Elusive extremal graphs

TL;DR

This paper challenges a conjecture in extremal graph theory by providing a counterexample, showing that not all extremal problems have a finitely forcible optimal solution, and extends the analysis to broader constraint types.

Contribution

It presents a counterexample to Lovasz's conjecture on the finite forcibility of extremal graph solutions and generalizes the approach to other constraint settings.

Findings

Counterexample disproves the conjecture

Techniques extend to broader constraint types

Highlights limitations of current extremal graph theory methods

Abstract

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz 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.