappointed, by the arxiv Combinatorics moderators (that include Victor Reiner) for not seeing a red flag, or at the very least putting it in GM. So what if it was written by David Jackson and Bruce Richmond? When they write yet-another-paper with yet-another-result, of course, they should be trusted, but humans being humans, and so prone to self-deception, when they claim such a breakthrough, the arxiv moderators should be on their guard.

We have all, at times, believed that we proved some major open problems, but luckily usually spot the mistake ourselves, or consult with friends and colleagues before going public.

One example of someone who (falsely) believed to have proven a major conjecture, was Jean Merlin (1876-1914), who died believing that he proved the Goldbach conjecture. The "Merlin sieve" was the precursor, and inspiration, of Viggo Brun's famous sieve. Sadly Merlin was killed in World-War-I, and his failed proof (probably as homage to a war hero) was published (Bulletin de Sciences Mathematiques, 39:121–136, 1915.), annotated by Hadamard, who pointed out the error. Incidentally, Merlin's "day job" was astronomer, and he was a colleague (and collaborator, and very possibly a good friend) of an astronomer by the name of Philippe Flajolet, that by coincidence, is the grandfather (see here) of our beloved late guru Philippe Flajolet, who is cited in the above-mentioned Jackson-Richmond erroneous proof.

Let us all learn from the Jackson-Richmond mistake of premature posting of a major breakthrough, and exercise some self-control. A good rôle model is Norbert Blum, also with a formerly impeccable reputation (and for a long time, world-record holder of Circuit complexity lower bound). Blum believed that he proved that P ≠ NP, but shortly after retracted it. David Jackson and Bruce Richmond should post v2, admitting that v1 was in error, and explaining the pitfalls. The sooner the better, since it is embarrassing to the whole Combinatorics (and Mathematics) Community.

Happy p(p(1)^p(1))*p(p(p(1)^p(1)))^p(1) .

ADDED JAN. 13, 2023: I am happy to report that this paper was retracted .

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