Extremal graph theory and finite forcibility

We study the uniqueness of optimal solutions to extremal graph theory problems. Our main result is a counterexample to the following conjecture of Lovász, which is often referred to as saying that “every extremal graph theory problem has a finitely forcible optimum”: every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints such that the resulting set is satisfied by an asymptotically unique graph.