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Let and let be the family of all -uniform hypergraphs with vertices and edges. Is it true that
Let and let be the family of all -uniform hypergraphs with vertices and edges for some . Is it true that
Source: erdosproblems.com/1076
An accepted solution exists. The statement is true.
The site's label is PROVED (page last edited 7 October 2025), and it describes the corrected Statement: the site's remark credits the asymptotic version to Bohman and Warnke [BoWa19] and to Glock, Kühn, Lo and Osthus [GKLO20], whose lower bound , with the upper bound that linearity gives, proves it (Glock–Kühn–Lo–Osthus 2018, Bohman–Warnke 2018). The site's wording, with the single family of -graphs with vertices and edges, is refuted at (Glock 2019), at (Glock–Joos–Kim–Kühn–Lichev–Pikhurko 2024), at (Glock–Kim–Lichev–Pikhurko–Sun 2026) and at (Pikhurko–Sun 2026), all refereed, and at by the Lean file Alexeev 2026; those five claim pages are rejected, as they answer the site's wording, not the corrected Statement.
The site's wording defines as the single family of -graphs with vertices and edges, so that is the Brown–Erdős–Sós function of [BES73], and that is what Erdős printed: display (13) of [Er74c], pp. 80–81, guesses that for every , with the hedge that the conjecture "may easily turn out to be nonsense", and Erdős's "only argument in favour", the theorem that edges force a - or a -configuration, concerns two configurations forbidden together. Under that wording the displayed asymptotic is false: the limit is at ([Gl19]), at ([GJKKLP24]), , and at ([GKLPS26]) and at least at ([PiSu26]), all refereed, and the file for the problem in Boris Alexeev's lean-proofs collection refutes the case with explicit systems of density ; only the lower bound holds, for every , by [BoWa19] and [GKLO20]. The site's curator, Thomas Bloom, reads the problem as the approximate form of Problem 207, in which every -configuration with is forbidden at once. The site's commentary (page last edited 7 October 2025, after Zach Hunter's thread comment of 6 October 2025 calling the problem "essentially a weaker version of" Problem 207) says that for satisfying the right divisibility conditions the extremal number is known exactly and that "the asymptotic version asked for here" was proved independently by Bohman and Warnke and by Glock, Kühn, Lo and Osthus, and it labels the problem PROVED. Each of those statements is true of the family and false of the single family: a -graph avoiding is linear, so for every , the two credited papers give the matching lower bound, and a Steiner triple system of high girth (Problem 207, proved by Kwan, Sah, Sawhney and Simkin) gives the exact value for large admissible . The corrected Statement replaces "with vertices and edges" by "with vertices and edges for some " and changes nothing else. It follows Bloom's reading; nothing in Erdős's text points to it, and the Brown–Erdős–Sós literature treats the single-family question as Erdős's. The answer to the site's wording is no, by the four refereed papers; their results are correct, but they answer the printed wording (a single family ), not the corrected Statement (the cumulative family), so their claim pages are kept and rejected and do not count toward the problem's standing, and Alexeev's file is rejected for the same reason. The answer to the corrected Statement is yes, and the problem is proved. Collin Yuanjie Ren's Lean submission states the corrected Statement and assembles its proof from the formalized theorem on Problem 207; the lean-proofs file states the site's wording. Neither is built here.