Treglown's equitable acyclic colouring conjecture
Statement
Treglown conjectured, in a complementary form, that for every positive integer every digraph with for all has an equitable acyclic -colouring. This implies the acyclic colouring versions of the Hajnal-Szemeredi theorem for digraphs proved by Czygrinow, DeBiasio, Kierstead and Molla, which in turn imply the original Hajnal-Szemeredi theorem for graphs. Proved: a short reduction shows the conjecture follows directly from the original Hajnal-Szemeredi theorem, and a modification of it gives a polynomial-time algorithm for finding such a colouring.
Record
Comments
No person has examined this. Nothing here has been checked at all. say whether it holds →
proof attempt · #1
ChatGPT 5.6 Sol, with Louis DeBiasio and Hal KiersteadThe record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.
The acknowledgements describe the model producing the argument and then correcting its own novelty claim: "The reduction given in this paper arose during a discussion between the first author and ChatGPT 5.6 Sol attempting to locate the bottleneck in extending the results of [3] to prove Conjecture 1.2. Instead of locating the bottleneck, the chatbot gave a clever proof which shows that Conjecture 1.2 reduces to the original Hajnal-Szemeredi theorem. After further discussion about the originality of this idea, the chatbot identified earlier work of Aboulker, Oijid, Petit, Rocton, and Simon" in which the reduction is implicit. The central idea of the paper came from the model, which places it at the discovered tier, with the caveat about priority recorded in the result note.
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