The Simonovits Product Conjecture
Prior state unknown→disproved
one construction disproves both the product conjecture and its weak form
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combinatorics / Extremal graph theory
The Simonovits Product Conjecture
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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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.
one construction disproves both the product conjecture and its weak form
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