A $d$-regular (simple finite) graph $G=(V,E)$ with diameter $k$ is a Moore graph if $$ |V| = 1 + d \sum_{i=0}^{k-1} (d-1)^i. $$ It is known from the Hoffman-Singleton theorem and results of Damerall that any Moore graph with $d>2$ and $k >1$ will either be the Petersen graph, the Hoffman-Singleton graph or a hypothetical Moore graph with $d=57$ and $k=2$. At the end of 2020, a paper has appeared on the arXiv (https://arxiv.org/abs/2010.13443) that claims to have fully solved the problem by proving that no Moore graph can be 57-regular. Since the paper is in Russian I have no easy way to verify this, and (somewhat strangely) I have seen no news anywhere else that this problem has been solved. My question is; does anyone know if the proof given in the cited arXiv paper is correct or not?
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7$\begingroup$ This site has generally refused to pass judgement on papers/preprints. $\endgroup$– Gerry MyersonCommented Jul 7, 2021 at 12:29
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7$\begingroup$ I believe this talk by the author on his work is in English: youtube.com/watch?v=Q2Uq28TRy0Y $\endgroup$– Gerry MyersonCommented Jul 7, 2021 at 12:35
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$\begingroup$ Thank you for the link! $\endgroup$– S. DewarCommented Jul 7, 2021 at 18:07
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$\begingroup$ Google translate gives a pretty good translation of the pdf document. you would have to add the math statement yourself but it's doable. Note that the v2 seems to include the comment made by Vidali at the end of the youtube video. $\endgroup$– Thomas LesgourguesCommented Jul 8, 2021 at 2:18
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4$\begingroup$ A paper of Vance Faber just published on the arXiv claims to find an error in the paper of Makhnev. $\endgroup$– Brendan McKayCommented Oct 19, 2022 at 6:25
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