---

Claim. For every fixed $\ell\ge4$ there are $n_\ell$ and $\beta_\ell>0$ such that for every $n\ge n_\ell$ some $3$-uniform hypergraph on $n$ vertices has at least $(1-n^{-\beta_\ell})\,n^2/6$ edges and girth larger than $\ell$, where the girth is the least $g\ge4$ such that some $g$ vertices span at least $g-2$ edges; so no $j$ vertices span $j-2$ or more edges for any $4\le j\le\ell$. This is Theorem 1.3 of Bohman and Warnke 2019. The construction is the high-girth triple process, which adds uniformly random triples subject to keeping the girth above $\ell$, analyzed by the differential equation method (Theorem 2.4). The same result was obtained independently by Glock, Kühn, Lo and Osthus (their claim page), whose page states what the two results cover and do not cover.

Covers. The theorem gives the lower bound $(1/6-o(1))n^2$ for the corrected Statement's family, and linearity gives the upper bound $\binom n2/3$, so the theorem settles the corrected Statement for every $k\ge5$. For the single family that the site's wording defines, the upper bound fails at every $k$ from $5$ to $10$ (Glock's page, the (6,4) page, the (7,5), (8,6) and (9,7) page, the (10,8) page), results that answer only that wording.

Acceptance. Refereed: T. Bohman and L. Warnke, Large girth approximate Steiner triple systems, J. Lond. Math. Soc. (2) 100 (2019), no. 3, 895–913, published online 10 June 2019 after the arXiv posting of 3 August 2018. Reviewed: the site's curator, Thomas Bloom, marks Problem 1076 proved and credits the asymptotic version to this paper [BoWa19] and to Glock, Kühn, Lo and Osthus [GKLO20], reading the question as the approximate form of Problem 207, the reading the corrected Statement adopts (problem page last edited 7 October 2025). The card records the theorem from the paper; its proof is unreviewed.

Formalizations. Collin Yuanjie Ren's Lean 4 submission, linked above, states and assembles the corrected Statement, deriving its lower bound from the formalized theorem of Kwan, Sah, Sawhney and Simkin on Problem 207 rather than from this paper's argument, and credits these authors among its informal sources; the Glock–Kühn–Lo–Osthus page describes it. The corpus has not built this submission, so this page lists no formalized evidence.