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.