A finite forbidden family with superlinear surplus and non-join extremal graphs

We give a common counterexample to two product-structure conjectures in extremal graph theory. More precisely, we construct a fixed nonempty finite family $\mathcal L$ with $p(\mathcal L)=2$ such that, for some $c>0$, $$\operatorname{ex}(n,\mathcal L)>t_2(n)+cn^{3/2}$$ for every sufficiently large $n$. Nevertheless, at every such order there is an $\mathcal L$-extremal graph with connected complement, and hence with no nontrivial join decomposition. This superlinear surplus also forces the decomposition family of $\mathcal L$ to contain no forest. The construction uses an endpoint-injective repair operation with a finite obstruction family whose extremal number and equality cases admit exact descriptions. These properties disprove both conjectures.