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Claim. For $r=3$ and $n\le9$, every $3$-uniform hypergraph on $n$ vertices is the union of at most $\mathrm{ex}_3(n;K_4^3)$ copies of $K_3^3$ (single edges) and $K_4^3$ (tetrahedra), no two sharing an edge: the question of Problem 719 restricted to these parameters. Equivalently, a $3$-graph with $m$ edges and a largest edge-disjoint tetrahedron packing of size $\nu$ satisfies $m-3\nu\le\mathrm{ex}_3(n;K_4^3)$. As the forum entry and the deposit's abstract describe the manuscript, The Erdős–Sauer clique decomposition problem for 3-graphs: verification through nine vertices and local packing reductions, the cases $n\le7$ follow from a near-Turán reduction together with the exact small Turán numbers, a range the abstract says an earlier working report had independently verified; the case $n=8$ is proved with the Steiner system $S(3,4,8)$ by a direct packing argument, and $n=9$ with an $S(3,4,10)$, a random relabeling and a count of collisions among the missing triples. The manuscript also proves lemmas on how edges pack and are covered near a largest set of edge-disjoint tetrahedra, which give structural hypotheses under which the inequality holds in general; the entry's notes say the general $r=3$ conjecture is not claimed.

Submission note. Posted to erdosproblems.com as a proof claim by Rafik Zeraoulia (account Rafikzeraoulia2025) on 7 September 2026, giving "OpenAI GPT-5.6 Thinking" as the AI used:

The paper verifies the r=3 Erdős–Sauer clique-decomposition bound for all 3-uniform hypergraphs on at most nine vertices. The cases n=8 and n=9 are proved using Steiner systems: S(3,4,8) gives a direct packing argument for n=8, while n=9 uses an S(3,4,10), random relabelling, and a collision count for missing triples. The paper also develops local packing-covering lemmas around a maximum tetrahedron packing, giving structural sufficient conditions for the conjectured inequality. Notes: The general r=3 conjecture is not claimed to be solved; the new finite verification extends the known range through n=9.

Covers. The case $r=3$ for every $n\le9$. It says nothing about $r=3$ with $n\ge10$ or about any $r\ge4$.

Claimant. Rafik Zeraoulia, who deposited the manuscript on Zenodo on 7 September 2026 (DOI 10.5281/zenodo.22568303, version 1.1, CC BY 4.0) and posted the claim on the erdosproblems.com forum the same day, naming OpenAI GPT-5.6 Thinking as the tool. The claimant's comment of 8 February 2026 in the problem's discussion thread had reported an exhaustive check of the inequality for all $3$-graphs on $6$ vertices, with $\mathrm{ex}_3(6;K_4^3)=14$; a thread comment has no page of its own.

Acceptance. None: the preprint is not refereed, no outside reviewer has endorsed it, the entry has no comments, and the site labels the problem OPEN. This account follows the Zenodo abstract and the forum entry.