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Claim. The statement fails for every $r\ge5$. The surviving manuscript's Theorem 1 states that for every $\gamma>5^{-5}$ there are $\epsilon>0$ and arbitrarily large $5$-uniform hypergraphs $H$ with $e(H)\ge(1+\epsilon)(v(H)/5)^5$ and $e(H[S])<\gamma|S|^5$ for every nonempty $S\subseteq V(H)$, so no constant $c_5>5^{-5}$ can force a dense subgraph; adjoining common vertices to every edge lifts the counterexample to every larger uniformity. Since Problem 1075 asks for such a constant at every $r\ge3$, a counterexample at one $r$ refutes it. The base construction is a sequence of explicit $5$-uniform hypergraphs $G_n$ on $2n+7$ vertices whose unnormalized Lagrangians satisfy $5^{-5}(1+1/(4(3n)^3))\le\lambda(G_n)\le5^{-5}+1/(4!\,n)$, obtained by coupling two cubic forms around a cycle; blow-ups of the $G_n$ give the counterexample. The manuscript leaves the cases $r=3,4$ open. The forum entry's summary describes an earlier version at uniformity $16$: $16$-uniform hypergraphs whose Lagrangians approach $16^{-16}$ from above, built from a cyclic pair of link systems, whose blow-ups have every subgraph below any prescribed density above $16^{-16}$; it does not say which other uniformities that version covered. Erdős's theorem ([Er64f] on the problem page) that $c_r=r^{-r}$ itself works stands beside the claim.

Submission note. Posted to erdosproblems.com as a proof claim by Qiyuan Gu (account fireflysentinel) on 5 September 2026, giving "GPT-6 Astra, GPT 5.6 Sol, Claude Opus 5" as the AI used:

We disprove Erdős Problem 1075 by constructing 16-uniform hypergraphs whose Lagrangians approach 16⁻¹⁶ from above, while all blow-ups have every subgraph below any prescribed density γ > 16⁻¹⁶. The construction uses a cyclic pair of link systems: closing the cycle gives a small gain, whereas opening it forces a larger quantitative loss, yielding the required Lagrangian bound. Notes: GPT-6 Astra was used to generate the mathematical proofs and draft the manuscript. GPT-5.6 Sol and Claude Opus 5 were used only for editorial review of the exposition. GPT-6 Astra was run in a research environment containing earlier results produced by GPT-5.6 Sol and Claude Opus 5, but those earlier results did not contribute to the final mathematical arguments. The author reviewed the final manuscript and takes full responsibility for its content.

Claimant. Qiyuan Gu, named by the forum entry, which was posted on 5 September 2026 under the username AlexErdosProblem and names GPT-6 Astra, GPT 5.6 Sol and Claude Opus 5 as its tools; the manuscript's own declaration says GPT-6 Astra proposed the construction and the estimates, drafted the text and generated the Lean formalization. The forum entry linked Zenodo record 22380117 (version v2 of concept record 10.5281/zenodo.22367904, published 5 September 2026 under Gu's name); that record, v1 (record 22367905) and v3 (record 22421171) were removed from Zenodo on 23 September 2026. The concept record survives as version v7, record 22667633 (published 8 September 2026), "Counterexamples to Erdős Problem 1075", whose creator is given as Anonymous, so the claimant's name rests on the forum entry alone. That record carries the manuscript and a Lean 4 archive (erdos-1075-lean.zip) that the manuscript presents as a formalization of its proof. This account rests on the manuscript's abstract and main statements; the corpus has not built the Lean archive, so no formalized evidence is listed.

Acceptance. None: the manuscript is not refereed, no outside reviewer has endorsed it, and the site's label is OPEN (page last edited 5 October 2025).