Simonovits Product Conjecture

For every finite family $\mathcal L$ of ordinary forbidden graphs, let $p(\mathcal L)=\min\{\chi(H):H\in\mathcal L\}-1$ . The Simonovits Product Conjecture asserts that, for all sufficiently large integers $n$ , there exists an $n$ -vertex $\mathcal L$ -free graph $G$ with $e(G)=\operatorname{ex}(n,\mathcal L)$ such that $G$ is the complete join $G_1\vee\cdots\vee G_{p(\mathcal L)}$ of $p(\mathcal L)$ graphs, each of positive order.

Progress summary

Partially solved

A new paper disproves a weakened version of the conjecture, while the full conjecture remains open.

The Simonovits Product Conjecture concerns product extremizers in forbidden-subgraph theory. The latest work studies a weaker requirement asking only for one extremizer at every sufficiently large order.

August 2026 counterexample

Xu and Chuandong give a fixed finite forbidden family for which no three-factor product extremizers exist, disproving the weakened existence-only formulation. This is genuine negative progress, but it does not settle the full conjecture.

Current status (as of August 2026): the existence-only three-factor formulation is disproved, while the Simonovits Product Conjecture in its other forms remains open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

A finite forbidden family with superlinear surplus and no three-factor product extremizers — arXiv — Xu, Chuandong

extremal-graph-theory

2 days ago