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Claim. The largest number of edges of a $3$-uniform hypergraph on $n$ vertices in which no three edges span at most five vertices is $(1\pm o(1))\,n^2/5$ (Glock 2019, main theorem, as the arXiv abstract and the Crossref record state it; the paper is not held in the library). In the site's notation, $\mathcal F_5$ is the family of $3$-graphs with $5$ vertices and $3$ edges, and a $3$-graph contains a member of $\mathcal F_5$ exactly when some three of its edges span at most five vertices, a member with an isolated vertex being three edges on four vertices. Hence

$$\mathrm{ex}_3(n,\mathcal F_5)=\Bigl(\frac15+o(1)\Bigr)n^2,$$

not $(1+o(1))n^2/6$: the displayed asymptotic fails at $k=5$ with $\mathcal F_k$ the single family the site's wording defines, so the site's wording, an assertion about every $k\ge5$, is false. The transfer from the theorem to the site's wording is the identification of the two forbidden families just made. The theorem says nothing about $k\ge6$; the limit at $k=6$ is $7/36$ (the (6,4) page), a second refutation of that wording.

Why it is rejected. The result is correct, but it answers the site's wording, the single family $\mathcal F_5$, not the corrected Statement of Problem 1076, whose family is cumulative: under the corrected Statement a $3$-graph avoiding $\mathcal F_4\cup\mathcal F_5$ is linear, so the theorem says nothing against it, and the page does not count toward the problem's standing. The problem page's Notes credit the result.

Acceptance. Refereed: S. Glock, Triple systems with no three triples spanning at most five points, Bull. Lond. Math. Soc. 51 (2019), no. 2, 230–236, published online 27 November 2018 after the arXiv posting of 6 September 2018. The paper's abstract presents the result as the case $k=5$ of Brown, Erdős and Sós's conjecture that the limit of $f^{(3)}(n;k,k-2)/n^2$ exists. The site does not cite the paper on this problem and its curator makes no statement about it; the proof is unreviewed.

Formalizations. None of this theorem is known. The file for the problem in Boris Alexeev's lean-proofs collection names Glock as its informal author in a header added on 23 August 2026, but its own docstring describes it as a self-contained disproof that does not rely on Glock's approximate packing theorem: it proves the weaker lower bound $92/529>1/6$ at $k=5$ by an explicit construction. It is an independent proof and has its own claim page, Alexeev 2026.