Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Claim. For integers and let be the maximum number of edges of an -vertex graph with no cycle having a vertex incident with at least chords. Theorem 1.3 of Xiaozheng Chen and Bo Ning, On Erdős Problem 767: Cycles with Chords, arXiv:2609.15330v1 (14 September 2026; 22 pages), states that for all and
(the abstract's display omits the outer "max"; the paper's Theorem 1.3 prints it). For the theorem gives for all , and the threshold is sharp: Construction 3.3 (pp. 4--5), a set of vertices identified with and carrying the differences (plus a half-shift matching when is odd), joined completely to an independent set of vertices, has edges and no cycle with chords at one vertex, and Remark 3.4 takes at to get edges. The lower bound for small is a second, nearly regular construction (Section 3); the upper bound (Section 5) builds on the stability theorem of Ma and Ning (Combinatorica 40 (2020), 105--147) for Bondy's theorem on long cycles. The paper restates Jiang's theorem as its Theorem 1.2 with the hypotheses and , credits Erdős's conjecture of the formula for to his 1969 paper, says that Lewin disproved that conjecture, citing B. Bollobás, Extremal Graph Theory (1978), p. 398, Problem 12, and takes Bollobás's question with an unspecified threshold from Problem 13 of the same page; its reference [19] is Pósa's Problem 127, from which the case of the formula follows. The preprint was named in the site's discussion thread on 15 September 2026, and the site's reference list did not carry it as of 2026-09-18.
For Problem 767, the formula gives for all large , for every (the case is Pósa's for , on its claim page), so it settles the question a second time, after Jiang, and names the exact threshold at which the equality begins, which Jiang's bounds from above. Since equals for and exceeds it from on, the sharpness places the failure of Erdős's range at , in agreement with Erdős's 1975 report that Lewin's examples exist for large .
AI assistance and formalization. The paper's declaration of AI usage
(p. 21) says that the work began while the authors were using GPT-5.5 Pro,
that the authors proved the cases and a stability version of Jiang's
theorem themselves, and that, given those manuscripts, the system's second
proposed general conjecture became the main theorem; the authors wrote and
checked the final text. The same declaration says that every original result
of the paper, Theorem 1.3 included, has been formalized and machine-checked
in Lean 4, with the source at the repository linked above (its one commit,
of 10 September 2026, is the pin; the repository describes itself as a Lean 4
formalization for Erdős Problem 767). The repository's README says that its
development targets the authors' final manuscript, that its public claim is
the verification of the paper's internal conclusions conditional on nine
cited literature theorems packaged as hypotheses, which it does not prove,
and that theorem_1_3 is its entry point. Because the development declares
itself a formalization of this paper's result, it is a link on this page and
not its own claim. This corpus has not built, audited or kernel-checked it,
so it gives no formalized evidence.
Standing. Claimed. A preprint with its authors' Lean development: no refereed version is recorded, no outside reviewer has published an examination, and the Lean is unbuilt here. The problem's standing rests on the refereed claim above.