Elusive extremal graphs

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovász conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the resulting set is satisfied by an asymptotically unique graph. This statement is often referred to as saying that ‘every extremal graph theory problem has a finitely forcible optimum’. We present a counterexample to the conjecture. Our techniques also extend to a more general setting involving other types of constraints.