Problem detail · source-aware
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Simonovits conjectured that if a forbidden family $\mathcal{F}$ with $p(\mathcal{F}) > 1$ has extremal number exceeding the Turan bound by a superlinear surplus, then its extremal graphs are joins of $p$ graphs, each extremal for a family of chromatic number two. Disproved by a fixed finite family $\mathcal{L}$ with $p(\mathcal{L}) = 2$ and $\mathrm{ex}(n,\mathcal{L}) > t_2(n) + cn^{3/2}$ that nevertheless has, at every large order, an extremal graph with connected complement and hence no nontrivial join decomposition. The same construction disproves the Weak Product Conjecture of Furedi and Simonovits.
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The paper's comment credits the counterexample to GPT-5.6 Sol, found during a Codex project devoted to the Product Conjecture. The exact extremal-number and equality-case analysis around it is the author's.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
Single-author arXiv preprint; not yet peer-reviewed.
Correctness: unknown · statement fidelity: unaudited · peer review: none
one construction disproves both the product conjecture and its weak form
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.