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Claim. Write $f^{(r)}(n;s,j)$ for the largest number of edges of an $r$-uniform hypergraph on $n$ vertices containing no $j$ distinct edges that span at most $s$ vertices, and $\pi(r,t,j)$ for the limit of $n^{-t}f^{(r)}(n;(r-t)j+t,j)$, the paper's notation with its $k$, the number of edges, written $j$ here to keep erdosproblems.com's $k$. Theorem 1.1 of Chao, Huang and Liu states that $\pi(r,t,j)=2/(t!\,(2\binom rt-1))$ for all integers $r>t\ge2$ and every odd $j\ge3$, which at $r=3$, $t=2$ is $1/5$. Theorem 1.2 states that $\pi(3,2,j)>1/6$ for every even $j\ge4$; its proof bounds $\liminf_{n\to\infty}n^{-2}f^{(3)}(n;j+2,j)$ below by $1/6+1/(2|J_\ell|)$, with $J_\ell$ the reserved pairs of an explicit recursive seed (Definition 3.19, Proposition 3.20), so it does not need the existence of the limit, which Delcourt and Postle proved. A $3$-graph contains a member of erdosproblems.com's $\mathcal F_k$, the $3$-graphs with $k$ vertices and $k-2$ edges, exactly when some $k-2$ of its edges span at most $k$ vertices, so $\mathrm{ex}_3(n,\mathcal F_k)=f^{(3)}(n;k,k-2)$, the case $j=k-2$, of the same parity as $k$. Hence

$$\mathrm{ex}_3(n,\mathcal F_k)=\Bigl(\frac15+o(1)\Bigr)n^2 \quad\text{for odd }k\ge5, \qquad \liminf_{n\to\infty}\frac{\mathrm{ex}_3(n,\mathcal F_k)}{n^2}>\frac16 \quad\text{for even }k\ge6,$$

and the displayed asymptotic fails at every $k\ge5$ with $\mathcal F_k$ the single family the erdosproblems.com wording defines, so the erdosproblems.com wording, an assertion about every $k\ge5$, is false. The transfer from the theorems to the erdosproblems.com wording is the identification of the forbidden families just made. The values agree with the refereed ones: the limit $1/5$ at $k=5$, $7$ and $9$, and the limits $7/36$ and $61/330$ and the lower bound $3/16$ at $k=6$, $8$ and $10$, all above $1/6$. For even $k$ the paper's concluding remarks place the limit in $(1/6,1/5]$, and its Question 5.1 leaves the value open from $k=10$ on. The theorems are new on this problem at every $k\ge11$; the refutations at $k=5$ to $10$ are on Glock's page, the (6,4) page, the (7,5), (8,6) and (9,7) page and the (10,8) page, and the case $k=11$ was proved independently on the (11,9) page.

The refutation uses only the lower bounds: $1/5-o(1)$ for odd $k$, from the paper's Theorem 3.6, and the bound above $1/6$ for even $k$. Both come from seeds fed to the paper's Lemmas 3.4 and 3.5, its form of the packing theorem of Glock, Joos, Kim, Kühn, Lichev and Pikhurko (their Theorem 3.1). The paper's AI statement says that the lower bounds were obtained without AI assistance and that the authors used ChatGPT-5.6 as a collaborative tool for the upper bounds and the manuscript.

Why it is rejected. It answers the erdosproblems.com wording, the single families $\mathcal F_k$, not the modified Statement of Problem 1076, whose family is cumulative: under the modified Statement a $3$-graph avoiding $\mathcal F_4\cup\dots\cup\mathcal F_k$ is linear, so the theorems say nothing against it at any $k$, and the page does not count toward the problem's standing. The problem page's Notes credit the result.

Acceptance. None documented. The paper is an arXiv preprint, read in its first version (29 September 2026, 22 pages): the abstract, Theorems 1.1 and 1.2 (p. 2), the AI-usage paragraph (p. 5), the statements of Lemmas 3.4 and 3.5 and Theorem 3.6 (p. 12), the proof of Theorem 1.2 (Subsection 3.2) and the concluding remarks (Section 5). erdosproblems.com does not cite it on this problem, and its curator makes no statement about it. The proof is unreviewed, and no formalization is known.