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Claim. Ting-Wei Chao, Xinqi Huang and Hong Liu, Asymptotics of the Brown–Erdős–Sós problem at integer exponents, arXiv:2609.38115v1 (29 September 2026, 22 pages, the only version), write $f^{(r)}(n;s,k)$ for the largest number of edges of an $n$-vertex $r$-graph in which no $k$ distinct edges span at most $s$ vertices. Their $s$ is erdosproblems.com's vertex count $k$ and their $k$ is erdosproblems.com's edge count $s$, so their function is erdosproblems.com's $\mathrm{ex}_r(n,\mathcal F)$, as the Formulation note of Problem 1157 explains; this page uses erdosproblems.com's letters. When $(rs-k)/(s-1)$ is an integer $t$ with $2\le t<r$, that is $k=(r-t)s+t$, the order of $\mathrm{ex}_r(n,\mathcal F)$ is $n^t$, and the paper asks for $\pi(r,t,s)=\lim_{n\to\infty}n^{-t}\mathrm{ex}_r(n,\mathcal F)$. Theorem 1.1 (p. 2): for all $r>t\ge2$ and $s\ge2$,

$$\pi(r,t,s)=\frac{2}{t!\,\bigl(2\binom rt-1\bigr)}\ \text{for odd }s,\qquad \pi(r,t,s)=\frac{1}{t!\binom rt}\ \text{for even }s\text{ and }r\ge4.$$

For $3$-graphs ($r=3$, $t=2$, $k=s+2$) this gives $\pi(3,2,s)=1/5$ for every odd $s$. The theorem does not cover $3$-graphs with even $s$. There the limit exists by Delcourt and Postle's theorem, which the paper cites, and $\pi(3,2,2)=1/6$ is Rödl's packing value; the paper notes (p. 2) that with Delcourt and Postle's theorem, Theorem 1.1 gives the existence of the limit for all $r>t\ge2$ and $s\ge2$. Theorem 1.2 (p. 2): $\pi(3,2,s)>1/6$ for every even $s\ge4$. With Theorems 2.3 and 2.4 this gives $1/6<\pi(3,2,s)\le1/5$ (Section 5, p. 20), and Question 5.1 asks for the value at even $s\ge8$.

The upper bounds are in Section 2. Theorem 2.3 (p. 5) deletes at most $(s-1)^2\binom n{t-1}$ edges to destroy every small dense subgraph, which the paper calls a block. Theorem 2.4 (p. 6), for every $s$, counts the $t$-sets covered by edges and the crossing $t$-sets of diamonds (two edges meeting in $t$ vertices) through an auxiliary forest, and bounds the remaining edges by $\frac{2}{2\binom rt-1}\binom nt$. Theorem 2.6 (p. 7), for even $s$ and $r\ge4$, attaches two uncovered $t$-sets to each diamond and gives the bound $\binom nt/\binom rt$. The lower bounds are in Section 3. For even $s$ a high-girth partial Steiner system from the packing theorem of Delcourt and Postle suffices (display (3), p. 11). For odd $s$, Proposition 3.7 (p. 13) builds finite seeds whose packing ratio approaches $2/(2\binom rt-1)$, starting from one diamond and applying a random alteration (Lemma 3.8) finitely often. Lemma 3.4 (p. 12) then turns a seed into $n$-vertex hypergraphs. It is a strengthened form of Theorem 3.1 of Glock, Joos, Kim, Kühn, Lichev and Pikhurko, which the paper says is implicit in that theorem's proof (footnote 1, p. 12). Theorem 1.2 (pp. 17–19) feeds Lemma 3.4 with a recursive seed grown from the seven-edge configuration of that paper (Definition 3.19, Proposition 3.20). The paper's statement of AI use (p. 5) says that the lower bounds were found without AI, that ChatGPT-5.6 served as a collaborative tool for the upper bounds and the manuscript, and that the authors verified all arguments. At $r=3$ the odd-$s$ value $1/5$ agrees with the refereed limits at $s=3$, $5$ and $7$ that Problem 1076 records.

Covers. The first-order asymptotics of $\mathrm{ex}_r(n,\mathcal F)$ on the integer-exponent family $k=(r-t)s+t$ with $r>t\ge2$ and $s\ge2$, one vertex below the range $k\ge(r-t)s+t+1$ of the Brown–Erdős–Sós conjecture, except $3$-graphs with even $s$: there the value at $s=2$ is Rödl's, and for even $s\ge4$ only $1/6<\pi(3,2,s)\le1/5$ is shown. The $3$-graph part, $k=s+2$, is the case erdosproblems.com routes to Problem 1076. Not covered: the exact values of $\mathrm{ex}_r(n,\mathcal F)$, every other triple $(r,k,s)$, the conjecture itself, and the linear case $t=1$, which the paper calls elementary and cites. The problem stays open.

Depends on. No page of this wiki. The lower bounds use two published or posted results as black boxes: Theorem 3.1 of Glock, Joos, Kim, Kühn, Lichev and Pikhurko, On the $(6,4)$-problem of Brown, Erdős, and Sós, Proc. Amer. Math. Soc. Ser. B 11 (2024), 173–186, in the strengthened form above, and Theorem 1.3 of Delcourt and Postle, arXiv:2204.08981, on high-girth packings. The upper bounds are self-contained.

Standing. Claimed: an arXiv preprint with no refereed version, Lean proof or outside review known to this corpus. Read depth: the statements of Theorems 1.1, 1.2, 2.3, 2.4, 2.6 and 3.6 and of Lemma 3.4 were checked against the problem's statement. The proof of Theorem 2.3 and the closing count of Theorem 2.4 were checked here; the acyclicity arguments of Theorems 2.4 and 2.6, the random alteration of Lemmas 3.8 and 3.13 and the proof of Proposition 3.20 were read through but not checked. Independently, Bodnár, Pikhurko, Sun, Wang and Zeng (arXiv:2610.03323v1, 2 October 2026, Theorem 1.1) claim the single value $\pi(3,2,9)=1/5$ by another construction; that agreement is not review. The erdosproblems.com page for the problem was last edited on 24 January 2026, before the result, and no proof claim on erdosproblems.com records it; Bloom's catalog labeled the problem OPEN.